Operator matrix representatives

This form, which is analogous to that of Eq. (13), exhibits symbolically the fact that only the second part, i.e., a complex operator (matrix), represents the non-Hermitian nature of the problem in a complete function space of square-integrable functions, where the state-specific expectation value of H(r) is the real Eq and that of K re ) is the complex self-energy Translated into the language of function spaces, it is evident that diagonalization of the real H(r) on a space of real square-integrable functions yields ( Tq/Eo). It is then necessary to find a practical way to incorporate into the complete calculation the equivalent to the effects of the complex operator (matrix) K(rc ). 

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... 

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... 

This matrix represents an effective operator that still has to act on the bending functions/ (p),/ (p). A generalization of (24) to the case when the kinetic energy operator (i.e., the coefficients 7 and A) has a different form in the... 

Wife some patience it is found feat fee operation [3 represented by the matrix. L... 

The matrix [Tij] should not be confused with the operator T although the same symbol is commonly used for an operator and a matrix representing it. 

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 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... 

According to the rule for matrix multiplication introduced earlier, each element of y is calculated as the scalar product between c and the corresponding column of A. These linear operations are represented exactly by the following system of inhomogeneous linear equations ... 

Note that because the identity operation must always be represented by the identity matrix, the trace of the matrix representing the identity is equal to the dimensionahty of the representation. 

The matrix elements (8.35) in the uncoupled space-fixed basis can be most easily evaluated if all interaction operators are represented as uncoupled products of spherical tensors, with each tensor defined in the space-fixed coordinate system. Since the Hamiltonian is always a scalar operator, we can write any interaction in the Hamiltonian as a sum... 

The matrix representing the effects of the / operation is similarly generated from the equations ... 

A brief summary of the mathematical notation adopted throughout this text is in order. Scalar quantities, whether constants or variables, are represented by italic characters. Vectors and matrices are represented by boldface characters (individual matrix elements are scalar, however, and thus are represented by italic characters that are indexed by subscript(s) identifying the particular element). Quantum mechanical operators are represented by italic characters if diey have scalar expectation values and boldface characters if their expectation values are vectors or matrices (or if they are typically constructed as matrices for computational purposes). The only deliberate exception to the above rules is that quantities represented by Greek characters typically are made neither italic nor boldface, irrespective of their scalar or vector/matrix nature. 

The functions (2.50) are called basis functions The matrices F, G,. .. are called matrix representatives of the operators F, G,. .. in the

specific form of the matrix representation of a set of operators depends on the basis chosen. Equation (2.53) shows that the effect of the operator G on the basis functions is determined by the matrix elements GkJ. Since an arbitrary well-behaved function can be expanded using the complete set (2.50), knowledge of the matrix G allows one to determine the effect of the operator G on an arbitrary function. Thus, knowledge of the square matrix G is fully equivalent to knowledge of the corresponding operator G. Since G is a Hermitian operator, its matrix elements satisfy Gij = (GJi). Hence the matrix G representing G is a Hermitian matrix (Section 2.1). 

For a one-electron spin problem, let the operator G have the matrix representative... 

The operator q2 is the square of the operator q. Since matrix representatives of operators obey the same relations as the operators (Section 2.3), the q2 matrix [whose elements are ] is the square of the q matrix [whose elements are (4.45)]. Hence, using the matrix-multiplication rule (2.11), we have... 

Let Rao be the matrix representing the symmetry operation R in the representation rAG. Equation (9.64) means that there is a similarity transformation that transforms RAO to RAO, where RAO is in block-diagonal form, the blocks being the matrices R],R2,...,Rfc of the irreducible representations ri,r2,...,ri. Let rAO denote the block-diagonal representation equivalent to TAO and let A be the matrix of the similarity transformation that converts the matrices of I AO to TAO Rao = A, RaoA. We form the following linear combinations of the AOs ... 

Hence these three operators are represented in rAO by the unit matrix of order 2. The remaining operators interchange 1 sa and 1 ... 

Each of these operators is represented by the square matrix in (9.70). The two-dimensional representation TA0 thus has the characters... 

We will not bother to write out the four 7x7 matrices of TAO since we only need their traces. Clearly, for each AO that is unchanged by the symmetry operator Or, we get a contribution of +1 to the trace of the matrix representing R in rAo for example, see (9.69). For each AO that is... 

It is relatively easy to deduce the four one-dimensional representations. As in every group, there must be the so-called totally symmetric representation, in which every symmetry operation is represented by the one-dimensional matrix 1. At this point, we have in hand the following part of the character table ... 

Example 6.1-1 This example describes a bonding in tetrahedral AB4 molecules. The numbering of the B atoms is shown in Figure 6.1. Denote by ar a unit vector oriented from A along the bond between A and Br. With ((TX matrix representatives (MRs) T(T) from T(rr (a V(T) since we only need the character system -/T of the representation To-. Every ar that transforms into itself under a symmetry operator T contributes +1 to the character of that MR T(T), while every oy that transforms into point group. The values of, t( T) for the point group Td are given in Table 6.1. This is a reducible representation, and to reduce it we use the prescription... 

The spin-orbit coupling term in the Hamiltonian induces the coupling of the orbital and spin angular momenta to give a total angular momentum J = L + S. This results in a splitting of the Russell-Saunders multiplets into their components, each of which is labeled by the appropriate value of the total angular momentum quantum number J. The character of the matrix representative (MR) of the operator R(0 n) in the coupled representation is... 

To find the symmetry of the normal modes we study the transformation of the atomic displacements xL y, z,, i 0,1,2,3, by setting up a local basis set e,i ci2 el3 on each of the four atoms. A sufficient number of these basis vectors are shown in Figure 9.1. The point group of this molecule is D3h and the character table for D3h is in Appendix A3. In Table 9.1 we give the classes of D3h a particular member R of each class the number of atoms NR that are invariant under any symmetry operator in that class the 3x3 sub-matrix r, (R) for the basis (e,i el2 cl31 (which is a 3 x 3 block of the complete matrix representative for the basis (eoi. .. e331) the characters for the representation T, and the characters for... 

II) If G contains a subgroup H then the choice of the set of poles made for G should be such that rule I is still valid for H, otherwise the representations of G will not subduce properly to those of H. Subduction means the omission of those elements of G that are not members of H and properly means that the matrix representatives (MRs) of the operators in a particular class have the same characters in H as they do in G. 

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