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1.
I. I. Guseinov 《Journal of mathematical chemistry》2008,44(1):197-205
The new formulas are obtained for complete orthonormal sets of exponential type vector orbitals of a particle with spin 1
in coordinate, momentum and four-dimensional spaces using the properties of spherical vectors and complete orthonormal scalar
basis sets of -exponential type orbitals ( -ETO), -momentum space orbitals ( -MSO) and -hyperspherical harmonics ( -HSH) introduced by the author for particles with spin s = 0, where These vector orbitals are complete without the inclusion of the continuum and, therefore, their group of transformation is
the four-dimensional rotation group of O(4). For overlap integrals over vector Slater orbitals with the same screening constant
the analytical relations in coordinate space are also derived. It should be noted that the new idea presented in this study
is the combination of spherical vectors with complete orthonormal scalar sets for radial parts of -orbitals. 相似文献
2.
I. I. Guseinov 《Journal of mathematical chemistry》2009,45(4):946-952
The new analytical relations for the relativistic vector wave functions and Slater type relativistic vector orbitals in coordinate,
momentum and four-dimensional spaces are derived using the properties of quasirelativistic vector spherical harmonics introduced
by the author in previous paper (I.I. Guseinov, J. Math. Chem., 44, 197 (2008)) and complete orthonormal scalar basis sets
of nonrelativistic ψ
α
-exponential type orbitals (ψ
α
-ETO), -momentum space orbitals (-MSO) and z
α
-hyperspherical harmonics (z
α
-HSH). The 6-component relativistic vector wave functions obtained are complete without the inclusion of the continuum. The
relativistic vector wave function sets and Slater type relativistic vector orbitals are expressed through the corresponding
quasirelativistic vector wave functions and Slater vector orbitals, respectively. The analytical formulas are also derived
for overlap integrals over Slater type relativistic vector orbitals with the same screening constants in coordinate space. 相似文献
3.
Tomislav P. Živković 《Journal of mathematical chemistry》2008,43(2):518-600
Interaction of a finite quantum system that contains ρ eigenvalues and eigenstates with an infinite quantum system that contains a single one-parameter eigenvalue band is considered. A new approach for the treatment of the combined system
is developed. This system contains embedded eigenstates with continuous eigenvalues , and, in addition, it may contain isolated eigenstates with discrete eigenvalues . Two ρ × ρ eigenvalue equations, a generic eigenvalue equation and a fractional shift eigenvalue equation are derived. It is shown that all properties of the system that interacts with the system can be expressed in terms of the solutions to those two equations. The suggested method produces correct results, however
strong the interaction between quantum systems and . In the case of the weak interaction this method reproduces results that are usually obtained within the formalism of the
perturbation expansion approach. However, if the interaction is strong one may encounter new phenomena with much more complex
behavior. This is also the region where standard perturbation expansion fails. The method is illustrated with an example of
a two-dimensional system that interacts with the infinite system that contains a single one-parameter eigenvalue band. It is shown that all relevant completeness relations are satisfied,
however strong the interaction between those two systems. This provides a strong verification of the suggested method. 相似文献
4.
Radosław Szmytkowski 《Journal of mathematical chemistry》2007,42(3):397-413
We present a comprehensive table of recurrence and differential relations obeyed by spin one-half spherical spinors (spinor
spherical harmonics) Ωκ μ(n) used in relativistic atomic, molecular, and solid state physics, as well as in relativistic quantum chemistry. First, we
list finite expansions in the spherical spinor basis of the expressions A·B Ωκμ(n) and A·(B×C) Ωκμ(n), where A, B, and C are either of the following vectors or vector operators: n=r/r (the radial unit vector), e
0, e
±1 (the spherical, or cyclic, versors), (the 2×2 Pauli matrix vector), (the dimensionless orbital angular momentum operator; I is the 2×2 unit matrix), (the dimensionless total angular momentum operator). Then, we list finite expansions in the spherical spinor basis of the
expressions A·B F(r)Ωκμ(n) and A·(B×C) F(r)Ωκμ(n), where at least one of the objects A, B, C is the nabla operator , while the remaining ones are chosen from the set . 相似文献
5.
I. I. Guseinov 《Journal of mathematical chemistry》2009,45(4):1032-1040
The new one- and two-electron nonrelativistic and quasirelativistic basic functions are introduced. The general analytical
relations in terms of basic functions suggested are derived for the non- and quasi-relativistic atomic integrals over complete
orthonormal sets of -exponential type orbitals introduced by the author, where α = 1,0, − 1, − 2, . . . The relationships obtained are valid for the arbitrary values of quantum numbers and screening constants
of orbitals. 相似文献
6.
