Interesting properties of Thomas–Fermi kinetic and Parr electron–electron‐repulsion DFT energy functional generated compact one‐electron density approximation for ground‐state electronic energy of molecular systems

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a (nonlinear, approximate) density functional of three spatial dimension one‐electron density for an N‐electron system, which is tractable in the practice, is a long desired goal in electronic structure calculation. If the Thomas‐Fermi kinetic energy (～∫ρ^5/3d r ₁) and Parr electron–electron repulsion energy (～∫ρ^4/3d r ₁) main‐term functionals are accepted, and they should, the later described, compact one‐electron density approximation for calculating ground state electronic energy from the 2nd Hohenberg–Kohn theorem is also noticeable, because it is a certain consequence of the aforementioned two basic functionals. Its two parameters have been fitted to neutral and ionic atoms, which are transferable to molecules when one uses it for estimating ground‐state electronic energy. The convergence is proportional to the number of nuclei (M) needing low disc space usage and numerical integration. Its properties are discussed and compared with known ab initio methods, and for energy differences (here atomic ionization potentials) it is comparable or sometimes gives better result than those. It does not reach the chemical accuracy for total electronic energy, but beside its amusing simplicity, it is interesting in theoretical point of view, and can serve as generator function for more accurate one‐electron density models. © 2008 Wiley Periodicals, Inc. J Comput Chem 2009     

Interrelations among x-ray scattering,electron densities,and ionization potentials

From the general formalism of elastic x-ray scattering and a few meaningful assumptions we have shown that for an atom, ∫ρ²dτ is an experimentally measurable quantity related to the intensity scattered by an element. We have labeled this quantity (p), the “average electron density.” If ψ obeys the virial theorem, within the Thomas–Fermi approximation we show (within a multiplicative constant) that 〈ρ〉 is a lower bound to (∑ ionization potentials )^3/2. Thus, the scattered intensity of x rays is related quantitatively to the energy of the scattering atoms. Inequalities have been developed to express these relationships and have been confirmed for the more exact Hartree–Fock wave functions.     

Self-consistent Thomas–Fermi–Dirac theory,extended by Gell-Mann and Brueckner correlation,in terms of density n and its two reduced gradients ∇2n/n and ∇n/n

We prove the following results, relevant for the density functional theory: the Thomas–Fermi–Dirac theory, generalized to include the contribution due to the high electron density result of Gell-Mann and Brueckner for the correlation energy, is shown to lead to a differential equation for the self-consistent ground-state density n( r ) in atoms and molecules in the form F(n, { ∇ n/n}², ∇²n/n)=1, where the function F is given explicitly. A straightforward extension yields a similar result for the equation determining the Pauli plus exchange–correlation potential and for the divergence of the many-electron force. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 145–149, 1998     

Note on the Fermi–Amaldi correction for the Thomas–Fermi theory of atoms

The aim of this note is to ascertain the importance of the Fermi–Amaldi (FA ) correction for the Thomas–Fermi (TF ) theory of atoms. For this purpose, an analytic trial electron density has been chosen and the Thomas–Fermi–Amaldi (TFA ) energy expression has been minimized by the Ritz method for the closed-shell atoms Ne, Ar, Kr, Xe, and Rn. The variationally obtained electron densities have then been used for calculating the diamagnetic susceptibilities of these atoms. The calculated values show only a very small improvement over values calculated by Jensen by an analogous procedure from the TF energy expression.     

Thomas–Fermi term as the simplest correction to the Weizsacker term

The nature of the correlation function G(1,2) appearing in the definition of the reduced first order density operator γ′(1,2) = ρ(1)^1/2ρ(2)^1/2G(1,2) is analyzed. It is shown that when G(1,2) is expanded in terms of plane waves in the context of a single-determinant approximation to the wave function, the correction to the Weizsacker term in the kinetic energy density expression is the Thomas–Fermi term.     

