Ab initio interpretation of Hund's rule for the methylene molecule: Variational optimization of its molecular geometries and energy component analysis

Hund's spin‐multiplicity rule for the ground state of the methylene molecule CH₂ is interpreted by Hartree‐Fock (HF) and multi‐reference configuration interaction (MRCI) methods. The stabilization of the triplet ground state ( ³B₁) relative to the second singlet excited state ( ¹B₁) is ascribed to the greater electron‐nucleus attraction energy that is gained at the cost of increasing the electron‐electron repulsion energy and with the aid of a reduction in the nucleus‐nucleus repulsion energy. The highest spin‐multiplicity in the ground state of CH₂ is accompanied by a set of three characteristic features, i.e., elongation of the internuclear distances, reduction in the bond angle, and contraction of the valence electron density distribution around the nuclei involving expansion of the core electron density distribution. The present calculations fulfill the virial theorem to an accuracy of ?V/T = 2.000 for both HF and MRCI. Accordingly, the molecular geometries are optimized for each of the two states. The inclusion of correlation by MRCI method reduces the energy splitting between the two states by about 14%. The energy splitting is analyzed by the correlational virial theorem 2T^c + V^c = 0 to make a clear interpretation of the correlation effect.© 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Core and valence regions of molecules

In real space, the partitioning of a molecule into valence and core regions is rooted in the picture of `valence electrons' in the field of `effective cores' consisting of nuclei partially shielded by inner-shell `core' electrons. The appropriate valence kinetic and potential energies, T ^v and V ^v, respectively, are obtained by subtracting the pertinent parts associated with the cores from the corresponding conventional total T and V energies but, far from T ^v+V ^v, the physically correct valence-region energy is . This result differs markedly from T ^v+V ^v because E ^v, T ^v and V ^v do not obey the virial theorem. Received: 21 August 1996 / Accepted: 14 November 1996     

Obtaining self–consistent wave functions which satisfy the virial theorem

The virial theorem has played an important role in applying quantum mechanics to chemical problems. It has served as one criterion of a satisfactory wave function and its consequences on chemical bonding, molecular structure, and substituent effects have been analyzed extensively. A common method of gaining compliance with the virial theorem is to introduce a “scale” factor which adjusts all distances by a factor η. Optimizing the scale factor through the variational principle produces a wave function satisfying the virial theorem. In the present paper it is shown that when this “scaling” procedure is applied to self-consistent wave functions, the virial theorem can be satisfied, but self-consistency is lost. Scaling generally has a small effect on the total energy, but the effects on the energy components (T, V_ne, V_ee, V_nn) can be two to three orders of magnitude larger and in the range of tens to hundreds of kcal. Consequently, for applications where the energy components are useful, it is highly desirable to obtain wave functions which satisfy the virial theorem and are self-consistent. In the present paper, a simple, inexpensive extrapolation technique is reported which requires one integral evaluation and two SCF cycles to achieve convergence. Applications to atoms and small molecules are reported.     

Relativistic virial theorem for atoms

A relativistic virial theorem is derived for atoms in a general manner. The virial ratio consists of the usual V/T term and a correction term W/T, where T, V, and W are the kinetic energy, the potential energy, and correction terms, respectively. Explicit forms of W are presented for four specific nuclear potential models. Numerical calculations for a uniform nuclear charge model show that the magnitude of the correction term W/T increases with increasing atomic numbers and that it modifies the ratio V/T considerably for atoms with large atomic numbers in particular. Received: 21 November 2000 / Accepted: 8 January 2001 / Published online: 3 April 2001     

Correlation energy contributions to reaction heats

New, more accurate, Hartree-Fock limit energies (E_HF) for ethane and ethylene are obtained from SCF total molecular energie using Ermler and Kern's procedure. These results, together with E_HF values for other small closed shell molecules, are employed to calculate correlation energy (E_c) contributions to reaction heats. Cancellation to within 98% of the total E_c involved, and often to more than 99%, is found for a wide variety of chemical reactions, which strongly suggests that there are systematic regularities in the contribution to E_c from the different kinds of electron pairs in the valence shell. Assuming trictly localized pairs occupying orbitals having strongly directional character, E_c for the valence shell is evaluated in terms of E_c per lone pair, E_c per X? H bond, and E_c per X/X shared pair for Ne and for molecules containing first row atoms, where X is C, N, O, and F.     