Zhang Shimin 《Journal of mathematical chemistry》2006,39(2):441-450
To increase inert substance i will make the equilibrium translation rate α
j
of reactant j decrease if ∑
i
ν
i
< 0 or increase if ∑
i
ν
i
> 0. When
or
, to increase non-inert substance i will make α
j
increase if i is reactant (i ≠ j) or decrease if i is resultant. When
has maximum if i is reactant (i≠ j) or minimum if i is resultant. If i is reactant,
(x
r
0 is “optimum proportion” of reactant) 相似文献
7.
F. Ivanauskas I. Kaunietis V. Laurinavičius J. Razumienė R. Šimkus 《Journal of mathematical chemistry》2008,43(4):1516-1526
Plate-gap model of enzyme doped porous electrode was utilized in order to calculate apparent Michaelis constants () and apparent maximal currents () of modeled amperometric biosensor for the wide range of given reaction/diffusion parameters. It was found that of plate-gap biosensor linearly depends on when rates of enzymatic reaction are lower than critical. Theoretically predicted linear correlation between apparent parameters
was observed experimentally for the case of carbon paste electrodes, which were modified by PQQ-dependent alcohol dehydrogenases.
At overcritical rates (or apparent maximal currents), is practically independent on Michaelis constant of soluble enzyme. Therefore, apparent Michaelis constant can be regarded
as biosensor’s topology representing parameter which, in fact, is not related to the specificity of enzyme kinetics. High
and rate-independent values of indicate that reaction proceeds at substrate-exposed top layer of the gap. In this case, reaction–diffusion system formally
is stratified into separate reaction (top) and diffusion (bottom) zones. Topology of such reaction–diffusion system reminds
“inverted” planar electrode, which contains diffusion layer below reaction layer. The net effect of plate-gap topology of
working electrode on apparent Michaelis constant is similar to the effect of diffusion layer covering enzymatic planar electrode. 相似文献
8.
Radosław Szmytkowski 《Journal of mathematical chemistry》2009,46(1):231-260
The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, , is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of , where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated
Legendre function of the second kind of integer degree and order, . In particular, we arrive at formulas which generalize to the case of (0 ≤ m ≤ n) the well-known Christoffel’s representation of the Legendre function of the second kind, Q
n
(z). The derivatives and , all with m > n, are also evaluated. 相似文献
9.
The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let
denote the set of trees on n vertices and diameter d,
. Yan and Ye [Appl. Math. Lett. 18 (2005) 1046–1052] have recently determined the unique tree in
with minimal energy. In this article, the trees in
with second-minimal energy are characterizedAMS Subject Classification: 05C50, 05C35 相似文献
10.
Tomislav P. Živković 《Journal of mathematical chemistry》2006,39(1):151-176
Time-dependent properties of a state
that interacts with an infinite dimensional quantum system
containing several one-parameter eigenvalue bands
are considered. This is done by a new mathematical method that produces correct results, however strong the interaction between
the state
and the system
. It is shown that in the case of the weak interaction one obtains standard results that are usually obtained within the formalism
of the perturbation expansion method. In particular, if the eigenvalue E of the state
is embedded inside the range
of the unperturbed eigenvalues, time evolution of the state
that is initially prepared in the state
has typical exponential decay behavior. One also reproduces standard results concerning probabilities of the transition of
the state
at infinite time (t=∞) into various eigenvalue bands. However, if the interaction is strong, one finds much more complex and much more complicated
behavior. 相似文献
11.
Within the Hartree-Fock framework, the spinless two-electron density function Γ (r
1, r
2) consists of direct Γdi (r
1, r
2) and exchange Γex (r
1, r
2) parts. Accordingly, the inner and outer radii in many-electron systems are rigorously separated into the direct and exchange contributions, i.e., and . It is generally shown that and , where is the usual average radius of an electron. Numerical examinations of the direct and exchange contributions for the 102 atoms
from He to Lr in their ground states find that the electron exchange works to decrease and increase . However, the exchange parts are very small and the direct parts essentially govern the inner and outer radii. 相似文献
12.
On the example of the molecular ion, we show that spherically distorted s-type orbitals possessing angular dependent orbital exponents, even in a minimal basis may lead to total energies the accuracy
of which is comparable with the ones obtained by fully numerical (‘complete basis’) calculations. 相似文献
13.