Kinetic study of the reactions of atomic chlorine with several volatile organic compounds at 240–340 K

The rate constant for the reactions of atomic chlorine with 1,4‐dioxane (k₁), cyclohexane (k₂), cyclohexane‐d₁₂(k₃), and n‐octane (k₄) has been determined at 240–340 K using the relative rate/discharge fast flow/mass spectrometer (RR/DF/MS) technique developed in our laboratory. Essentially, no temperature dependence for these reactions was observed over this temperature range, with an average of k₁ = (1.91 ± 0.20) × 10^?10 cm³ molecule^?1 s^?1, k₂ = (2.91 ± 0.31) × 10^?10 cm³ molecule^?1 s^?1, k₃ = (2.73 ± 0.30) × 10^?10 cm³ molecule^?1 s^?1, and k₄ = (3.22 ± 0.36) × 10^?10 cm³ molecule^?1 s^?1, respectively. The kinetic isotope effect of the reaction of cyclohexane with atomic chlorine has also been determined to be 1.14 by directly monitoring the decay of both cyclohexane and cyclohexane‐d₁₂ in the presence of chlorine atoms, which is consistent with the literature value of 1.20. © 2006 Wiley Periodicals, Inc. Int J Chem Kinet 38: 386–398, 2006     

Addendum to “variational principle for obtaining approximate analytical solutions of the temperature-perturbed Thomas–Fermi equation for compressed atoms”

In the present work the total energy of a Ne atom at T = 0 K is calculated as a function of a spherical container radius. The calculation is based on the Thomas–Fermi (TF ) equation, which is solved approximately by an equivalent variational principle. The effect of an approximate exchange correction on the variational TF energy values is investigated.     

Ionic bonding of lanthanides,as influenced by d‐ and f‐atomic orbitals,by core–shells and by relativity

Lanthanide trihalide molecules LnX₃ (X = F, Cl, Br, I) were quantum chemically investigated, in particular detail for Ln = Lu (lutetium). We applied density functional theory (DFT) at the nonrelativistic and scalar and SO‐coupled relativistic levels, and also the ab initio coupled cluster approach. The chemically active electron shells of the lanthanide atoms comprise the 5d and 6s (and 6p) valence atomic orbitals (AO) and also the filled inner 4f semivalence and outer 5p semicore shells. Four different frozen‐core approximations for Lu were compared: the (1s²–4d¹⁰) [Pd] medium core, the [Pd+5s²5p⁶ = Xe] and [Pd+4f¹⁴] large cores, and the [Pd+4f¹⁴+5s²5p⁶] very large core. The errors of Lu? X bonding are more serious on freezing the 5p⁶ shell than the 4f¹⁴ shell, more serious upon core‐freezing than on the effective‐core‐potential approximation. The Ln? X distances correlate linearly with the AO radii of the ionic outer shells, Ln³⁺‐5p⁶ and X^?‐np⁶, characteristic for dominantly ionic Ln³⁺‐X^? binding. The heavier halogen atoms also bind covalently with the Ln‐5d shell. Scalar relativistic effects contract and destabilize the Lu? X bonds, spin orbit coupling hardly affects the geometries but the bond energies, owing to SO effects in the free atoms. The relativistic changes of bond energy BE, bond length R_e, bond force k, and bond stretching frequency v_s do not follow the simple rules of Badger and Gordy (R_e～BE～k～v_s). The so‐called degeneracy‐driven covalence, meaning strong mixing of accidentally near‐degenerate, nearly nonoverlapping AOs without BE contribution is critically discussed. © 2015 Wiley Periodicals, Inc.     

Information entropies of many-electron systems

The Boltzmann–Shannon (BS ) information entropy S_ρ = ∫ ρ(r)log ρ(r)dr measures the spread or extent of the one-electron density ρ(r), which is the basic variable of the density function theory of the many electron systems. This quantity cannot be analytically computed, not even for simple quantum mechanical systems such as, e.g., the harmonic oscillator (HO ) and the hydrogen atom (HA ) in arbitrary excited states. Here, we first review (i) the present knowledge and open problems in the analytical determination of the BS entropies for the HO and HA systems in both position and momentum spaces and (ii) the known rigorous lower and upper bounds to the position and momentum BS entropies of many-electron systems in terms of the radial expectation values in the corresponding space. Then, we find general inequalities which relate the BS entropies and various density functionals. Particular cases of these results are rigorous relationships of the BS entropies and some relevant density functionals (e.g., the Thomas–Fermi kinetic energy, the Dirac–Slater exchange energy, the average electron density) for finite many-electron systems. © 1995 John Wiley & Sons, Inc.     