Applicability of cubic equation of state mixing rules on correlation of excess molar volume of non-electrolyte binary mixtures – Part II

The excess molar volume (V?^E) data of the 24 binary highly non-ideal mixtures containing dicyclic ethers (593 data points) were correlated by the Peng–Robinson–Stryjek–Vera (PRSV) cubic equation of state (CEOS) coupled with two different classes of mixing rules: (i) the composition dependent van der Waals (vdW) mixing rule and (ii) the excess free energy mixing rules (CEOS/G?^E) based on the approach of the Gupta–Rasmunssen–Fredenslund (GRF), as well as the Twu–Coon–Bluck–Tilton (TCBT) mixing rule; both rules with the NRTL equation as the G?^E model. The results obtained by these models show that the type of applied mixing rules, including the number and position of interaction parameters are of great importance for a satisfactory correlation of V?^E data. The GRF mixing rules gave mostly satisfactory results for V?^E correlation of the non-ideal binary systems available at one isotherm of 298.15?K, while for the correlation in temperature range from 288.15 to 308.15?K the TCBT model can be recommended.     

Nonbonding pairs in cyclic thioethers: Electrostatic modeling and ab initio calculations for complexes of 2,5-dihydrothiophene,thietane, and thiirane with hydrogen fluoride

Electrostatic potential energies V(ϕ) of a non-perturbing, protonic charge at fixed distances r from the S atom in three cyclic thioethers were examined as functions of the angles ϕ made by the r-vector with the C₂ axis (thiirane and 2,5-dihydrothiophene) or the local C₂ axis (thietane). The electrostatic PE V_HF(ϕ) of HF (HF modelled as an extended electric dipole) was also calculated and the results compared with geometries of the thioether⋯HF complexes calculated at the CCSD(T)-F12c/cc-pVTZ-F12 level. The latter reveal angular deviations θ ∼10-20° of the S⋯H F nuclei from collinearity in directions suggesting secondary interactions of F with H atom(s) of the rings. Angles ϕ made by the S⋯H hydrogen bond with the C₂ (or local C₂) axes in the complexes are systematically larger (∼4-9°) than indicated by the V_HF(ϕ) functions. Minima in the simple V(ϕ) versus ϕ functions occur at values smaller (∼5-10°) than those in the V_HF(ϕ) curves.     

Excess Molar Volume and Viscosity of Isobutyric Acid + Water Binary Mixtures Near and Far Away from the Critical Temperature

The excess molar volume V^E, shear viscosity deviation Δη and excess Gibbs energy of activation ΔG^∗E of viscous flow have been investigated by using density (ρ) and shear viscosity (η) measurements for isobutyric acid + water (IBA+W) mixtures over the entire range of mole fractions at five different temperatures, both near and close to the critical temperature (2.055K ≤ (T−T_c)≤ 13.055K). The results were also fitted with the Redlich–Kister equation. This system exhibited very large negative values of V^E and very large positive values of Δη due to increased hydrogen bonding interactions and correlation length between unlike molecules in the critical region and to very large differences between the molar volumes of the pure components at low temperatures. The activation parameters ΔH^∗ and ΔS^∗ have been also calculated and show that the critical region has an important effect on the volumetric properties.     

Löwdin correlation energy density of the inhomogeneous electron liquid in some closed-shell molecules at equilibrium geometry

Using existing theoretical studies, we point out that the dominant variable in determining Löwdin correlation energies per electron E _c /N of isoelectronic series of molecules at equilibrium is the total number of electrons. This turns out to be E _c /N = ?0.033 ± 0.003 a.u. for CH₄, NH₃, H₂O and HF (N = 10), and E _c /N = ?0.039 ± 0.007 for some 20 Si-containing molecules in the series SiX _n Y _m . Following earlier work of March and Wind on atoms, some proposals are then made as to a possible explanation of such behaviour. A test is proposed, via low-order Møller–Plesset perturbation theory, as to whether the Löwdin correlation energy density ε _c (r) is, albeit approximately, a local functional ε _c (ρ) of the ground-state density for molecules at equilibrium. Such an LDA assumption would imply that ε _c (ρ) is quantitatively linear in ρ(r), for closed-shell molecules at equilibrium, at least for the light atomic components treated here. This, in turn implies that the dominant effect of the Löwdin correlation energy for closed-shell molecules at equilibrium is merely to shift the chemical potential.     

