Lower Bounds to Ground-State Eigenvalues

A new lower bound method is presented for ground-state eigenvalues which relies on the Eckart inequality. The method bears similarity to the variational method and the Temple lower bound formula. Restrictions are that an exactly soluble base Hamiltonian must be available with a positive-semidefinite perturbation to the Hamiltonian of interest. A sample calculation shows that our method is able to sometimes best the Temple bound and also perform rigorously when the Temple bound does not.     

Equivalence of Two Lower Bound Methods

Two lower bound methods are compared: the inverse method and a generalized variance method. As the generalized variance method is a generalization of the simple method of variance, the inverse method is shown to be a generalization of the Temple formula. We show that for the same a priori information and basis set, the generalized variance and inverse methods are equivalent.     

Bounds to electronic expectation values for atomic and molecular systems

A generalization of a method to calculate lower bounds to expectation values of non‐negative observables is presented. We consider bounds to three electronic expectation values 〈r²〉, 〈r〉, and 〈r^?1〉 in the helium atom as an example. For both 〈r²〉 and 〈r〉, we are able to calculate improved lower bounds. The lower bound to 〈r^?1〉 does not improve, but we are able to calculate an upper bound which is much closer to the expectation value than the lower bound. Although our generalization allows for improved bounds and/or upper bounds, these bounds to general observables are much less precise than energy bounds and even the expectation values calculated from variational wave functions. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Surpassing the Temple Lower Bound

A non-traditional derivation of the Temple lower bound formula shows how the Temple bound can be improved using expectation values of H ³. Improved upper bounds result as a byproduct as well. Since expectation values of H ² are already quite difficult in analytical calculations we also introduce an approximate lower bound which avoids the need for H ³ but which is rigorous only in a limiting sense. Examples of the two bounds are given using the lithium and perturbed hydrogen atoms.     

Perfect Matchings in Bipartite Lattice Animals: Lower Bounds and Realizability

In the first part of this paper we establish sharp lower bounds on the number of perfect matchings in benzenoid graphs and polyominoes. The results are then used to determine which integers can appear as the number of perfect matchings of infinitely many benzenoids and/or polyominoes. Finally, we consider the problem of concealed non-Kekuléan polyominoes. It is shown that the smallest such polyomino has 15 squares, and that such polyominoes on n squares exist for all n ≥ 15.     

Upper Bound to the Expectation Value of the Squared Hamiltonian

An upper bound to the expectation value of the squared Hamiltonian H ² is derived which relies on replacing products of certain operators with products of the matrix representations of said operators to reduce the computational demands of H ². An example is given which shows the strength of the bound and an application with the Temple lower bound is shown.     

Computing upper and lower bounds using Monte Carlo methods

We computed lower bounds for some ground and excited states of the helium atom using variational Monte Carlo and the Weinstein, Temple, and Stevenson formulas. For these systems, the Temple and Stevenson bounds are approximately the same and have similar standard deviations. The Weinstein bounds are much further from the actual energies and have much larger standard deviations. We also investigated the reliability of these lower bound formulas when nonorthogonal wave functions are used. The Temple formula has been shown to produce an accurate lower bound even under such circumstances, while the Weinstein and Stevenson formulas are shown to yield incorrect results. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

A nonlinear programming approach to lower bounds for the ground‐state energy of helium

Lower‐bound estimates for the ground‐state energy of the helium atom are determined using nonlinear programming techniques. Optimized lower bounds are determined for single‐particle, radially correlated, and general correlated wave functions. The local nature of the method employed makes it a very severe test of the accuracy of the wave function. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 455–463, 1999     

Computation, Properties and Resonance Topology of Benzenoid Monoradicals and Polyradicals and the Eigenvectors Belonging to Their Zero Eigenvalues

Radical benzenoid structures, i.e., those which cannot have all electrons paired, are known to possess much larger structure counts than closed-shell benzenoids of similar size. Building on our previous work, we report methods for calculating eigenvectors, eigenvalues, and structure counts for benzenoid radicals, diradicals, and radicals of higher multiplicity. When a series of such species is constructed by repeated addition of an aufbau unit, structure counts can usually be expressed as a polynomial in one or two variables. Structure counts for radical series generated by repeated circumscribing, however, cannot.     

快速多模型回归分析方法的研究

逐步回归[1]是变量选择的常用方法．由逐步回归法得到的数学表达式不一定是最优数学模型，所以有其局限性．假如有m个变量x；，x。，……，x。，我们希望得到分别包含1个变量、2个变量、……、直至m个变量的最优回归方程，可以采用组合算法C：，对每个子集的所有组合进行回归分析，然后从C：个回归方程中选出最优的数学模型．由于随着变量个数m的增加，总的回归次数以2”的形式倍增，运算量大，时间长．此时可采用快速多模型（LeaPsandbounds）回归分析［‘j．此法为一种快速多模型回归，据此可以得到一组含有不同变量个数的最优方程．该…     

New Ni(II)Cu(II)Ni(II) trinuclear complexes with an 5=3/2 high-spin ground

Four new heterotrinuclear complexes have been synthesized and characterized, namely {[Ni(L)₂]₂[Cu(opba)]}(ClO₄)₂, where opba denotes o-phenylenebis(oxamato) and L stands for 1,10-phenanthroline(phen) (1), 5-nitro-l,10-phenanthroline(NO₂-phen) (2), 2,2′-bipyridyl(bpy) (S) and 4,4′-dimethyl-2,2′-bipyridyl(Me₂bpy) (4). The temperature dependence of the magnetic susceptibility of {[Ni(phen)₂]₂[Cu(opba)]}(ClO₄)₂3H₂O has been studied in the 4–300 K range, giving the exchange integral J—109 cm^?1. The HMT vs. T plot exhibits a minimum at about 100 K, characteristic of this kind of coupled polymetallic complex with an irregular spin-state structure.     

