Matrix representation of operators

Matrix representation of operators. - If an operator T is defined on the space A = (x), then the quantity TXi is also an element of this space and may be expanded in the form... 

Table 4.5 The multiplication table for the C2 point group using the matrix representation of operators for the x and y basis.

Dyall KG, Grant IP and Wilson S 1984 Matrix representation of operator products. J. Phys. B 17, 493-503. 

To obtain equations that are independent of it is necessary to consider the different contributions to the derivative density. As for any matrix representation of operators, it is possible to split the contributions into different subspace projections (compare Eq. [108]) ... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

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... 

A number of procedures have been proposed to map a wave function onto a function that has the form of a phase-space distribution. Of these, the oldest and best known is the Wigner function [137,138]. (See [139] for an exposition using Louiville space.) For a review of this, and other distributions, see [140]. The quantum mechanical density matrix is a matrix representation of the density operator... 

For the following basis of funetions (T 2p p and F2p ), construet the matrix representation of the Lx operator (use the ladder operator representation of Lx). Verify that... 

From this general result, it is clear that the matrix representation of the operator is given by... 

It is easy to see that we ean form the matrix representation of any linear operator for any eomplete basis in any spaee. To do so, we aet on eaeh basis funetion with the operator and express the resulting funetion as a linear eombination of the original basis funetions. The eoeffieients that arise when we express the operator aeting on the funetions in terms of the original funetions form the the matrix representation of the operator. 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

Fock Space Representation of Operators.—Let F be some operator that neither creates nor destroys particles, and is a known function in configuration space for N particles. In symbols such an operator must by definition have the following matrix elements in Fock space ... 

Now let us use the set, <0> to form a matrix representation of some operator Q at time hi assuming that Q is not explicitly a function of time. The expectation value of Q in the various states, changes in time only by virtue of the time-dependence of the state vectors used in the representation. However, because this dependence is equivalent to a unitary transformation, the matrix at time t is derived from the matrix at time t0 by such a unitary transformation, and we know that this cannot change the trace of the matrix. Thus if Q — WXR our result entails that it is not possible to change the ensemble average of R, which is just the trace of Q. 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

This step is similar to what we have done in equation (7-7) where we obtained the matrix representation of the Kohn-Sham operator. If we insert expression (7-14) for the charge density in terms of the LCAO functions and make use of the density matrix P defined in equation (7-15), we arrive at... 

If a proper rotation is combined with a reflection with respect to the axis of rotation, it is called an improper rotation The matrix representation of silvan operation is found simply by replacing 1 by -1 in Eq. (104), The Scftfllfllies symbol for an improper rotation by y is S /tp- Hence, matrix the representation of an improper counter-clockwise rotation by y is of the form ... 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

The matrix representation of the spin operator requires the spin state of a particle to be represented by row vectors, commonly interpreted as spin up or down. An arbitrary state function J must be represented as a superposition of spin up and spin down states... 

Let us consider the simple case of the H atom and its variational approximation at the standard HF/3-21G level, for which we can follow a few of the steps in terms of corresponding density-matrix manipulations. After symmetrically orthogonalizing the two basis orbitals of the 3-21G set to obtain orthonormal basis functions A s and dA, we obtain the corresponding AO form of the density operator (i.e., the 2 x 2 matrix representation of y in the... 

Another method that may be used to generate the projection operator involves the use a matrix representation of the operator. In particular, we will use the orthogonal representation. First we must assign a Yamanouchi symbol to each tableau we have created. This is done by going through the numbers from 1 to n in each tableau and writing down in which row the number occurs. Thus if we assign names to the above tableaux ... 

Now we may find the matrix representation, U, of the operators. The dimensions of the matrices will be the same as the dimensions of the irreducible representation used. The matrix representation of the identity operator, U E), will of course be the identity matrix. If it is noted that any permutation may be written as a product of transpositions (permutations of order 2), and any transposition may be written as a product of elementary transpositions p p + 1) [74], then it is only nessesary to find matrix representations of the elementary transpositions. The diagonal elements of the elementary transposition p p + 1) are given by... 

Li is the matrix representation of the lattice Liouvillian in the space of the basis operators, 1 is a unit (super)operator and Ci are projection vectors representing the operators of Eq. (32) in the same space. The... 

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