Evaluation of moments and their application in Hückel molecular orbital theory

Moments of acyclic carbon chains are treated in a systematic way. Explicit formulas in terms of connectivities are tabulated up to ₁₄. This facilitates the evaluation of moments by simply counting the numbers of various fragments involved. The total electron energy is analyzed by means of moments and the meaning of additivity is interpreted. An approximate formula for E is parametrized by truncation, preserving only five bond parameters. Based on these, we attempt to better rationalize and reformulte the concept of aromaticity.Formerly used names: Kiang, Yuan-sun; Tang, Au-chin     

Stability and reactivities based on moment analysis

The truncated expansion of a function ¦x¦ is used to obtain the total Hückel -electron energy partitioned into various sums, in terms of moments as well as molecular fragments. The additivity is in general satisfactory for acyclic and cyclic systems, which exhibit a regularity called the generalized Hückel rule which indicates whether a fragment plays the role of stabilization or of destabilization. A unified treatment based on the energy partitioning is proposed for rationalizing aromaticity, reactivities and bond lengths of conjugated hydrocarbons. The relationships between molecular properties and topology can be deduced from inspecting, enumerating or summing the relative contributions of various fragments. Also known as: Yuan-sun Kiang     

The problem of unphysical states in the theory of intermolecular interactions

It was shown by Claverie that the interactions between atoms and molecules make unphysical electronic solutions of the Schradinger equation accessible in perturbation calculations of intermolecular interactions, accessible in the sense that the perturbation expansion is likely to converge to an unphysical solution if it converges at all. This is a difficult problem because there are generally an infinite number of unphysical states with energies below that of the physical ground state. We have carried out configuration interaction calculations on LiH of both physical and unphysical states. They show that avoided crossings occur between physical and unphysical energy levels as the interaction between the two atoms is turned on, i.e. as the expansion parameter is increased from 0 to 1. The avoided crossing for the lowest energy state occurs for < 0.8, implying that the perturbation expansion will diverge for larger values of . The behavior of the energy levels as functions of . is shown to be understandable in terms of a two-state model. In the remainder of the paper, we concentrate on designing effective Hamiltonians which have physical solutions identical to those of the Schrödinger equation, but which have no unphysical states of lower energy than the physical ground state. We find that we must incorporate ideas from the Hirschfelder-Silbey perturbation theory, as modified by Polymeropoulos and Adams, to arrive finally at an effective Hamiltonian which promises to have the desired properties, namely, that all unphysical states be higher in energy than the physical bound states, that the first and higher order corrections to the energy vanish in the limitR = . that the leading terms of the asymptotic 1/R expansion of the energy be given correctly in second order, and that the overlap between the zeroth order wave function and the corresponding eigenfunction of the effective Hamiltonian be close to one.     

Remark on the moment expansion of total π-electron energy

Summary The truncated expansion of the function |x| was frequently used to express the total Hückel -electron energy (E) in terms of moments. We now present an identity which connectsE with an infinite series of moments. This series is convergent. Lower and upper bounds forE are obtained, based on the same infinite moment expansion.     

An ab initio study of the keto-enol tautomerism

The keto-enol tautomerism is studied using an approximative HF method outlined in the appendix. The following results are obtained: (1) The experimentally observed alternance of G in acyclic monoketones could not be reproduced. (2) The stabilization of C=C double bonds, especially of conjugated double bonds, by CH₃- or -CH₂- groups is responsible for the observed difference between acyclic and cyclic 1.2-diketones, e.g. for the different enol content of diacetyl and cyclopentane-1.2-dione. (3) The enols of 1.2-diketones contain a hydrogen bond which differs from the hydrogen bond in enols of 1.3-diketones. (4) A system of two conjugated C=O double bonds is not favoured compared to a system of two C=O bonds which are separated by one (or more) -CH₂- group. (5) 5-ring enols with a C=C double bond in the ring are more stable than one would expect by an energy estimation from acyclic compounds.     

On subspectral acyclic molecular graphs

Based on the contraction and expansion of graphs, the subspectrality of acyclic molecular graphs is treated in a systematic way. The graphs containing eigenvalues 0, t 1 and ± 2 are discussed in detail, with emphasis on how to detect and construct the concealed species which can neither be recognized by symmetry considerations nor by the Heilbronner procedure. As a consequence, graphs and their counts have been given for species with numbers of vertices less than 16.Also known as Yuan-sun Kiang.     

