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Elements matrix

For the case of a double-D coil we multiply each matrix element with an element shifted by a constant distance of the same line. This is done in x- and y-direction. The distance between the two elements is the correlation length X for filtering in x-direction and a second correlation length for the movement in y-direction. Thus one gets two new matrices Ax and Ax for the filtering from the left to the right (positiv x-direction) and vice versa (negativ x-direction). [Pg.261]

The off-diagonal elements in this representation of h and v are the zero vector of lengtii (for h) and matrix elements which couple the zeroth-order ground-state eigenfunction members of the set q (for v) ... [Pg.47]

Under the assumption that the matrix elements can be treated as constants, they can be factored out of the integral. This is a good approximation for most crystals. By comparison with equation Al.3.84. it is possible to define a fiinction similar to the density of states. In this case, since both valence and conduction band states are included, the fiinction is called the joint density of states ... [Pg.119]

In solving the secular equation it is important to know which of the off-diagonal matrix elements " I wanish since this will enable us to simplify the equation. [Pg.160]

Equation (A 1.6.94) is called the KHD expression for the polarizability, a. Inspection of the denominators indicates that the first temi is the resonant temi and the second temi is tire non-resonant temi. Note the product of Franck-Condon factors in the numerator one corresponding to the amplitude for excitation and the other to the amplitude for emission. The KHD fonnula is sometimes called the siim-over-states fonnula, since fonnally it requires a sum over all intennediate states j, each intennediate state participating according to how far it is from resonance and the size of the matrix elements that coimect it to the states i. and The KHD fonnula is fiilly equivalent to the time domain fonnula, equation (Al.6.92). and can be derived from the latter in a straightforward way. However, the time domain fonnula can be much more convenient, particularly as one detunes from resonance, since one can exploit the fact that the effective dynamic becomes shorter and shorter as the detuning is increased. [Pg.252]

For themial unimolecular reactions with bimolecular collisional activation steps and for bimolecular reactions, more specifically one takes the limit of tire time evolution operator for - co and t —> + co to describe isolated binary collision events. The corresponding matrix representation of f)is called the scattering matrix or S-matrix with matrix elements... [Pg.773]

The physical interpretation of the scattering matrix elements is best understood in tenns of its square modulus... [Pg.773]

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... [Pg.790]

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. [Pg.968]

The present derivation can easily be generalized to systems with an arbitrary number of internal degrees of freedom, and it leads to coupled channel equations identical with equation (A3.11.63). where the coupling temis (A3.11.62) are expressed as matrix elements of the interaction potential using states which depend on these internal degrees of... [Pg.973]

To solve for the Y, we begin by solving a reference problem wherein the coupling matrix is assumed diagonal with constant couplings within each step. (These could be accomplished by diagonalizing U, but it would be better to avoid this work and use the diagonal U matrix elements.) Then, in temis of the reference U (which we call Uj), we have... [Pg.986]

This converts the calculation of S to the evaluation of matrix elements together with linear algebra operations. Generalizations of this theory to multichaimel calculations exist and lead to a result of more or less tire same form. [Pg.989]

State I ) m the electronic ground state. In principle, other possibilities may also be conceived for the preparation step, as discussed in section A3.13.1, section A3.13.2 and section A3.13.3. In order to detemiine superposition coefficients within a realistic experimental set-up using irradiation, the following questions need to be answered (1) Wliat are the eigenstates (2) What are the electric dipole transition matrix elements (3) What is the orientation of the molecule with respect to the laboratory fixed (Imearly or circularly) polarized electric field vector of the radiation The first question requires knowledge of the potential energy surface, or... [Pg.1059]

In turn, an expression for is obtained, which, in the frequency domain, consists of a numerator containing a product of (.s + 1) transition moment matrix elements and a denominator of. s complex energy... [Pg.1182]

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... [Pg.1192]

As for the Imear response, the transitions occur tlnough the electric-dipole operator and are characterized by the matrix elements hr equation Bl.5.30, the energy denominators involve the energy differences... [Pg.1274]

Luce T A and Bennemann K H 1998 Nonlinear optical response of noble metals determined from first-principles electronic structures and wave functions calculation of transition matrix elements P/rys. Rev. B 58 15 821-6... [Pg.1302]

In the language of quanPim meehanies, the time-dependent B -field provides a perturbation with a nonvanishing matrix element joining the stationary states a) and P). If the rotating field is written in temis of an amplitude a perturbing temi in tlie Hamiltonian is obtained... [Pg.1550]

The operators and have matrix elements between states a) and p) or, in general, between states Mg) and Mg l) and eonsequently induee transitions between levels adjaeent in energy. [Pg.1550]


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6, orbital perturbation matrix element