Effects of exchange on equilibrium bond lengths of heavy,almost spherical,tetrahedral molecules XH4

The simplest nonrelativistic density functional theory, namely, that of Thomas, Fermi, and Dirac, is used to study the effect of exchange on the equilibrium bond lengths of heavy tetrahedral molecules XH₄. In particular, the limiting bond length as the central atom X becomes infinitely heavy is shown to be reduced by exchange by some 0.34 Å. Comparison with experiment shows that the main features of the bond-length variation through the series CH₄? PbH₄ are reflected by the Thomas–Fermi–Dirac predictions, though the bond lengths remain too large for finite atomic number Z of the central atom. Therefore, a semiempirical correction is proposed, which, however, tends to zero as Z tends to infinity.     

Variational principle for obtaining approximate analytical solutions of the temperature-perturbed Thomas–Fermi equation for compressed atoms

A temperature correction to the Thomas–Fermi (TF ) model of a neutral compressed atom has been given by Marshak and Bethe. The aim of the present work is to point out that, by formulating a variational principle, one may obtain approximate analytical solutions of the temperature-perturbed TF equation. As a test of the proposed theory, the total energy of a Ne atom, confined to a spherical container, has been calculated as a function of the container radius at T = 0°K. Comparison of the data obtained with energy values calculated by Ludeña, using a self-consistent-field Hartree–Fock approach, shows reasonable agreement. It is, therefore, concluded that the variational approach suggested in this paper is a sound one.     

Kinetics of the bromate–bromide reaction at high bromide concentrations

At bromide concentrations higher than 0.1 M, a second term must be added to the classical rate law of the bromate–bromide reaction that becomes ?d[BrO₃^?]/dt = [BrO₃^?][H⁺]²(k₁[Br^?] + k₂[Br^?]²). In perchloric solutions at 25°C, k₁ = 2.18 dm³ mol^?3 s^?1 and k₂ = 0.65 dm⁴ mol^?4 s^?1 at 1 M ionic strength and k₁ = 2.60 dm³ mol³ s^?1and k₂ = 1.05 dm⁴ mol^?4 s^?1 at 2 M ionic strength. A mechanism explaining this rate law, with Br₂O₂ as key intermediate species, is proposed. Errors that may occur when using the Guggenheim method are discussed. © 2006 Wiley Periodicals, Inc. Int J Chem Kinet 39: 17–21, 2007     

Analytic form of Thomas–Fermi–Dirac dielectric function for III–V compound semiconductors

The nonlinear Thomas–Fermi–Dirac equation is solved variationally for positive and negative point charges in various III–V compound semiconductors. The parameters are presented, and the results are compared with those obtained numerically.     

Radical-initiated homo- and copolymerization of α-methylene-γ-butyrolactone

The kinetics of α-methylene-γ-butyrolactone (α-MBL) homopolymerization was investigated in N,N-dimethylformamide (DMF) with azobis(isobutyronitrile) as initiator. The rate of polymerization (R_p) was expresed by R_p = k[AIBN]^0.54[α-MBL]^1.1 and the overall activation energy was calculated as 76.1 kJ/mol. Kinetic constants for α-MBL polymerization were obtained as follows: k_p/k_t^1/2 = 0.161 L^1/2 mol^?1/2·s^?1/2; 2fk_d = 2.18 × 10^?5 s^?1. The relative reactivity ratios of α-MBL(M₂) copolymerization with styrene (r₁ = 0.14, r₂ = 0.87) were obtained. Applying the Q–e scheme led to Q = 2.2 and e = 0.65. These Q and e values for α-MBL are higher than those for MMA     

Relativistic theory for indirect nuclear spin–spin couplings within the polarization propagator approach

We present a relativistic theory for the nuclear spin–spin coupling tensor within the polarization propagator approach using the particle-hole Dirac–Coulomb–Breit Hamiltonian and the full four-component wave function. We give explicit expressions for the coupling tensor in the random-phase approximation, neglecting the Breit interaction. A purely relativistic perturbative electron–nuclear Hamiltonian is used and it is shown how the single relativistic contribution to the coupling tensor reduces to Ramsey's three second-order terms (Fermi contact, spin–dipole, and paramagnetic spin–orbit) in the nonrelativistic limit. The principal propagator becomes complex and the leading property integrals mix atomic orbitals of different parity. The well-known propagator expressions for the coupling tensor in the nonrelativistic limit is obtained neglecting terms of the order c^?n (n ? 1). © 1993 John Wiley & Sons, Inc.     

Thomas–Fermi theory from a perturbative treatment of the hydrogenic limit energy density functional: A possible link to local density functional theory

A perturbation expansion which connects the hydrogenic limit energy density functional to the Thomas–Fermi functional is discussed. This perturbation series, where the Coulomb energy density functional is treated as the perturbation to the hydrogenic limit functional, is, in fact, the q = (N/Z) expansion of Thomas–Fermi theory. A truncated form of the first-order correction to the functional provides further insight into the model which treats the ground state energy as a local functional of the electron density.     