A finite expansion method for the calculation and interpretation of molecular electrostatic potentials

Because it is useful to have the molecular electrostatic potential as an element in a complex scheme to assess the toxicity of large molecules, efficient and reliable methods are needed for the calculation and characterization of these potentials. A multicenter multipole expansion of the molecular electron charge density calculated with a limited Gaussian basis set is shown here to have only a finite number of nonzero terms from which the molecular electrostatic potential can be calculated. The discrete contributions to the electrostatic potentials from the terms of this expansion provide a physically meaningful decomposition of the potential and a means for its characterization. With pyrrole as an example, the electrostatic potential calculated from this finite expansion of the electron density is compared to that obtained from exact calculations from the same wave function. Good agreement is obtained at distances greater than 1.5 A from any atom in the molecule. In contrast, rearrangement of the terms into an expansion corresponding only to Mulliken atomic charges and dipoles yields a decomposition that produces electrostatic potentials which agree less well with the exact potential. This discrepancy is attributable to the neglect of terms due to higher moments.     

Calculation of the entropy of random coil polymers with the hypothetical scanning Monte Carlo method

Hypothetical scanning Monte Carlo (HSMC) is a method for calculating the absolute entropy S and free energy F from a given MC trajectory developed recently and applied to liquid argon, TIP3P water, and peptides. In this paper HSMC is extended to random coil polymers by applying it to self-avoiding walks on a square lattice--a simple but difficult model due to strong excluded volume interactions. With HSMC the probability of a given chain is obtained as a product of transition probabilities calculated for each bond by MC simulations and a counting formula. This probability is exact in the sense that it is based on all the interactions of the system and the only approximation is due to finite sampling. The method provides rigorous upper and lower bounds for F, which can be obtained from a very small sample and even from a single chain conformation. HSMC is independent of existing techniques and thus constitutes an independent research tool. The HSMC results are compared to those obtained by other methods, and its application to complex lattice chain models is discussed; we emphasize its ability to treat any type of boundary conditions for which a reference state (with known free energy) might be difficult to define for a thermodynamic integration process. Finally, we stress that the capability of HSMC to extract the absolute entropy from a given sample is important for studying relaxation processes, such as protein folding.     

Growing self avoiding walk trees

The Growing self avoiding walk model (GSAW) was proposed to explain statistical mechanics of the growth process in polymerization. Close examination of the model reveals that it is suited only for addition polymerization where only one monomer unit is added to the growing polymer chain. In this paper we propose a model for step growth or condensation polymerisation, where all of the monomer is converted first to dimer, all dimers are converted to tetramer etc. In the calculation of probabilities of the walker taking a step in a specified direction in the GSAW model the probability is the reciprocal of the number of steps available if the walker looks one step ahead. It is seen that our model for condensation polymerization GSAW can be generalized to the walker looking a finite number of steps ahead (instead of one only) for the purpose of calculation of probabilities. This is explained in detail in the first section of the paper. In the second section of the paper we explain the use of the Depth first search (DFS) algorithm in the calculation of probabilities and moments for the generalized growing self avoiding walk model. In the third section of the paper we report the exact values of the mean square end to end distances and relevant survival probabilities for the model. When GSAW was first proposed there was a controversy over whether it belonged to the same universality class as the True self avoiding walk $\left( \upsilon =\frac{2}{d+1}\right) $ or that of the self avoiding walk $\left( \upsilon =\frac{3}{d+2}\right) $ . At that time it was conclusively shown that the GSAW model cannot belong to the same universality class as the true self avoiding walk model. However it was never conclusively shown that $\upsilon =\frac{3}{d+2}$ for GSAW. In this paper we propose two methods of studying this problem. One method is rigorous analysis of the DFS algorithm. We explain how this algorithm can be used to study restricted random walks with finite memory and self avoiding walks. The second method proposed by us is a detailed analysis of the generalised GSAW model proposed by us. This paper is to be viewed as an introductory paper on a new model that should be of interest to both chemists and mathematicians.     

Acyclic systems with extremal Hückel π-electron energy

The main result of the present work is the proof that among acyclic polyenes C _n H _n+2, the linear isomer H₂C=(CH) _n–2=CH₂ has maximal HMO -electron energy. The 1,1-divinyl isomer (H₂C=CH)₂C(CH) _n–6=CH₂ has maximal -energy among branched acyclic systems. Among trees withn vertices, the star has minimal energy. A number of additional inequalities for HMO total -electron energy of acyclic conjugated systems are proved.     

