Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Permutations

One way of proceeding is shown in the flow diagram of figure 2 for the ease of = 8, P = 3. The operation labeled PERMUTE rearranges the sequence of data. The /th member is placed into theyth position where] is calculated from i as follows... [Pg.183]

The purpose of permutation is the same as with the EFT, namely the bisect of the data sequence progressively until data pairs are reached. By definition whenA = 2... [Pg.183]

The following discussion of wastage profiles is a result of observations made during many years of inspection. The classifications have been kept relatively broad as many permutations of these basic classes have been observed. It is also nor uncommon for two types of wastage profile to be evident on the same tube. [Pg.1033]

Each entry is the product of first applying the permutation at the top of the column and then applying the permutation at the left end of the row. [Pg.144]

Note the order of the subscripts on D[R] which follows from the fact that we use the N-convention of (equation A1.4.56) to define the effect of a permutation on a function. [Pg.182]

The so-ealled Slater-Condon rules express the matrix elements of any one-eleetron (F) plus two-eleetron (G) additive operator between pairs of antisymmetrized spin-orbital produets that have been arranged (by permuting spin-orbital ordering) to be in so-ealled maximal eoineidenee. Onee in this order, the matrix elements between two sueh Slater determinants (labelled >and are summarized as follows ... [Pg.2196]

Consider a triatomic system with the three nuclei labeled A, Ap, and Ay. Let the arrangement channel -1- A A be called the X arrangement channel, where Xvk is a cyclic permutation of apy. Let Rx,r be the Jacobi vectors associated with this arrangement channel, where r is the vector from A to and the vector from the center of mass of AyA to A . Let R i, rx be the corresponding mass-scaled Jacobi coordinates defined by... [Pg.206]

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). [Pg.330]

A symmetry that holds for any system is the permutational symmetry of the polyelectronic wave function. Electrons are fermions and indistinguishable, and therefore the exchange of any two pairs must invert the phase of the wave function. This symmetry holds, of course, not only to pericyclic reactions. [Pg.344]

With 4) containing a normalization factor and all permutations over the atomic orbital wave functions i (1 = 1,2,... 2n). Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, 4>b, has the fonn ... [Pg.391]

PERMUTATIONAL SYMMETRY AND THE ROLE OF NUCLEAR SPIN IN THE VIBRATIONAL SPECTRA OF MOLECULES IN DOUBLY DEGENERATE ELECTRONIC STATES ... [Pg.551]

IT. Total Molecular Wave Functdon TIT. Group Theoretical Considerations TV. Permutational Symmetry of Total Wave Function V. Permutational Symmetry of Nuclear Spin Function VT. Permutational Symmetry of Electronic Wave Function VIT. Permutational Symmetry of Rovibronic and Vibronic Wave Functions VIIT. Permutational Symmetry of Rotational Wave Function IX. Permutational Symmetry of Vibrational Wave Function X. Case Studies Lis and Other Systems... [Pg.551]

We will now explain the meaning of the word identical used above. Physically, it is meant for particles that possess the same intrinsic attributes, namely, static mass, charge, and spin. If such particles possess the same intrinsic attributes (as many as we know so far), then we refer to them as physically identical. There is also another kind of identity, which is commonly refeiTed to as chemical identity [56]. As discussed in the next paragraph, this is an important concept that must be steessed when discussing the permutational properties of nuclei in molecules. [Pg.566]

Let us examine a special but more practical case where the total molecular Hamiltonian, H, can be separated to an electronic part, W,.(r,s Ro), as is the case in the usual BO approximation. Consequendy, the total molecular wave function fl(R, i,r,s) is given by the product of a nuclear wave function X uc(R, i) and an electronic wave function v / (r, s Ro). We may then talk separately about the permutational properties of the subsystem consisting of electrons, and the subsystemfs) formed of identical nuclei. Thus, the following commutative laws Pe,Hg =0 and =0 must be satisfied X =... [Pg.568]

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... [Pg.568]

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. [Pg.569]


