Difference equations, matrix representation

While the computational work for setting up the matrix representation R of p(r) scales formally as N4, this can be cut down to N3 using again the trick introduced in section 7-3 by expanding the density in terms of an atom centered, orthonormalized auxiliary basis set cok (recall equation (7-25)). Let us review this simplification under a slightly different perspective. The starting point is again... 

Equation (15) implies that the 2-RDM and 2-HRDM matrices contain the same information. Indeed, these matrices are two of the three different matrix representations of the 2-RDM on the two-body space, the third one being the second-order G-matrix (2-G) [16]. This matrix, which may be written [24, 25]... 

The only assumption, in addition to Bohr s conjecture, is that the electron appears as a continuous fluid that carries an indivisible charge. As already shown, Bohr s conjecture, in this case, amounts to the representation of angular momentum by an operator L —> ihd/dp, shown to be equivalent to the fundamental quantum operator of wave mechanics, p —> —ihd/dq, or the difference equation (pq — qp) = —ih(I), the assumption by which the quantum condition enters into matrix mechanics. In view of this parallel, Heisenberg s claim [13] (page 262), quoted below, appears rather extravagent ... 

An even simpler approach is possible for so(4) that avoids the solution of difference equations and gives the matrix representation, Eqs. (52) and (53), directly in terms of 6-j symbols, which can be easily evaluated (Biedenharn and Louck, 1981b). If we define... 

It should be emphasized that the matrix representation becomes possible due to the Euler integration of the differential equations, yielding appropriate difference equations. Thus, flow systems incorporating heat and mass transfer processes as well as chennical reactions can easily be treated by Markov chains where the matrix P becomes "automatic" to construct, once gaining enough experience. In addition, flow systems are presented in unified description via state vector and a one-step transition probability matrix. 

The discretization scheme, which leads to an error 0 h ) for second-order differential equations (without first derivative) with the lowest number of points in the difference equation, is the method frequently attributed to Nu-merov [494,499]. It can be efficiently employed for the transformed Poisson Eq. (9.232). In this approach, the second derivative at grid point Sjt is approximated by the second central finite difference at this point, corrected to order h, and requires values at three contiguous points (see appendix G for details). Finally, we obtain tri-diagonal band matrix representations for both the second derivative and the coefficient function of the differential equation. The resulting matrix A and the inhomogeneity vector g are then... 

The group elements (to) e SO(3, IR) are represented in the specific Hilbert space L S) in principle by the same unitary operators as given in Equation [7] though the corresponding Hilbert spaces are different. The matrix elements of the 21 + 1 [-dimensional irreducible representations of SO(3, IR) are given by ... 

H is called the matrix representation of the Hamiltonian operator. A matrix representation of an operator is a matrix of integral values arranged in rows and columns according to the basis functions. Clearly, the values in a matrix representation are dependent on the functions that were selected for the basis set. A different basis set implies a different matrix representation. Wherever it is important to keep track of the basis used in the representation, a superscript is added to the designation of the matrix, for example, W, and it identifies the particular function set. Now, the quantity in Equation C.19 can be written with the coefficient vectors and the Hamiltonian matrix in a very simple form. 

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]) ... 

The diagonal matrix elements ( p Al/ p ) are the effective potential energy surface that governs nuclear motion. From Equations 1.10 and 1.23, it is evident that the vibrational wavefunction x differs from the adiabatic wavefunction x As long as the basis set

electronic space, the CA basis is perfectly adequate (independent of the choice of qg). The two matrix representations 1.8 and (1.20) are merely two different representations of the same operator. 

Using GTO bases, it cannot be expected that the variational representations of the electron waves are snfficiently accnrate far ontside the so-called molecular region , i.e. the rather limited region of space where the potential clearly deviates from the asymptotic Conlomb form. Therefore the phaseshifts of the pwc basis states cannot be obtained from the analysis of their long-range behaviour, as was done in previous works with the STOCOS bases. In the present approach, this analysis may be avoided since the K-matrix techniqne allows to determine, by equation [3] below, the phase-shift difference between the eigenfunctions of Hp and the auxiliary basis functions... 

The A-representability constraints presented in this chapter can also be applied to computational methods based on the variational optimization of the reduced density matrix subject to necessary conditions for A-representability. Because of their hierarchical structure, the (g, R) conditions are also directly applicable to computational approaches based on the contracted Schrodinger equation. For example, consider the (2, 4) contracted Schrodinger equation. Requiring that the reconstmcted 4-matrix in the (2, 4) contracted Schrodinger equation satisfies the (4, 4) conditions is sufficient to ensure that the 2-matrix satisfies the rather stringent (2, 4) conditions. Conversely, if the 2-matrix does not satisfy the (2, 4) conditions, then it is impossible to construct a 4-matrix that is consistent with this 2-matrix and also satisfies the (4, 4) conditions. It seems that the (g, R) conditions provide important constraints for maintaining consistency at different levels of the contracted Schrodinger equation hierarchy. 

Simplification of secular equations. Because the Hamiltonian is totally symmetric - that is, for a molecule of C2v symmetry such as H2O, of symmetry species Ai - the matrix elements Hij = ipi, Ti. ipj) as well as the overlap integrals Sij = (tpi, ipj) will be equal to zero unless the direct product representation r. contains Ai. This is the basis for the assertion that states of different symmetry do not mix. ... 

The question, what conditions are to be fulfilled by a density matrix to be the image of a wave function, that is, to describe a real physical system is opened till today. The contracted Schrodinger-equations derived for different order reduced density matrices by H. Nakatsui [1] give opportunity to determine density matrices by a non-variational way. The equations contain density matrices of different order, and the relationships needed for the exact solutions are not yet known in spite of the intensive research activity [2,3]. Recently perturbation theory corrections were published for correcting the error of the energy obtained by minimizing the density matrix directly applying the known conditions of N-representability [4], and... 

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