Eigenvalues of graphs with threefold symmetry

In many cases, the spectrum of one graph contains the entire spectrum of a second, smaller graph. The larger (composite) graph and the smaller (component) graph are said to be subspectral. Rules are given for constructing two subspectral components of a composite graph which has threefold symmetry such that the eigenvalues of one component and the eigenvalues of the second component taken twice comprise the complete spectrum of the composite graph. The mathematical basis for these rules is shown to be a unitary transformation upon the eigenvalue equation of the adjacency matrix of the composite graph by the matrix which represents threefold rotation.     

New lower bound formula from the local energy

A new formula to determine a lower bound to the ground‐state energy of an electronic system is presented making use of the local energy. Two‐electron atoms are considered and each lower bound is not only superior to the traditional lower bound, which is just the minimum of the local energy, but is also closer to the exact energy than the Hamiltonian expectation value that uses the same trial function. In this method, we have removed the restriction that the trial function be non‐zero. Lower bound methods using the local energy have the advantage that the expectation value of the squared Hamiltonian or another potentially difficult integral does not have to be calculated as in many other lower bound methods. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

低热固相合成类金刚石型配位聚合物Zn(INA)2

用低热固相合成法制备了配位聚合物Zn(INA)2(HINA=异烟酸),并用X射线粉末衍射,SEM等方法对产物进行了表征.该合成方法在近室温下进行,不需溶剂的参与,反应迅速,且产率接近100%,是一种高效简便的绿色合成方法.制得的配位聚合物Zn(INA)2具有四面体结构的四配位Zn2+中心,形成了类似金刚石结构的三维框架,并具有较好的热稳定性.     

Lower scaling approximation to EOM‐CCSD: A critical assessment of the ionization problem

In this article, we investigate the performance of different approximate variants of the EOM‐CCSD method for calculation of ionization potential (IP), as compared to EOM‐CCSDT reference values. None of the lower scaling approximations to the EOM‐CCSD method give a consistent performance for valence, inner valence, and core ionization, favoring one, or the other depending on the nature of the approximation used. The parent EOMIP‐CCSD method gives superior performance for valence IP but can show large errors for inner valence and core ionization. The problem is particularly severe for core‐ionization, where even the EOMIP‐CCSDT method cannot provide quantitative accuracy.     

Bounds to electron-repulsion energies

A review of rigorous bounds to electron-repulsion integrals for atoms and molecules is presented. Inequalities involving direct (classical) as well as indirect (quantal) Coulomb energies are discussed. This is followed by an account of two-electron integrals in a Hartree Fock context over Gaussian basis-sets. Novel rigorous bounds to these integrals are derived and tested for some organic molecules. Connections are established with the density-based inequalities presented earlier. The present results are expected to enhance the efficiency of a generalab initio Gaussian program and yet have a sound theoretical footing.     

CASSCF study into the mechanism for predissociation of the allyl radical

A theoretical study has been carried out on the allyl radical in its ground and first excited electronic states. Complete active space self‐consistent field (CASSCF) calculations show the presence of a conical intersection between the ground and first excited electronic states (～400 cm^?1 above the adiabatic excited state energy), reached by decreasing the C? C? C angle and twisting the C? C bonds. The presence of this conical intersection provides a likely explanation for the very rapid predissociation in the excited electronic state. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Effect of doping on the ground‐state spin of stacked organometallic ferromagnets

The effective Hamiltonians that describe the low‐lying energy states of some stacked organometallic ferromagnets with dopants have been derived. It is shown that the lattice stack with N unit cells and p conduction electrons has the ground‐state spin S₀=(N+p)/2 at p<N and S₀≤(3N−p)/2 at N≤p<2N. This means that the ferromagnetic ground state of metallocene‐based donor–acceptor stack is unstable to the donor doping and is stable to the acceptor doping. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 511–517, 1999     

Approximate lower bounds of the Weinstein and Temple variety

By using the Weinstein interval or coupling the Temple lower bound to a variational upper bound one can in principle construct an error bar about the ground‐state energy of an electronic system. Unfortunately there are theoretical and calculational issues which complicate this endeavor so that at best only an upper bound to the electronic energy has been practical in systems with more than a few electrons. The calculational issue is the complexity of 〈H²〉 which is necessary in the Temple or Weinstein approach. In this work we provide a way to approximate the 〈H²〉 to any desired accuracy using much simpler 〈H〉‐like information so that the lower bound calculations are more practical. The helium atom is used as a testing ground in which we obtain approximate error bars for the ground‐state energy of [?2.904230, ?2.903721] hartree using the variational energy with the Temple lower bound and [?2.919098, ?2.888344] hartree for the Weinstein interval. For comparison, the slightly larger error bars using the exact value of 〈H²〉 are: [?2.904358, ?2.903721] hartree and [?2.919765, ?2.887677] hartree, respectively. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007