See other pages where Permutations is mentioned: [Pg.141]    [Pg.142]    [Pg.171]    [Pg.171]    [Pg.172]    [Pg.173]    [Pg.182]    [Pg.11]    [Pg.29]    [Pg.31]    [Pg.210]    [Pg.330]    [Pg.337]    [Pg.342]    [Pg.345]    [Pg.357]    [Pg.393]    [Pg.393]    [Pg.460]    [Pg.553]    [Pg.553]    [Pg.555]    [Pg.557]    [Pg.559]    [Pg.561]    [Pg.563]    [Pg.565]    [Pg.566]    [Pg.566]    [Pg.567]    [Pg.567]    [Pg.568]    [Pg.569]    [Pg.569]    [Pg.569]   
See also in sourсe #XX -- [ Pg.58 , Pg.79 , Pg.81 , Pg.301 , Pg.387 ]

See also in sourсe #XX -- [ Pg.62 ]

See also in sourсe #XX -- [ Pg.13 ]

See also in sourсe #XX -- [ Pg.64 ]

See also in sourсe #XX -- [ Pg.3 ]

See also in sourсe #XX -- [ Pg.89 ]

See also in sourсe #XX -- [ Pg.22 , Pg.24 , Pg.293 , Pg.605 , Pg.613 ]

See also in sourсe #XX -- [ Pg.34 ]

See also in sourсe #XX -- [ Pg.31 ]

See also in sourсe #XX -- [ Pg.89 ]

See also in sourсe #XX -- [ Pg.81 , Pg.82 ]

See also in sourсe #XX -- [ Pg.238 , Pg.286 ]

See also in sourсe #XX -- [ Pg.214 ]

See also in sourсe #XX -- [ Pg.7 ]

See also in sourсe #XX -- [ Pg.19 ]

See also in sourсe #XX -- [ Pg.225 ]

See also in sourсe #XX -- [ Pg.91 , Pg.95 , Pg.97 , Pg.328 , Pg.334 ]




SEARCH



A Construction from Pairs of Permutations

A circular permutation

Algebra permutations

Ammonia Permutation symmetry

Antisymmetric/antisymmetries permutations

Antisymmetry, permutational

Bose-Einstein statistics, permutational

Bose-Einstein statistics, permutational function

Building blocks, permutations

Butyllithium permutational halogen/metal

Characters from orbits permutation

Charge-mass-permutation

Charge-permutation reactions

Circular permutation

Circular permutations, protein function

Complete nuclear permutation group

Complete nuclear permutation inversion

Component analysis, permutations

Conical intersections permutational symmetry

Conserved Landscape Permutations

Constructions from Claw-Intractable Pairs of Permutations

Counting permutational isomers

Cyclic permutation

Diatomic molecules permutational symmetry

Displacement reactions, nucleophilic permutational rearrangement

Electron nuclear dynamics permutational symmetry

Electron permutation symmetry

Electron spin, permutational symmetry

Electronic wave function, permutational

Electronic wave function, permutational symmetry

Factorials, Permutations, and Combinations

Function families Cases of factoring and claw-intractable permutation pairs

Functions, Permutations, and Their Applications

GMR family of permutation pairs

Group theory permutations

Halogen/metal permutations

Homonuclear molecules, permutational

Homonuclear molecules, permutational electronic wave function

Homonuclear molecules, permutational symmetry

Hydrogen/metal permutations

Identical particles permutation operators

Index-permutation symmetry

Invariance with respect to permutation of identical particles (fermions and bosons)