Tomislav P. ivkovi 《Journal of mathematical chemistry》2009,45(4):1060-1101
Mathematical formalism of the Low Rank Perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic
approximation with a standard assumption that force field does not change under isotopic substitutions. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at ρ atomic sites is considered. In the LRP approach vibrational frequencies ω
k
and normal modes of the isotopomer B are expressed in terms of the vibrational frequencies ν
i
and normal modes of the parent molecule A. In those relations complete specification of the normal modes is not required. Only amplitudes at sites τ affected by the isotopic substitutions and in the coordinate direction s (s = x, y, z) are needed. Out-of-plane vibrations of the (H,D)-benzene isotopomers are considered. Standard error of the LRP frequencies
with respect to the DFT frequencies is on average . This error is due to the uncertainty of the input data (± 0.5 cm−1) and in the absence of those uncertainties and in the harmonic approximation it should disappear. In comparing with experiment,
one finds that LRP frequencies reproduces experimental frequencies of (H,D)-benzene isotopomers better () than scaled DFT frequencies () which are designed to minimize (by frequency scaling technique) this error. In addition, LRP is conceptually and numerically
simple and it also provides a new insight in the vibrational isotope effect in the harmonic approximation. 相似文献
14.
Tomislav P. Živković 《Journal of mathematical chemistry》2006,39(2):295-344
A new method for the exact solution of the interaction of an isolated state
with an infinite dimensional quantum system §∞ b
containing several one-parameter eigenvalue bands
is developed. Unlike standard perturbation expansion approach, this method produces correct results however strong the interaction
between the state
and the system
. It is shown that in the case of the weak interaction this method correctly reproduces standard results obtained within the
formalism of the perturbation expansion method. In particular, due to the interaction with the system
, eigenvalue E of the state
shifts to a new position. In addition, if this eigenvalue is embedded inside the range
of the unperturbed eigenvalues, this shifted eigenvalue broadens and spectral distribution of the state
has the shape of the universal resonance curve. However, if the interaction is strong, one finds much more complex and much
more complicated behavior 相似文献
15.
Let λ1 (G) and Δ (G), respectively, denote the largest eigenvalue and the maximum degree of a graph G. Let
be the set of trees with perfect matchings on 2m vertices, and
. Among the trees in
, we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest
eigenvalue when
. Furthermore, it is proved that, for two trees T
1 and T
2 in
(m≥ 4), if
and Δ (T
1) > Δ (T
2), then λ1 (T
1) > λ1 (T
2). 相似文献
16.
This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs)
for the evaluation of auxiliary function which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated
with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function is completely general for p
a
≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function is compared with other method. 相似文献
17.
The mathematical models of the ion transport problem in a potential field are anayzed. Ion transport is regarded as the superposition
of diffusion and convection. In the case of pure diffusion model the classical Gottrell’s result is studied for a constant
as well as for the time dependent Dirichlet data at the electrode. Comparative analysis of the current response
and the classical Gottrellian
is given on the obtained explicit formulas. The approach is extended to find out the current response
corresponding to the diffusion-convection model. The relationship between the current response
and Gottrellian
is obtained in explicit form. This relationship permits one to compare pure diffusion and diffusion-convection models, including
asymptotic behaviour of current response and an influence of the convection coefficient. The theoretical result are illustrated
by numerical examples.
相似文献
18.
Numerical, adaptive algorithm evaluating the overlap integrals between the Numerical Type Orbitals (NTO) is presented. The
described algorithm exploits the properties of the prolate ellipsoidal coordinates, which are the natural choice for two-center
overlap integrals. The algorithm is designed for numerical atomic orbitals with the finite support. Since the cusp singularity
of the atomic orbitals vanish in the prolate ellipsoidal coordinate system, the adaptive integration algorithm in generates small number of subdivisions. The efficiency and reliability of the algorithm is demonstrated for the overlap integrals
evaluated for the selected pairs of Slater Type Orbitals (STO). 相似文献
19.
Wen-Huan Wang An Chang Lian-Zhu Zhang Dong-Qiang Lu 《Journal of mathematical chemistry》2006,39(1):231-241
The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings,
of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by
such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of
have been discussed.
MSC 2000: 05C17, 05C35 相似文献
20.
In this article, we derived several new recurrence relations of the rotation matrix elements by using Gauss’ recurrence formulas
for hypergeometric functions and auxiliary functions , and . The aim of this contribution is to obtain general algorithm to compute the rotation matrix elements, paying attention to
the use recurrence relationships of the auxiliary functions that allow the treatment of the functions with high angular momentum
quantum numbers. 相似文献