On the optimal mixing of the exchange energy and the electron-electron interaction part of the exchange-correlation energy

The optimal mixing coefficient C of the exchange energy E_x and the electron-electron interaction part of the exchange-correlation energy W¹_xc in the formula for the total exchange-correlation energy E_xc was expressed through the ratio of the kinetic T_c and potential W_c contributions to the correlation energy E_c. This expression can be derived from a Heavyside step function model of the dependence of W^λ_xc on the coupling parameter of the electron interaction λ. Values of T_c and W_c obtained from ab initio wave functions were used to estimate C for a number of atoms and molecules. A strong dependence of T_c, W_c, and C on the bond distance was demonstrated for the case of the H₂ molecule. T_c and C approach zero in the bond-dissociation limit; so for an electron-pair bond, the admixing of exact exchange to obtain an accurate E_xc is strongly dependent on the bond length and has to disappear for weak interaction/large bond distances. The potential of the exchange-correlation hole constructed for H₂ from an ab initio second-order density matrix was compared with its generalized gradient approximation (GGA). © 1996 John Wiley & Sons, Inc.     

Study of chemical bonding in H2 and HF molecules: Wave function and density functional theory (DFT) parameters approach

Balint Kurti's Fourier grid Hamiltonian method is employed to obtain the molecular wave function and equilibrium bond length for H₂ and HF molecules. The density functional theory parameter, namely, the chemical hardness (η) value, was determined for some diatomic hydride molecules using this wave function and the results are found to be in good agreement with the values obtained from the ab initio HF–SCF method. A new formula for chemical hardness (η=1/2Dr, where D is the proportionality constant and r is the internuclear distance) is introduced in binding energy and change of hardness equations to determine the chemical hardness and chemical potential values for different bond lengths. The binding energy and change of hardness values are calculated for H₂, H, H, HF, HF⁺, and HF⁻ molecules and the bond stability is discussed. Finally, the concept of an atom in a molecule is examined in the context of DFT parameters and comparison is made between an atom in a molecule and the isolated atom. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 662–669, 2000     

Application of potential constants: Empirical determination of molecular energy components for diatomic molecules

The potential (force) constant of a diatomic molecule is applied to determine the molecular energy components such as the electronic kinetic (T) and electrostatic potential (V) energies. The theoretical framework of the method is constructed from T and V representations of the quantum mechanical virial theorem. To confirm the utility of the method developed here, the calculated molecular energy components of diatomic molecules are compared with available Hartree-Fock data. It is concluded that the present method is simple and powerful for evaluating the molecular energy components of various diatomic molecules.     

Density and excess molar volumes of binary mixtures of sulfolane with methanol,n -propanol,n -butanol,and n -pentanol at 298.15–323.15 K and atmospheric pressure

Densities, ρ and excess molar volumes, V?^E of the binary mixtures of sulfolane, +methanol, +n-propanol,?+n-butanol, and +n-pentanol were measured at temperatures 298.15, 303.15, 308.15, 313.15, and 318.15?K, respectively, covering the whole composition range except methanol at 303.15–323.15?K. The V ^E for the systems were found to be negative and large in magnitude. The values of V ^E of the sulfolane, +n-butanol and sulfolane, +n-pentanol mixtures are being positive at lower and higher mole fractions of the alkanols (x ₂). The magnitudes of the V ^E values of the mixtures are in the order sulfolane?+?methanol?>?sulfolane?+?n-propanol?>?sulfolane?+?n-butanol?>?sulfolane?+?n-pentanol. The observed values of V ^E for the mixtures have been explained in terms of (i) effects due to the differences in chain length of the alcohols, (ii) dipole–dipole interactions between the polar molecules, and (iii) geometric effect due to the differences in molar volume of the component molecules. These are more noticeable in the case of lower alcohols. All these properties have been expressed satisfactorily by appropriate polynomials.     