Invariant operators, permutational symmetry

Inverse permutation, definition

Irreducible representations permutational symmetry

Isomer notation, permutational

Isomer permutational

Isomerism permutational analysis

Isomerization, permutational, of pentavalent

Isomerization, permutational, of pentavalent phosphorus compounds

Iterated permutation

Jahn-Teller effect permutational symmetry

Kramers permutation operator

Libraries of permutational isomers

Line-up permutation

Metalation permutational interconversions

Mi permutation

Molecular potential permutation symmetry

Motion general permutation

Near-degenerate states, permutational

Non-adiabatic coupling permutational symmetry

Nonlinear molecules permutational symmetry

Nuclear permutation operator

Nuclear permutational symmetry

Nucleotide sequence permutation

Operator permutation

Ordered sequences, permutations

Overlap Matrices and the Neglect of Some Permutations

Parity permutational

Particle permutation

Permutability

Permutability

Permutability definition

Permutation Groups and Point Group Symmetries

Permutation cycle structure

Permutation descriptor

Permutation diagram

Permutation formula

Permutation functionals

Permutation group

Permutation implications

Permutation index

Permutation inverse

Permutation matrix

Permutation motion

Permutation of Indices

Permutation of electrons

Permutation operator case functions

Permutation operator definition

Permutation pair

Permutation parity

Permutation problem

Permutation product form

Permutation representation

Permutation rules

Permutation signature

Permutation symbol

Permutation symmetry

Permutation symmetry implications

Permutation symmetry of rotational levels

Permutation test

Permutation tubes

Permutation-inversion operation

Permutational

Permutational

Permutational hydrogen/metal

Permutational interconversions

Permutational interconversions halogen/metal

Permutational interconversions hydrogen/metal

Permutational isomerism

Permutational isomerization

Permutational isomers by content

Permutational symmetry

Permutational symmetry GBO approximation and geometric phase

Permutational symmetry Jahn-Teller theorem

Permutational symmetry adiabatic states, conical intersections

Permutational symmetry antilinear operator properties

Permutational symmetry degenerate states chemistry, xiii

Permutational symmetry dynamic Jahn-Teller and geometric

Permutational symmetry dynamic Jahn-Teller and geometric phase

Permutational symmetry effects

Permutational symmetry electron/nuclear spin effects

Permutational symmetry energy functional form

Permutational symmetry format

Permutational symmetry group theoretical issues

Permutational symmetry group theoretical properties

Permutational symmetry levels

Permutational symmetry nuclear dynamics

Permutational symmetry nuclear spin function

Permutational symmetry of the basis

Permutational symmetry phase effects

Permutational symmetry phase-change rule

Permutational symmetry potential energy surfaces

Permutational symmetry rotational wave function

Permutational symmetry rovibronic/vibronic wave functions

Permutational symmetry theoretical background

Permutational symmetry two-dimensional Hilbert space model

Permutational symmetry vibrational wave function

Permutational symmetry, 3 isotopomers

Permutational symmetry, GBO

Permutational symmetry, adiabatic states

Permutational symmetry, dynamic Jahn-Teller

Permutational symmetry, dynamic Jahn-Teller and geometric phase effects

Permutational symmetry, total molecular

Permutational symmetry, total molecular wave function

Permutations algorithms

Permutations and combinations

Permutations of identical particles

Permutations only

Permutations, Combinations, and Probability

Permute

Permute

Permuted

Permuted

Permuted adaptive partitioning

Permuted cyclically

Phosphorus compounds, pentavalent, turnstile rearrangement and pseudoration in permutational

Phosphorus compounds, pentavalent, turnstile rearrangement and pseudoration in permutational isomerization

Phosphorus compounds, pentavalent, turnstile rearrangement and pseudoration permutational isomerization

Phosphorus compounds, pentavalent, turnstile rearrangement and pseudorotation in permutational isomerization

Phosphorus compounds, pentavalent, turnstile rearrangement and pseudorotation permutational isomerization

Planar molecules, permutational symmetry

Probability densities, permutational symmetry

Protein circular permutation [

RS-permutation

Random permuted blocks

Reference configuration permutational symmetry

Relation to Permutation Matrices

Representation generalized permutation

Ring permutation

Rotational wave function, permutational

Rovibronic wave function, permutational

Rovibronic wave function, permutational symmetry

Schrodinger equation permutational symmetry

Space-fixed coordinates, permutational

Space-inversion operator, permutational

Spin Permutation Formalism for Hubbard Model with Infinite Repulsion

Spin Permutation Technique in the Theory of Strongly Correlated Electron Systems

Spin and Permutation Symmetry

Spin function, permutational symmetry

Spin multiplicity, permutational symmetry

Spin permutation

Spin-orbit coupling permutational symmetry

Strong claw-intractable family of permutation

Strong claw-intractable family of permutation pairs

Superpositioning by PERMutations,

Symmetric properties permutational symmetry

Symmetry permutation group

Terms Permutation and Combination with Examples

The nuclear permutation operator for a homonuclear diatomic molecule

The unitarity of permutations

Tin/lithium permutation

Total molecular wave function, permutational

Translational motion general permutation

Trap-door one-way permutation

Triatomic molecules permutational symmetry

Variable classification permutations

Vibrational wave function, permutational

Wave function permutational symmetry

Weak claw-intractable family of permutation

Weak claw-intractable family of permutation pairs

© 2024 chempedia.info