Thermal reactions of ethane-d6–acetylene and D2–acetylene mixtures behind shock waves

A study of the thermal decomposition of an acetylene–ethane-d₆ mixture indicates that the rate constant for hydrogen abstraction from acetylene by methyl is more than 20 times less than for abstraction from ethane. Isotopic exchange is initiated by a rapid reaction between product D atoms and C₂H₂. A series of experiments involving the reactions of a D₂–acetylene mixture indicated that a molecular exchange process was also occurring, and it was shown that d[C₂HD]/dt = k[D₂]^0.7[C₂H₂]^0.3, effective activation energy = 15.8 kcal/mol. This mechanism made an insignificant contribution to isotope exchange in C₂H₂–C₂D₆ mixtures.     

Kinetics of the gas‐phase reactions of cyclo‐CF2CFXCHXCHX – (X = H,F, Cl) with OH radicals at 253–328 K

Rate constants were determined for the reactions of OH radicals with halogenated cyclobutanes cyclo‐CF₂CF₂CHFCH₂? (k₁), trans‐cyclo‐CF₂CF₂CHClCHF? (k₂), cyclo‐CF₂CFClCH₂CH₂? (k₃), trans‐cyclo‐CF₂CFClCHClCH₂? (k₄), and cis‐cyclo‐CF₂CFClCHClCH₂? (k₅) by using a relative rate method. OH radicals were prepared by photolysis of ozone at a UV wavelength (254 nm) in 200 Torr of a sample reference H₂O? O₃? O₂? He gas mixture in an 11.5‐dm³ temperature‐controlled reaction chamber. Rate constants of k₁ = (5.52 ± 1.32) × 10^?13 exp[–(1050 ± 70)/T], k₂ = (3.37 ± 0.88) × 10^?13 exp[–(850 ± 80)/T], k₃ = (9.54 ± 4.34) × 10^?13 exp[–(1000 ± 140)/T], k₄ = (5.47 ± 0.90) × 10^?13 exp[–(720 ± 50)/T], and k₅ = (5.21 ± 0.88) × 10^?13 exp[–(630 ± 50)/T] cm³ molecule^?1 s^?1 were obtained at 253–328 K. The errors reported are ± 2 standard deviations, and represent precision only. Potential systematic errors associated with uncertainties in the reference rate constants could add an additional 10%–15% uncertainty to the uncertainty of k₁–k₅. The reactivity trends of these OH radical reactions were analyzed by using a collision theory–based kinetic equation. The rate constants k₁–k₅ as well as those of related halogenated cyclobutane analogues were found to be strongly correlated with their C? H bond dissociation enthalpies. We consider the dominant tropospheric loss process for the halogenated cyclobutanes studied here to be by reaction with the OH radicals, and atmospheric lifetimes of 3.2, 2.5, 1.5, 0.9, and 0.7 years are calculated for cyclo‐CF₂CF₂CHFCH₂? , trans‐cyclo‐CF₂CF₂CHClCHF? , cyclo‐CF₂CFClCH₂CH₂? , trans‐cyclo‐CF₂CFClCHClCH₂? , and cis‐cyclo‐CF₂CFClCHClCH₂? , respectively, by scaling from the lifetime of CH₃CCl₃. © 2009 Wiley Periodicals, Inc. Int J Chem Kinet 41: 532–542, 2009     

The elements of Flatland: Hartree-Fock atomic ground states in two dimensions for Z = 1–24

The Hartree–Fock problem in two dimensions (2D) has been solved for 1 ≤ Z ≤ 24 using a Gaussian basis and assuming r^?1 Coulomb interactions. The order of occupation of the one-electron states is like in the 3D case. The 1s shell is found to be particularly small and strongly bound, making the 2D hydrogen a “superhalogen” and the 2D He a “superinert gas.” In contrast to 3D, 4s¹3d² and 4s²3d³ configurations are preferred for the 2D “Sc” and “Cu,” respectively. The six first 2D atoms have stronger and the later ones weaker valence-bonding energies than do their 3D analogs. It is noted that the 2D Dirac energy expression for a hydrogenlike atom for m_j = l + 1/2 agrees with the 3D Klein–Gordon one.