Operators matrix representation

The fundamenfal argumenf underlying fhe work which is discussed in fhis chapter is that, whether in wavefunction or in operator-matrix representation, the physically and computationally appropriate symbolic form fhat must transcent theoretical approaches to the understanding of resonance sfates is... 

The relationships among symmetry operations, matrix representations, reducible and irreducible representations, and character tables are conveniently illustrated in a flowchart, as shown for C2v symmetry in Table 4.8. 

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... 

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... 

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... 

The operators F eorresponding to all physieally measurable quantities are Hermitian this means that their matrix representations obey (see Appendix C for a deseription of the bra I > and kef < notation used below) ... 

Here, DF represents the matrix produet of the density matrix Dj i and the matrix representation Fi j = <( )i F ( )j> of the F operator, both taken in the ([ij basis, and Tr represents the matrix traee operation. 

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... 

Here, Xr(R) is the eharaeter belonging to symmetry E for the symmetry operation R. Applying this projeetor to a determinental flinetion of the form ( )i( )j generates a sum of determinants with eoeffieients determined by the matrix representations Ri ... 

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. 

In quantum meehanies, physieally measurable quantities are represented by hermitian operators. Sueh operators R have matrix representations, in any basis spanning the spaee of funetions on whieh the R aet, that are hermitian ... 

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. 

We ean likewise write matrix representations for eaeh of the symmetry operations of the C3v point group ... 

Internally, molecules can be represented several different ways. One possibility is to use a bond-order matrix representation. A second possibility is to use a list of bonds. Matrices are convenient for carrying out mathematical operations, but they waste memory due to many zero entries corresponding to pairs of atoms that are not bonded. For this reason, bond lists are the more widely used technique. 

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. 

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. 

For any symmetry operator T = T 0) (rewritten r when operating on the domain of basis functions x)) for instance, the rotation-reflexion about the z-axis, with matrix representation... 

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 ... 

It should be noted that the trace of a matrix that represents a given geo] operation is equal to 2 cos y 1, the choice of signs is appropriate to or improper operations. Furthermore, it should be noted that the aim direction of rotation has no effect on the value of the trace, as a inverse sense corresponds only to a change in sign of the element sin y. TE se operations and their matrix representations will be employed in the following chapter, where the theory of groups is applied to the analysis of molecular symmetry. 

As indicated above there may be many equivalent matrix representations for a given operation in a point group. Although the form depends on the choice of basis coordinates, the character is Independent of such a choice. However, for each application there exists a particular set of basis coordinates in terms of which the representation matrix is reduced to block-diagonal form. This result is shown symbolically in Fig. 4. ft can be expressed mathematically by the relation... 

---