A+ spin-orbit matrix element

Algebraic solutions matrix elements

Algorithm constructing Hamiltonian matrix elements

Amplitude Scattering Matrix Elements

Angular Matrix Elements

Angular momenta operator matrix elements

Angular momentum matrix elements

Angular-momentum-adopted Gaussian matrix elements

Anharmonic coupling matrix elements

Anharmonic oscillator, matrix element

Approximations to the Coulomb-Breit matrix elements

Atomic radial matrix elements

Bardeen matrix element

Basic Matrix Elements

Basis functions matrix elements

Bond polarization matrix elements

Born-Oppenheimer approximation electronic, matrix elements

Bound-free dipole matrix elements

Breit Interaction Matrix Element

Breit matrix elements

Calculation of matrix elements

Charge operator, matrix elements

Chemical hardness matrix elements

Chemical source term element matrix

Cl Matrix Elements

Compliance matrix elements

Configuration interaction matrix elements

Configurational matrix element

Coriolis matrix elements

Coulomb interaction matrix elements

Coulomb matrix element

Coulomb operator, matrix elements

Coupling matrix element, variance

Coupling matrix elements

Cross Sections and Matrix Elements

Crude Born-Oppenheimer approximation matrix elements

Crystal field theory matrix elements

Crystal matrix elements

Current conservation matrix elements

Current matrix elements

Current matrix elements electromagnetic

Current matrix elements in the quark-parton model

Current operator, matrix elements

Currents hadronic matrix elements

Density matrices matrix elements

Density matrix diagonal elements

Density matrix elements

Density matrix elements master equation

Density matrix elements, transferability

Density operator matrix elements

Diagonal elements in a matrix

Diagonal elements of a matrix

Diagonal matrix elements

Diatomic molecules kinetic energy matrix elements

Diatomic molecules potential energy matrix elements

Dimensions matrix elements

Dipole approximation matrix element

Dipole matrix element

Dipole moment matrix element

Dipole transition matrix element

Dirac delta function matrix elements

Direction cosine matrix elements

Doubly reduced matrix element

ELEMENTS OF THE g MATRIX

Effective interactions matrix elements

Electric dipole Matrix element

Electric dipole moment, component matrix element

Electron coupling matrix element

Electron density matrix elements

Electron density matrix elements transferability

Electron matrix elements

Electron propagator spin matrix elements

Electron transfer matrix element

Electron-phonon matrix elements

Electron-vibrational matrix elements

Electronic Wavefunctions and Calculation of Matrix Elements

Electronic coupling matrix elements

Electronic matrix elements for

Electronic model, matrix elements

Electronic overlap matrix element

Electronic wavefunctions matrix elements between

Electrostatic interaction matrix elements

Element Connectivity Matrix

Element of a matrix

Element stiffness matrix

Element-species matrix

Elements balance, molecular matrix

Elements elemental matrix

Elements of the Secular- Matrix

Elements, of matrices

Evaluation of Matrix Elements

Evaluation of the nuclear derivative coupling matrix elements with canonical molecular orbitals

Exchange matrix element

Expectation values and matrix elements

Factorization, matrix elements

Fock operator diagonal matrix elements

Fock operator, matrix elements

Fock-matrix elements

Formulas for Hamiltonian and Overlap Matrix Elements in the PPD Algorithm

Franck-Condon factor matrix elements

Franck-Condon matrix element

G matrix elements

Hamiltonian Matrix Elements and Overlaps between Atomic Orbital-Based Determinants

Hamiltonian matrix elements

Hamiltonian operator matrix elements

Harmonic oscillator matrix elements

Hopping matrix element

Identifying Nonzero Matrix Elements

Induced dipole matrix element

Interaction matrix element

Interatomic matrix elements

Interbond matrix elements

Irreducible tensor operators matrix elements

Jacobian matrix elements

Jones Matrices for Simple Polarizing Elements

K-matrix element

Kane matrix element

Kinetic energy matrix elements

Kinetic energy matrix elements functions

Kohn-Sham Hamiltonian, matrix element

Kohn-Sham Hamiltonian, matrix element calculations

Kohn-Sham matrix elements

Local density functional Hamiltonian matrix elements

Magnetic dipole matrix element

Matrices referencing element

Matrix Elements and Perturbation Coefficients

Matrix Elements and Symmetry

Matrix Elements and the Wigner-Eckart Theorem

Matrix Elements of Operators

Matrix Elements under Time Reversal

Matrix element MALDI)