Guanidinium-Type resonance stabilization and its biological implications. 2. The doubly-extended-guanidine series

π-Electron delocalization in neutral and protonated “doubly-extended-guanidine,” (H₂N)₂C?N? CH?N—CH?NH, has been studied by ab initio methods at the self-consistent field (SCF) STO -3G and 3-21G levels for a large number of tautomeric, rotameric, pseudocyclic, and monocyclic (disubstituted triazine) forms. These π systems have been characterized in terms of a number of structural and energetic parameters: degree of single/bond character from bond lengths and π bond orders, electron distributions, and tautomer, rotamer, and protonation energies. The acyclic neutral forms exhibit largely alternant single–double bond patterns as predicted by classical bonding structures but with, however, significant deviations due to conjugation. The acyclic protonated forms exhibit bond patterns consistent with resonance delocalized structures extending over the whole molecule (“doubly-extended guanidinium”) or part of the molecule (“extended-guanidinium”) or guanidinium . All systems showed alternant charge distributions with electron-deficient carbons. The energy results have been analyzed in terms of possible contributions from steric interactions, lone-pair repulsions, purportive electrostatic interactions in pseudocyclic forms, overall π-system conformation (extended, kinked, or folded), and specific through-space π-overlap interactions in some pseudocyclic forms. It was found that these other interactions usually dominate the specifically π effects so that the general concept of preferential π delocalization in straight lines does not hold for the acyclic systems. Some interesting examples of pseudocyclic forms exhibiting strongly stabilizing intramolecular interactions attributed to π through-space coupling are identified. These systems with incipient-ring characteristics present intermediate bonding models between the acyclic and closed-ring π systems. The extent of stabilization of the guanidinium-type cations by resonance delocalization in cyclic systems depended on whether it reinforced or interfered with the overall ring delocalization.     

Applications of laser-induced breakdown spectrometry (LIBS) in surface analysis

The applicability of laser-induced breakdown spectrometry (LIBS) for surface analysis is presented in terms of its lateral and depth resolution. A pulsed N₂ laser at 337.1 nm (3.65 J/cm²) was used to irradiate solar cells employed for photovoltaic energy production. Laser produced plasmas were collected and detected using a charge-coupled device. An experimental device developed in the laboratory permits an exact synchronization of sample positioning using an XY motorized system with laser pulses. Multielement analysis with lateral resolution of up to 30 m is feasible with the present system. Three-dimensional capabilities of the system are used for studies on the distribution of carbon impurities at the surface of the solar cells.     

The Liu‐Parr power series expansion of the Pauli kinetic energy functional with the incorporation of shell‐inducing traits: Atoms

An approximate expression for the Pauli kinetic energy functional T_p is advanced in terms of the Liu‐Parr expansion [S. Liu, R.G. Parr, Phys. Rev. A 1997 , 55, 1792] which involves a power series of the one‐electron density. We use this explicit functional for T_p to compute the value of the noninteracting kinetic energy functional T_s of 34 atoms, from Li to Kr (and their positive and negative monoions). In particular, we examine the effect that a shell‐by‐shell mean‐square optimization of the expansion coefficients has on the kinetic energy values and explore the effect that the size of the expansion, given by the parameter n, has on the accuracy of the approximation. The results yield a mean absolute percent error for 34 neutral atoms of 0.15, 0.08, 0.04, 0.03, and 0.01 for expansions with n = 3, 4, 5, 6, and 7, respectively (where ). We show that these results, which are the most accurate ones obtained to date for the representation of the noninteracting kinetic energy functional, stem from the imposition of shell‐inducing traits. We also compare these Liu‐Parr functionals with the exact but nonexplicit functional generated in the local‐scaling transformation version of DFT.     

Configuration interaction: Molecular orbitals for accurate calculations on diatomics

A procedure is developed to construct an optimum set of molecular orbitals (MO's) to be used in large scale configuration interaction expansion for diatomics. The set is optimum in the sense that a significant energy improvement can be obtained for a relatively short wavefunction expansion. Application of this methodology to the ground state of the LiH molecule gave an energy of –8.06345 a.u. for an expansion with 1852 terms obtained from a set with 16-, 12-, and 6-type MO's.     

