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Operational matrix

Comparison with Eq. (7) shows that the the non-adiabatic operator matrix, A, has been added. This is responsible for mixing the nuclear functions associated with different BO PES. [Pg.277]

The non-adiabatic operator matrix, A can be written as a sum of two terms a matrix of numbers, G, and a derivative operator matrix... [Pg.277]

The superaiatrix notation emphasizes the structure of the problem. Each diagonal operator drives a wavepaclcet, just as in the adiabatic case of Eq. (10), but here the motion of the wavepackets in different adiabatic states is mixed by the off-diagonal non-adiabatic operators. In practice, a single matrix is built for the operator, and a single vector for the wavepacket. The operator matrix elements in the basis set <() are... [Pg.279]

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... [Pg.279]

The Hamiltonian again has the basic form of Eq. (63). The system is described by the nuclear coordinates, Q, which are relative to a suitable nuclear configuration Q. In conbast to Section in.C, this may be any point in configmation space. As a diabatic representation has been assumed, the kinetic energy operator matrix, T, is diagonal with elements... [Pg.285]

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... [Pg.288]

The familiar BO approximation is obtained by ignoring the operators A completely. This results in the picture of the nuclei moving over the PES provided by the electrons, which are moving so as to instantaneously follow the nuclear motion. Another common level of approximation is to exclude the off-diagonal elements of this operator matrix. This is known as the Bom-Huang, or simply the adiabatic, approximation (see [250] for further details of the possible approximations and nomenclature associated with the nuclear Schrodinger equation). [Pg.313]

Thus the Jacobi procedure, by making many rotations of the elements of the operand matrix, ultimately arrives at the operator matrix that diagonalizes it. Mathematically, we can imagine one operator matr ix that would have diagonalized the operand matr ix R, all in one step... [Pg.207]

II. The Slater-Condon Rules Give Expressions for the Operator Matrix Elements Among the CSFs... [Pg.276]

Here, an effective one-electron operator matrix has Fock and energy-dependent, self-energy terms. Prom this matrix expression, one may abstract one-electron equations in terms of the generalized Fock and energy-dependent, self-energy operators ... [Pg.40]

At a fundamental level, it has been shown that PECD stems from interference between electric dipole operator matrix elements of adjacent continuum f values, and that consequently the chiral parameters depend on the sine rather than the cosine of the relative scattering phases. Generally, this provides a unique probe of the photoionization dynamics in chiral species. More than that, this sine dependence invests the hj parameter with a greatly enhanced response to small changes in scattering phase, and it is believed that this accounts for an extraordinary sensitivity to small conformational changes, or indeed to molecular substitutions, that have only a minimal impact on the other photoionization parameters. [Pg.319]

The density operator (matrix) is Hermitian and for an arbitrary countable basis may be represented by a square matrix, that may be infinite, and with elements... [Pg.461]

Suppose now that we are interested in a given state M that satisfies the condition stated in eq. (2). As we deal just with one eigenvector, i.e., the set of c coefficients, the dressing matrix need not be composed entirely of non-zero elements. In fact, the single state (SC) operator matrix is diagonal [8]. Its elements are calculated for each G Sq as... [Pg.90]

Since any operator can be written as the sum of Hermitian and anti-Hermitian operators, we can restrict our discussion to these two types only. Further, any operator can be written as a linear combination of irreducible symmetry operators, so we can restrict ourselves to irreducible tensor operators. An operator matrix 0(r, K) that transforms according to the symmetry (T, K) obeys the relationship... [Pg.132]

This operator matrix transforms as the a th row of the representation a( 0. We can therefore immediately use the analysis for non-totally symmetric operators to devise a skeleton symmetrization scheme based on the P4 list. The only problem is that the form of V shown is not invariant under the index permutations of T4, but the form... [Pg.133]

Dependence of the energy operator matrix elements on nuclear... [Pg.256]

The frequency matrix Qy and the memory function matrix Ty, in the relaxation equation are equivalent to the Liouville operator matrix Ly and the Uy matrix, respectively. The later two matrices were introduced by Kadanoff and Swift [37] (see Section V). Thus the frequency matrix can be identified with the static variables (the wavenumber-dependent thermodynamic quantities) associated with the nondissipative part, and the memory kernel matrix can be identified with the transport coefficients associated with the dissipative part. [Pg.94]

In general, the strain vector e is written using a differential operator matrix [A] and the displacement vector U ... [Pg.328]


See other pages where Operational matrix is mentioned: [Pg.314]    [Pg.447]    [Pg.289]    [Pg.651]    [Pg.103]    [Pg.335]    [Pg.618]    [Pg.619]    [Pg.238]    [Pg.279]    [Pg.286]    [Pg.419]    [Pg.221]    [Pg.58]    [Pg.618]    [Pg.619]    [Pg.588]    [Pg.129]    [Pg.133]    [Pg.346]    [Pg.360]    [Pg.361]    [Pg.79]   
See also in sourсe #XX -- [ Pg.275 ]

See also in sourсe #XX -- [ Pg.275 ]




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A1 Appendix Matrix Operations

Adjoint matrices operator

Angular momenta operator matrix elements

Charge operator, matrix elements

Coulomb operator, matrix elements

Current operator, matrix elements

Density matrix operator

Density operator matrix elements

Derivative Fock operator matrices

Elementary Matrix Operations

Elementary Operations and Properties of Matrices

Elimination, matrix operation

Excel matrix operations

Fock operator diagonal matrix elements

Fock operator matrix representation

Fock operator, matrix elements

Hamiltonian matrix operator

Hamiltonian operator matrix elements

Hermitian-symmetric matrix operator

Irreducible tensor operators matrix elements

Linear Operators and Transformation Matrices

Matrices and Matrix Operations

Matrices array operations

Matrices as Representations of Symmetry Operators

Matrices colon operator

Matrix Elements of Operators

Matrix and Spin Operators

Matrix computations, MATLAB operations

Matrix crystal symmetry operator representation

Matrix elements Breit operator

Matrix elements annihilation operator

Matrix elements charge-current operator

Matrix elements creation operator

Matrix elements of spherical tensor operators the Wigner-Eckart theorem

Matrix inverse operations

Matrix operations

Matrix operations

Matrix operations in Excel

Matrix operations in Matlab

Matrix representation 50 vector operators

Matrix representation of an operator

Matrix representation of operators

Matrix representatives of operators

Matrix row operations

Nonsingular matrix Operator

OPERATIONS WITH PARTITIONED MATRICES

Operating with R-matrices

Operational space inertia matrix

Operational space inertia matrix Method

Operational space inertia matrix inverse

Operations on Matrices

Operator Overlap matrix

Operator Pauli matrix

Operator and matrices

Operator matrix

Operator matrix

Operator matrix element

Operator matrix representation

Operator matrix representatives

Operators and matrix elements in second-quantization representation

Polarization properties operator matrix

Real operating matrix

Reduced matrix elements of tensor operators

Reduced matrix elements operators

Reduced matrix elements tensor operators

Relativistic Breit operator and its matrix elements

Review of scalar, vector, and matrix operations

Selected Topics in Matrix Operations and Numerical Methods for Solving Multivariable 15- 1 Storage of Large Sparse Matrices

Simultaneous Operations matrix

Some formulae and rules of operation on matrices

Statistic operator matrix)

Symmetry operations matrix representation

Symmetry operations, matrix

The Matrix as Operator

The Operational Space Inertia Matrix

Unitary matrix expansions of creation and annihilation operators

Vector operators, 50 algebra matrix representation

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