Long periods in slow-cooled linear and branched polyethylene: Part II

It is shown that the long periods L in slow-cooled polyethylene materials obey the general law L = L⁰ + αr_w, where r_w is the weight average dimension of the coil before crystallization, and L⁰ is a parameter of the order of l_c, the crystalline core thickness, which increases as the cooling rate V decreases. α is a parameter independent of M and V but decreasing with the number of long-chain branches per molecule. The two terms in the above relation are, respectively, the contributions of crystalline and amorphous layers. For cooling rates from 800°C/min to 0.2°C/min, it is shown that the temperature T_c of crystallization is constant; hence the change of morphology (long period, crystalline core thickness, crystallinity) cannot be explained by supercooling. The increase in long period and crystallite thickness in slow-cooled materials with decreasing cooling rate is interpreted in terms of annealing of the crystallized materials between the crystallization temperature T_c and the secondary transition temperature T_αc. Crystallization proceeds by a two-step process of solidification and annealing. During the annealing stage, the mobility of the chains in the crystalline phase is due to defects; the kinetics of thickening is then governed by the mobility (or nucleation) of the defects appearing above T_αc. In the proposed model of crystallization, the assumption that the energy of activation is proportional to T_αc explains the observed laws L ≡ l_c ≡ log t_a, where the annealing time t_a is equal to (T_c ? T_αc)/V. The model applies also to polymers crystallized from the melt and subsequently annealed.     

Hartree-Fock theory for negative ions

It is shown that HF computations which yield ?_i > 0 for an occupied MO do not minimize the HF energy. If ?_i > 0, which frequently occurs in the RHF treatment of negative ions, one can reduce ?_i to zero and simultaneously lower E_HF by an appropriate admixture of a continuum function to the corresponding MO ?. We propose a modification of the HF model that takes these facts into account. Applications to the systems O^2?, N^3?, C^4?, S^2?, O₂^2?, C₂^2? are reported and discussed.     

Chemiluminescence studies of vibrational-to-vibrational energy transfer: MF3 + HF

The vibrational-to-vibrational energy transfer process, MF³ + HF (ν = 0) → MF + HF³ (ν ? 9) is studied by means of a “triple beam” experiment. Vibrationally excited MF³ molecules are created at the intersection of crude crossed beams of M (M = Na, Mg) and F₂. The metal fluorides thus formed then cross an HF beam, where energy transfer occurs. This is observed by measuring the overtone emission from HF. Upper bounds on the reaction cross sections for M ÷ F₂ are measured to be 135 ± 20 A² for M = Na and 80 ± 15 A² for M = Mg, and laser induced fluorescence is used to determine the vibration energy distribution of MgF, which peaks at 2.6 eV. The chemiluminescence signal from the overtone emission indicates a large vibrational interconversion cross section, which is estimated to be ? 30 A².     

Rigidity percolation model of polymer fracture

A theory of the fracture of polymers with network microstructure was developed that was based on the vector, or rigidity percolation (RP) model of Kantor and Webman, in which the modulus, E, is related to the lattice bond fraction p, via E ～ [p ? p_c]^τ. The Hamiltonian for the lattice was replaced by the strain energy density function of the bulk polymer, U = σ²/2E, where σ is the applied stress and p was expressed in terms of the lattice perfection via the bond density ν, with the entanglement molecular weight, ν = ρ/M_e and appropriate measures of crosslink density for rubber, thermosets, and carbon nanotubes. The stored mechanical energy, U, was released by the random fracture of νD_o[p ? p_c] over stressed hot bonds of energy D_o ≈ 330 kJ/mol. The polymer fractured critically when p approached the percolation threshold p_c, and the net solution was obtained as σ = (2EνD_o [p ? p_c])^1/2 with a fracture energy, G_1c ～ [p ? p_c]. The fracture strength of amorphous and semicrystalline polymers in the bulk was well described by, σ = [ED_oρ/16 M_e]^1/2, or σ ≈ 4.6 GPa/M_e^1/2. Fracture by disentanglement was found to occur in a finite molecular weight range, M_c < M < M*, where M*/M_c ≈ 8, such that the critical draw ratio, λ_c = (M/M_c)^1/2, gave the molecular weight dependence of the fracture as G_1c ～ [(M/M_c)^1/2 ? 1]². The critical entanglement molecular weight, M_c, is related to the percolation threshold, p_c, via M_c = M_e/(1 ? p_c). Fracture by bond rupture was in accord with Flory's suggestion, G/G* = [1 ? M_c/M], where G* is the maximum fracture energy. Fracture of an ideal rubber with p = 1 was determined not to occur without strain hardening at λ > 4, such that the maximum stress, σ = E (λ ? 1/λ) = 3.75E. The fracture properties of rubber were found to behave as σ ～ ν, σ ～ E, and G_1c ～ ν. For highly crosslinked thermosets, it was predicted that σ ～ (Eν)^1/2, σ ～ (X ? X_c)^1/2, and G_1c ～ ν^?1/2, where X is the degree of reaction of the crosslinking groups and X_c defines the gelation point. When applied to carbon nanotubes (SWNT and MWNT) of diameter d and hexagonal bond density ν = j/b², the nominal stress as a function of diameter is σ(d) = [16 ED_o(p ? p_c) j/b]^1/2/d ≈ 211/d (GPa.nm) and the critical force, F_c(d) ≈ 166 d (nN/nm), in which j = 1.15, b = 0.142 nm, E ≈ 1 Tpa, and D_o = 518 kJ/mol. For polymer interfaces with Σ chains per unit area of length L and width X ～ L^1/2, G_1c is then ～ [p ? p_c], where p ～ ΣL/X. The results predicted by the RP fracture model were in good agreement with a considerable body of fracture data for linear polymers, rubbers, thermosets, and carbon nanotubes. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 168–183, 2005     