Asymptotic behavior of the primitive function of different “symmetry-adapted” perturbation schemes for the H ground state

The leading terms in the asymptotic 1/R expansion of the wave functions and energies of various “symmetry-adapted” perturbation schemes for intermolecular forces, as well as for the “polarization approximation” (PA ) are derived for the H ground state, both exactly (i.e., to infinite order in λ) and in the first two orders of the λ expansion. It is pointed out that only in the PA and the Hirschfelder–Silbey scheme is the formal primitive function Φ genuinely primitive, whereas Φ in the other schemes is asymmetric in a rather strange way. In this lack of genuine primitivity lies the reason why in these schemes the leading term of the 1/R expansion is only recovered to infinite order in λ and requires knowledge of the R^?4 term of the wave function, provided that one uses the “internal” energy expression. Four different energy expressions are compared that behave differently in the different schemes. The results obtained here are basis independent, but the implications of the basis completeness problem are discussed as well.     

The electronic structure of polymers by the FSGO (Floating Spherical Gaussian Orbital) method

Non-empirical band-structure calculations have been performed on polyethylene using two basis sets introduced by Christoffersen. Both basis sets had to be optimised with respect to the carbon-carbon framework bond in order to yield solutions within the nearest-neighbour approximations. The valence bands of polyethylene are well reproduced by both basis sets whilst the conduction bands are only in fair agreement with those produced by conventional gaussian calculations. The use of the unsplit basis set was considered unsatisfactory for the representation of the core bands. The effect of increasing the number of interacting unit cells on the energy terms is discussed. Some of the energy terms converge when five unit cells are used and almost all of the terms reach a constant value when nine unit cells are employed.     

Application of 2-point Padé approximants to the ground state of the 2-dimensional hydrogen atom in an external magnetic field

The 2-dimensional hydrogen atom in an external field r ² is important in certain problems of solid-state physics. The exact results for the ground state are obtained from the numerical solution of the radial Schrödinger equation and are compared with various 2-point Padé approximants. Terms to order ⁴ in the low-field RS expansion are combined with 5 terms in the high-field expansion in order to obtain the Padé approximants. The results indicate that the best approximants have relative errors in the range 10^–8 to 10^–5 throughout the interval = 0 4.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

High Dimensional Model Representations Generated from Low Dimensional Data Samples. I. mp-Cut-HDMR

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. For a high dimensional system, an output f(x) is commonly a function of many input variables x=|x ₁,x ₂,...,x _n } with n10² or larger. HDMR describes f(x) by a finite hierarchical correlated function expansion in terms of the input variables. Various forms of HDMR can be constructed for different purposes. Cut- and RS-HDMR are two particular HDMR expansions. Since the correlated functions in an HDMR expansion are optimal choices tailored to f(x) over the entire domain of x, the high order terms (usually larger than second order, or beyond pair cooperativity) in the expansion are often negligible. When the approximations given by the first and the second order Cut-HDMR correlated functions are not adequate, this paper presents a monomial based preconditioned HDMR method to represent the higher order terms of a Cut-HDMR expansion by expressions similar to the lower order ones with monomial multipliers. The accuracy of the Cut-HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of higher order terms. The mathematical foundations of monomial based preconditioned Cut-HDMR is presented along with an illustration of its applicability to an atmospheric chemical kinetics model.     

Use of kinetic plots for relative assessment of reactor throughput and energy consumption

The progress of high temperature processes is generally described in terms of variation of the degree of conversion () with time (t). The present paper outlines a procedure for making use of-t plots for comparative assessment of productivity and energy requirements for a test system with respect to a reference, on the basis of some simplifying assumptions. It is assumed that the throughput is inversely proportional to reaction time as in the case of batch reactors and plug flow reactors. It is also assumed that the energy requirement is a simple function of process temperature. The principles outlined is illustrated with reference to some laboratory data for reduction of iron oxide by coal.The authors wish to thank Prof. P. R. Rao, Director National Metallurgical Laboratory, Jamshedpur, India, for providing facilities for experimental work and for according permission to publish this work.     

On minimal energy ordering of acyclic conjugated molecules

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Gutman (Acyclic conjugated molecules, trees and their energies, J. Math. Chem. 1 (1987) 123–143) proposes two conjectures about the minimum of the energy of conjugated trees (trees with a perfect matching), which are verified by Zhang and Li (On acyclic conjugated molecules with minimal energies, Discrete Appl. Math. 92 (1999) 71–84). This paper focuses on the trees of conjugated hydrocarbon /EquationSource> trees in the class in the increasing order of their energies.