Definition matrix representation

If X is an operator on and if x can be written as x take the matrix X, with entries X,y, to be the matrix representation of x. With this definition, expectation values can be written... 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

The angular momentum components span a definite irreducible representation (IR) of the given point group (Table 9), and thus its matrix element vanishes unless the direct product of the IRs for the bra kets contains the IR of the Ifl-operator hence... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

You will find that many of the sources do not use exactly the same matrix representations for some of the product operators and rotation matrices. The exact form of the density matrix depends on the numbering of the spin states and on certain conventions that are not consistent in the literature. In the above examples, the definitions are consistent with the product operator methods and with themselves. 

Table III shows that definitions A, i = I-IIl, and Aj,, y = I-IV, have the same commutation properties as definitions A and A, respectively. This arises because the operators A, i = I-III, and j = I-IV, have the same matrix representation as, respectively, A and one particular operator A and, thus, are formally equivalent to the latter (although computationally the procedures are different). These equivalences are most simply established in the basis sets of the dgenvectors used to projluce the A. Table I shows that o( ...

The second one is an effective closed operator W which is defined to produce as one of its eigenvalues, the desired energy E on diagonalization of its matrix representation in the IMS spanned by <. The coefficients < come out as the components of its eigenvector. Thus W, by its very mode of definition, satisfies... 

The computational procedure involves obtaining the matrix representation of the symmetry operators of the (2n + 2) site chain in the direct product basis. The matrix representation of both J and P for the new sites in the Fock space is known from their definitions. Similarly, the matrix representations of the operators J and P for the left (right) part of the system at the first iteration are also known in the basis of the corresponding Fock space states. These are then transformed to the density matrix eigenvectors bcisis. The matrix representation of the symmetry operators of the full system in the direct product space are obtained as the direct product of the corresponding matrices ... 

This means that the poles of the Green s function, i.e., the molecular orbital energies, are obtained as eigenvalues of a negative definite matrix with elements -Kr Srs + Srs)Ks. The molecular orbital coeflficients can straightforwardly be inferred from the residues at the poles of this spectral representation. 

It is frequently effective to use block representation of parallel algorithms. For instance, a parallel version of the nested dissection algorithm of Section VIII.C for a symmetric positive-definite matrix A may rely on the following recursive factorization of the matrix Ao = PAP, where P is the permutation matrix that defines the elimination order (compare Sections III.G-I) ... 

When operators precedence constraints are not imposed within the establishment definition, establishments can be interpreted as causal links common in engineering applications. In engtneeiing practices, it is customary to represent causality in a matrix representation (Warfield 1973). In this research, we define two typ>es of causal links that result among operators and literals resp>ectively. These causal links are discussed in the following sections. 

The complete matrix representation of the Hamiltonian with isotropic and (anti-) symmetric anisotropic interactions in the uncoupled basis is directly obtained from the operations listed in Eq.3.91 and using the definitions of D/j in dij in Eq. 3.92... 

Fig. 5.13 Definition of the exchange integrals that appear in the matrix representation of the model space formed by the neutral determinants of the four-electron/four-orbital case. For the centro-symmetric case here considered Kn = K- K i K2a > Ku = K23...

It has been noted (footnote on p.l24) that p(xi xl) formally resembles a matrix element, in which Xi andxj play the part of (continuous) row and column indices in this sense it provides a particular representation of p. We now note that, on introducing any orthonormal set i/ r(JCi) > the array of coefficients appearing in (6.4.1) simply provides a true matrix representation of the operator p, in which Xj, x[ are replaced by the discrete indices r, s. This follows easily from the definition of the matrix elements of an operator since, using the orthonormality property,... 

For the real unitary matrix representations we have been considering, the matrix of the inverse transformation is obtained from the original matrix by simply interchanging rows and columns. By definition, therefore... 

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 we are ready to formulate the basic idea of the correction algorithm in order to correct the four-indexed operator f1, it is enough to correct the two-indexed operators fc and f 1 in the supermatrix representation (7.100). The real advantage of this proposal is its compatibility with any definite way of f2 and P7 correction [61, 294], The matrix inversion demanded in (7.99) is divided into two stages. In the fi, v subspace it is possible to find the inverse matrix analytically with the help of the Frobenius formula that is well known in matrix algebra [295]. The... 

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