Note on the atomic correlation energy

In the introductory section, we compare the total, kinetic, nuclear-electron, Coulomb, exchange, and correlation energies of ground-state atoms. From the analyses of the data, one can conclude that the Hartree-Fock (HF) model is notably good and might require only a small perturbation to become essentially an “accurate” model. For this reason and considering past literature, we present a semiempirical extension of the HF model. We start with a calibration of three independent models, each one with an effective Hamiltonian, which introduces a small perturbation on the kinetic, the nuclear-electron, or the Coulomb HF operators. The perturbations are expressed as very simple functions of products of orbital probability density. The three perturbations yield very equivalent results and the computed ground-state energies are reasonably near to the accurate nonrelativistic energies recently provided by E. Davidson and his collaborators for the 2–18 electron systems and the estimates by Clementi and his collaborators for the 19–54 electron systems. The first ionization potentials from He to Cs, the second ionization potentials from Li to Zn, and excitation energies for npⁿ, 3dⁿ, and 4s¹3dⁿ configurations are used as additional verification and validation. The above three effective Hamiltonians are then combined in order to redistribute the correlation energy correction in a way which exactly satisfies the virial theorem and maintains the HF energy ratios between kinetic, nuclear-electron, and electron-electron interaction energies; the resulting effective Hamiltonian, named “virial constrained,” yields good quality data comparable to those obtained from the three independent effective operators. Concerning excitation energies, these effective Hamiltonians yield values only in modest agreement with experimental data, even if definitively superior to HF computations. To further improve the computed excitation energies, we applied an empirical scaling in the vector coupling coefficient; this correction yields very reasonable excitations for all the configurations that we have considered. We conclude that the use of effective potentials to introduce small perturbations density-dependent onto the HF model constitutes a broad class of practical and reliable semiempirical solutions to atomic many-electron problems, can provide an alternative to popular proposals from density functional theory, and should prepare the ground for “generalized HF models.” © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 571–591, 1997     

Nonlinear dielectric effect in a critical solution

This paper presents measurements of Δ/E² versus T for the critical solution nitroethane in cyclohexane. The Δ/E² for the temperature near the critical temperature T_c has a positive value and satisfies the following dependence: Δ/E² ∝ 1/(T ? T_c)^λ where λ = 0.34.     

Relativistic virial theorem for diatomic molecules. Application to H 2 +

Summary The virial theorem for a molecule in the relativistic clamped-nuclei approximation is derived. The individual energy contributionsA (momentum energy),B (mass energy),T=A+B (kinetic energy) andV (potential energy) are expressed in terms ofE, E/R (derivate w.r.t. the nuclear coordinates) and the relativistic correction E/² (derivative w.r.t. Sommerfield's fine-structure constant ). IfE and E/R are known as functions of , then all individual energy terms are also known as functions of . As an example, numerical results for H ₂ ⁺ are presented. The relativistic and nonrelativistic potential energy curves and the paradoxical behavior of their different contributions are analyzed and interpreted in both the largeR and shortR ranges.Dedicated to Professor W. Kutzelnigg on the occasion of his 60th birthday