Matrix representation 50 vector operators

The problem is then reduced to the representation of the time-evolution operator [104,105]. For example, the Lanczos algorithm could be used to generate the eigenvalues of H, which can be used to set up the representation of the exponentiated operator. Again, the methods are based on matrix-vector operations, but now much larger steps are possible. 

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 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... 

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... 

Elements of second order reduced density matrix of a fermion system are written in geminal basis. Matrix elements are pointed out to be scalar product of special vectors. Based on elementary vector operations inequalities are formulated relating the density matrix elements. While the inequalities are based only on the features of scalar product, not the full information is exploited carried by the vectors D. Recently there are two object of research. The first is theoretical investigation of inequalities, reducibility of the large system of them. Further work may have the chance for reaching deeper insight of the so-called N-representability problem. The second object is a practical one examine the possibility of computational applications, associate conditions above with known methods and conditions for calculating density matrices. 

It is also possible to construct matrix representations by considering the effeot that the symmetry operations of a point group have on one or more sets of base veotors. We will consider two cases, both using the point group as an example (1) the set of base veotors eu e, and e introduced in 5-2 (2) three sets of mutually perpendioular base vectors, each located at the foot of a symmetric tripod. 

In Table 6-3.1 we show the matrices for all of the operations of the 8v point group using both real and complex p-orbitals as basis functions. For the operations Ct and Cj we have simply replaced 0 by 27 /3 and 4t /3 respectively in both eqn (6-3.1) and eqn (6-3.2). The matrices for the rejection operations have been obtained in a fashion similar to that used for the rotations. In carrying out these steps it has been assumed that plf p, and p lie along the vectors 6t, e8, and e, respectively (see Fig. 6-3.1). For obvious reasons the matrix representation in the real basis is identical to the one given in 5-3(2) and, further, the reader may verify for himself that the matrices using the complex basis obey the 8v group table (Table 3-4.1). 

Translation of the PtClJ- ion in the x and y directions can be represented by the two vectors shown on the platinum atom (Fig. 3.19). fn contrast to all of the cases we have so far considered, certain operations of the Du group lead to new orientations for both vectors that do not bear a simple +1 or — I relationship to the original positions. For example, under a clockwise C4 operation, the x vector is rotated to the + y direction, and the y vector is rotated to a -x position. The character for this operation is zero. (This arises because the diagonal elements of the matrix for this operation are all zero other elements in the matrix are nonzero but do not contribute to the character.) The S4 and operations lead to a similar muting of tbe x and y functions and also have characters of zero. Because of this mixing, the x and y functions are inseparable within the />4k symmetry group and arc said to transform as a doubly degenerate or two-dimensional representation. 

Together, these 16 product operators describe the 16 matrix elements in the 4 x 4 density matrix representation of a two-spin system (Chapter 10). In the matrix, each element represents coherence between (or superposition of) two spin states. As there are four spin states for a two-spin system (aid s i/3s, P as, and PiPs), there are 16 possible pairs of states, which can be superimposed or share coherence. The product operators are closer to the visually and geometrically concrete vector model representations, so in most cases they are preferable to writing down the 16 elements of the density matrix, especially as only a few of the elements are nonzero in most of the examples we discuss. 

Associated with each operator realization of a Lie algebra we generally have a vector space on which these operators act. For the realization given by L this might be either an abstract space of angular momentum states lm), 1 = 0, 1,... m = —/, — l + 1,..., l or a concrete realization of them as spherical harmonic functions Ylm(6, (j)). We can then consider the matrix elements of the operators with respect to this vector space of states and this leads to the important concept of a matrix representation of a Lie algebra. 

In general, Eq. (39a) holds for any component of V so it also holds for V . Thus, the matrix representation of a component of an so(3) vector operator has a particularly simple form. 

These two vectors may be used in the description of the stretching vibrations of the molecule. The molecular symmetry is C2h. Figure 4-7 helps to visualize the effects of the symmetry operations of this group on the selected basis. There are four symmetry operations in the C2h point group, E, C2, /, and ah. E leaves the basis unchanged, so the corresponding matrix representation is a unit matrix ... 

Let us take now a more complicated basis, and consider all the nuclear coordinates of HNNH shown in Figure 4-8a. These are the so-called Cartesian displacement vectors and will be discussed in Chapter 5 on molecular vibrations. Let us find the matrix representation of the crh operation (see Figure 4-8b). The horizontal mirror plane leaves all x and y coordinates unchanged while all z coordinates will go into their negative selves. In matrix notation this is expressed in the following way ... 

Symmetry operations on a position vector 6-3. Matrix representations for and... 

Converting the equations in the first part of this section to their basis set forms involves little more than replacing a wavefunction (or derivative wavefunction) with a vector of coefficients, a column vector if to the right of an operator, and replacing an operator by its matrix representation. 

Here mx is the column vector of the localized material property function m/,. d is the column vector of the field data, and G is the matrix representation of the corresponding linear operator Gxj or G/x-... 

As has been seen, the operation of forming the derivative of a vector is equivalent to a transformation of this vector into a new vector and that K is a matrix representation of this transformation. As one might expect, the n X n matrix K is not the only matrix that transforms vectors with n elements into their derivatives. Multiplying each side of Eq. (11) of text from the left by an arbitrary n X n matrix P, which has an inverse P (nonsingular), and using the fact that the unit matrix I = PP may be placed at any point in the equation without changing its value, we obtain... 

Once again, the operational application of this development to CASSCF wave functions means that the states (kets) are replaced by the configuration-interaction vectors, and the Hamiltonian operator by its matrix representation in the space of the configuration-state functions (see Appendix). 

The relation with the eigenvectors was made in the matrix representation by the statement that the column vectors of the matrix S that diagonalizes A are the components pr- of the eigenvectors with respect to the reference basis. Extracting the jth column vector of a matrix S is done by letting the matrix act on the coordinates ej of the jth basis element Sej = jth column of S. In terms of operators, this translates into the statement that the jth eigenvector can be expressed as... 

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