Matrix element between determinants

Matrix element between generalized product

Matrix element between hybrids

Matrix element between projected functions

Matrix element effects

Matrix element functions

Matrix element integration

Matrix element magnetic transition

Matrix element method

Matrix element of transitions

Matrix element overlap

Matrix element spin-orbit interaction

Matrix element spin-other-orbit

Matrix element transition gradient

Matrix element, meaning

Matrix elements 464 INDEX

Matrix elements 580 Subject

Matrix elements 8 decay probability, factorization

Matrix elements Breit operator

Matrix elements Zeeman perturbation

Matrix elements annihilation operator

Matrix elements basis

Matrix elements between Bloch sums

Matrix elements between antisymmetrized products

Matrix elements charge-current operator

Matrix elements computation

Matrix elements connecting different electronic configurations

Matrix elements contributions

Matrix elements creation operator

Matrix elements definition

Matrix elements dependence upon

Matrix elements evaluation

Matrix elements for

Matrix elements for composite systems

Matrix elements many-electron spin-orbit

Matrix elements molecular, factorization

Matrix elements moment integrals

Matrix elements of angular momentum

Matrix elements of harmonic oscillator

Matrix elements of spherical tensor operators the Wigner-Eckart theorem

Matrix elements of the quadrupole Hamiltonian

Matrix elements powers

Matrix elements projected basis

Matrix elements second quantization

Matrix elements spin-orbit, determination

Matrix elements symmetry

Matrix elements symmetry reduction

Matrix elements symmetry-adapted

Matrix elements, electrode-electrolyte interface

Matrix elements, pseudopotentials

Matrix nonzero elements

Momentum Matrix Elements of GaN

Momentum matrix elements

Morse potential matrix elements

Neutron scattering matrix elements

Non-adiabatic coupling matrix element

Non-adiabatic matrix elements

Nonadiabatic coupling matrix element

Nuclear matrix element

Obtaining Spin-Orbit Matrix Elements

Occurrence matrix nonzero elements

Off-Diagonal Matrix Elements of Total Hamiltonian between Unsymmetrized Basis Functions

Off-diagonal matrix elements

Operator matrix element

Operators and matrix elements in second-quantization representation

Orbital Energies and Interaction Matrix Elements

Perovskites matrix elements

Perturbation matrix elements

Photoelectron photoemission matrix element

Photoemission matrix elements

Photoionization matrix elements

Photon-atom interaction and photoionization matrix elements

Potential energy surface matrix elements

Potential matrix element

Potential matrix element evaluation

Potential matrix element reduced

Potential matrix element special cases

Potential matrix element spin-orbit

Potential matrix elements for

Primitive basis function matrix elements

Quantum chemical methods density matrix elements

Quasispin and isospin for relativistic matrix elements

Radial coupling matrix elements

Raman scattering matrix element

Recipes for Evaluation of Molecule-Fixed Angular Momentum Matrix Elements

Reduced matrix elements

Reduced matrix elements collision potentials

Reduced matrix elements interaction

Reduced matrix elements of tensor operators

Reduced matrix elements operators

Reduced matrix elements potential scattering

Reduced matrix elements tensor operators

Reduced matrix elements, rotational

Relation between particle and antiparticle matrix elements

Relativistic Breit operator and its matrix elements

Resonance energy as tunneling matrix element

Rotational overlap matrix element

Rotations/rotation matrix elements

Rules for matrix elements

S-matrix element

Scattering matrix elements

Self-energy matrix elements

Several forms of the dipole matrix element

Slater Matrix Element Rules

Solid State Matrix Elements

Spin-orbit diagonal matrix elements

Spin-orbit matrix elements

Spin-orbit perturbation matrix elements

Spin-other-orbit interaction matrix elements

Stark matrix element

Surfaces hamiltonian matrix elements

Surfaces hopping matrix elements

Surfaces overlap matrix elements

Symmetry relations between the matrix elements

T-matrix element

Tensor matrix elements

Tensorial products and their matrix elements

The G and F Matrix Elements of Typical Molecules

The Nuclear Coordinate Dependence of Matrix Elements

The coupling matrix element

Three-photon matrix element

Three-photon transition matrix elements

Tight-binding method matrix elements

Transfer matrix element

Transfer matrix element, definition

Transition dipole matrix elements states

Transition matrix elements

Transition metals hybridization matrix element

Transition-Monopole Treatments of Interaction Matrix Elements and Mixing with Charge-Transfer Transitions

Tunneling matrix element

Tunneling matrix element impurity

Tunneling matrix element theory

Tunneling matrix element, electron-transfer

Tunneling matrix element, electron-transfer effects

Tunneling matrix element, electron-transfer electronic coupling

Tunnelling matrix element

Two-Body Matrix Elements

Two-photon matrix elements

Valence bond theory matrix elements

Vibrational Assignment by the Matrix Element Method

Vibrational matrix element

Vibrational overlap matrix element

Vibronic matrix elements

Wave Functions and Matrix Elements of Bichromophores

Wave function expansions transition matrix elements

Wave functions and matrix elements

Wicks Theorem for the Evaluation of Matrix Elements

Wigner matrix elements

Wigner rotation matrix element

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