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221.
. Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis
by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry.
We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements
of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases.
A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred
to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical
operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches
to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used.
Received: 3 September 1996 / Accepted: 19 December 1996 相似文献
222.
An efficient angular momentum method is presented and used to derive analytic expressions for the vibration-rotational kinetic energy operator of polyatomic molecules.The vibration-rotational kinetic energy operator is expressed in terms of the total angular momentum operator J,the angular momentum operator J and the momentum operator p conjugate to Z in the molecule-fixed frame Not only the method of derivation is simpler than that in the previous work,but also the expressions ot the kinetic energy operators arc more compact.Particularly,the operator is easily applied to different vibrational or rovibrational problems of the polyatomic molecules by variations of matrix elements Gn of a mass-dependent constant symmetric matrix 相似文献
223.
Successive band-splitting transitions occur in the one-dimensional map xi+1=g(xi),i=0, 1, 2,... withg(x)=x, (0 x 1/2) –x +, (1/2 <x 1) as the parameter is changed from 2 to 1. The transition point fromN (=2n) bands to 2Nbands is given by=(2)1/N (n=0, 1,2,...). The time-correlation function
i=xix0/(x0)2,xi xi–xi is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition point=2,
i–[(10–42)/17] i,0-[(102-8)/51]i,1 + [(7 + 42)/17](–1)ie–yi, where2(–2) is the damping constant and vanishes at=2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly at=2. The asymptotic expression for
i has been obtained by deriving an analytic form of
i for a sequence of which accumulates to 2 from the above. Near the transition point=(2)1/N, the damping constant of
i fori N is given by
N=2(N-2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results. 相似文献
224.
Uppadine LH Gisselbrecht JP Kyritsakas N Nättinen K Rissanen K Lehn JM 《Chemistry (Weinheim an der Bergstrasse, Germany)》2005,11(8):2549-2565
An extended family of heterometallic [(M1)2(M2)2(L-)4](n+) [2x2] grid-type arrays 1-9 has been prepared. The three-tiered synthetic route encompasses regioselective, redox and enantioselective features and is based on the stepwise construction of heteroditopic hydrazone ligands A-C. These ligands contain ionisable NH and nonionisable NMe hydrazone units, which allows the metal redox properties to be controlled according to the charge on the ligand binding pocket. The 2-pyrimidine (R) and 6-pyridine (R') substituents have a significant effect on complex geometry and influence both the electrochemical and magnetic behaviour of the system. 1H NMR spectroscopic studies show that the Fe(II) ions in the grid can be low spin, high spin or spin crossover depending on the steric effect of substituents R and R'. This steric effect has been manipulated to construct an unusual array possessing two low-spin and two spin-crossover Fe(II) centres (grid 8). Electrochemical studies were performed for the grid-type arrays 1-9 and their respective mononuclear precursor complexes 10-13. The grids function as electron reservoirs and display up to eight monoelectronic, reversible reduction steps. These processes generally occur in pairs and are assigned to ligand-based reductions and to the Co(III)/Co(II) redox couple. Individual metal ions in the heterometallic grid motif can be selectively addressed electrochemically (e.g., either the Co(III) or Fe(II) ions can be targeted in grids 2 and 5). The Fe(II) oxidation potential is governed by the charge on the ligand binding unit, rather than the spin state, thus permitting facile electrochemical discrimination between the two types of Fe(II) centre in 7 or in 8. Such multistable heterometallic [2x2] gridlike arrays are of great interest for future supramolecular devices incorporating multilevel redox activity. 相似文献
225.
Summary. Parameters of the formalism [1–6] describing spin crossover in the solid state have been defined via molecular potentials in model systems of neutral and ionic complexes. In the first instance Lennard-Jones and electric dipole–dipole potentials have been used whereas in ionic systems Lennard-Jones and electric point-charge potentials have been used. Electric dipole–dipole interaction of neutral complexes brings about
a positive excess energy controlled by the difference of electric dipole moments of HS and LS molecules. Differences of the
order of Δμ = 1–2 D cause an abrupt spin crossover in systems with T1/2 = 100–150 K. Magnetic coupling contributes both to the excess energy and excess entropy, however the overall effect is equivalent
to a modest positive excess energy. Ionic systems in the absence of specific interactions are characterised by very small
excess energies corresponding to practically linear van’t Hoff plots. Detectable positive and negative excess energies in these systems may arise from interactions of ligands belonging
to neighbouring complexes. The HOMO–LUMO overlap in HS–LS pairs can bring about a nontrivial variation of the shape of transition
curves. Examples of regression analysis of experimental transition curves in terms of molecular potentials are given. 相似文献
226.
227.
N. S. Papageorgiou 《Journal of Optimization Theory and Applications》1991,68(2):305-320
In this paper, we consider a sequence of abstract optimal control problems by allowing the cost integrand, the partial differential operator, and the control constraint set all to vary simultaneously. Using the notions of -convergence of functions,G-convergence of operators, and Kuratowski-Mosco convergence of sets, we show that the values of the approximating problems converge to that of the limit problem. Also we show that a convergent sequence of optimal pairs for the approximating problems has a limit which is optimal for the limit problem. A concrete example of parabolic optimal control problems is worked out in detail.This research was supported by NSF Grant No. DMS-88-02688. 相似文献
228.
We consider the following (solitary) game: each node of a directed graph contains a pile of chips. A move consists of selecting a node with at least as many chips as its outdegree, and sending one chip along each outgoing edge to its neighbors. We extend to directed graphs several results on the undirected version obtained earlier by the authors, P. Shor, and G. Tardos, and we discuss some new topics such as periodicity, reachability, and probabilistic aspects.Among the new results specifically concerning digraphs, we relate the length of the shortest period of an infinite game to the length of the longest terminating game, and also to the access time of random walks on the same graph. These questions involve a study of the Laplace operator for directed graphs. We show that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable.Finally, we show how the basic properties of the probabilistic abacus can be derived from our results. 相似文献
229.
Let H=1/2+V on l2(B), whereB is the Bethe lattice andV(x),x
B, are i.i.d.r.v.'s with common probability distribution. It is shown that for distributions sufficiently close to the Cauchy distribution, the density of states(E) is analytic in a strip about the real axis. 相似文献
230.
Ann M. Schmiedekamp Anthony Ginnetti Brian Piccione Kevin Cannon M. Dominic Ryan 《International journal of quantum chemistry》2007,107(6):1415-1429
Octahedral, six‐coordinate Co2+ can exist in two spin states: S = 3/2 and S = 1/2. The difference in energy between high spin (S = 3/2) and low spin (S = 1/2) is dependent on both the ligand mix and coordination stereochemistry. B3LYP calculations on combinations of neutral imidazole, NH3, and H2O ligands show that low‐spin isomers are stabilized by axial H2O ligands and in structures that also include trans pairs of equatorial NH3 and protonated imidazole ligands, spin crossover structures are predicted from spin state energy differences. Occupied Co d orbitals from the DFT calculations provide a means of estimating effective ligand strength for homoleptic and mixed ligand combinations. These calculations suggest that in a labile biological system, a spin crossover environment can be created. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 相似文献