Natural iteration method

When we want to calculate for different composition, we do not fix pj in minimizing the free energy. We rather fix the chemical potential p. The reason is that when we use the chemical potential, we can prove the convergence of the natural iteration method. If we fix the composition instead, natural iteration method (NIM, mentioned below) does not necessarily converge. 

The set of equations in the previous section 4 is solved numerically using the NIM (natural iteration method). When we use the constraint Lagrange multipliers, the NIM consists of the "Major Iteration" and the "Minor Iteration". The major iteration solves the set of basic of equations in (11), the reduction relations in (2) and the normalization relation. It was proved that this iteration always converges from whatever initial guess values it starts with. 

Natural iteration method Nearest-neighbor approxmiation Near-neighbor shell Ni-Al... 

The examples in the previous section demonstrate that nonunique solutions to the equilibrium problem can occur when the modeler constrains the calculation by assuming equilibrium between the fluid and a mineral or gas phase. In each example, the nonuniqueness arises from the nature of the multicomponent equilibrium problem and the variety of species distributions that can exist in an aqueous fluid. When more than one root exists, the iteration method and its starting point control which root the software locates. 

Of course, the roots of Eq. (46a) may easily be found. The unknown quantities fi and 2m may be found by substituting hm from Eq. (46a) into the imaging equations (18) and normalization equation (3). Because of the nonlinear nature of these equations, a Newton-Raphson (or other iterative) method of solution would be necessary. 

The effectiveness of NSO s in reducing the expansion size in systems with more than two electrons is not as great and, in fact, for larger systems, their use is not practical. The loss in practicality is immediately obvious when one realizes that in order to obtain them, one must diagonalize the first-order density matrix of the exact wavefunction, i.e. a full configuration interaction must first be performed. Two methods have been introduced in order to regain the initial usefulness of natural orbitals the pseudonatural orbital method and the approximate or iterative natural orbital method. 

CCSD(T) method. The question then naturally arises as to how these methods can be extended to excited states. For the iterative methods, the extension is straightforward by analyzing the correspondence between terms in the CC equations and in H, one can define an H matrix for these methods, even though it is not exactly of the form of a similarity-transformed Hamiltonian. If one follows the linear-response approach, one arrives at the same matrix in the linear response theory, one starts from the CC equations, rather than the CC wave function, and no CC wave function is assumed. This matrix also arises in the equations for derivatives of CC amplitudes. In linear response theory, this matrix is sometimes called the Jacobian [19]. The upshot is that excited states for methods such as CCSDT-1, CCSDT-2, CCSDT-3, and CC3 can be obtained by solving eigenvalue equations in a manner similar to those for methods such as CCSD and CCSDT. 

The polarizable point dipole model has also been used in Monte Carlo simulations with single particle moves.When using the iterative method, a whole new set of dipoles must be computed after each molecule is moved. These updates can be made more efficient by storing the distances between all the particles, since most of them are unchanged, but this requires a lot of memory. The many-body nature of polarization makes it more amenable to molecular dynamics techniques, in which all particles move at once, compared to Monte Carlo methods where typically only one particle moves at a time. For nonpolarizable, pairwise-additive models, MC methods can be efficient because only the interactions involving the moved particle need to be recalculated [while the other (N - 1) x (]V - 1) interactions are unchanged]. For polarizable models, all N x N interactions are, in principle, altered when one particle moves. Consequently, exact polarizable MC calculations can be... 

Other iterative methods have been proposed that are less well justified. For example, one approach [30] employs equations (48) and (49) directly, with the nth approximation for g(T) substituted into the integrals to obtain the (n + l)st approximations for A and g(z). A variational approach also has been developed [25], based on the introduction of / = as a new variable, so that equation (46) can be written in a standard form, d ildrj = — Aco/r], None of these approaches circumvent the cold-boundary difficulty unless is modified suitably, for example, by the introduction of 0. The approximations to be discussed next bypass the cold-boundary difficulty in a natural way. 

A method that is Bayesian in nature is that proposed by Racine-Poon (38). The method uses the estimates of the individual parameters j and asymptotic variance matrix Vj obtained from the individual fits, with very weak assumptions about the prior distribution of the population parameters to calculate a posterior density function from which and Q can be obtained. In an iterative method suggested by Dempster et al. (39) the EM algorithm is used to calculate the posterior density function. Simulation studies in which several varying and realistic conditions were... 

Hartree -Fock or Self-Consistent Field (SCF) Method Spin Optimized Self-Consistent Field Method Configuration Interaction Iterative Natural Orbital Method Multi-Configuration SCF Many Body Perturbation Theory Valence-Bond Method Pair-Function or Geminal Method... 

In our implementation we have taken a different approach, for which it will be seen that the above sorts of terms can be treated exactly without posing computational problems. Our approach fits naturally within the iterative method of solution of the CPKS equations, in which it is not the individual elements of A which are calculated, but rather the contraction of these with the current approximate solution vectors. The XC contribution to these quantities may be expressed as... 

R. Ahhichs and E. Driessler, Determination of pair natural orbitals. A new method to solve the multiconfiguration Hartree-Eock problem for two-electron wave functions, Theor. Chim. Acta, 36 (1975) 275. W. Meyer, Theory of self-consistent electron pairs. An iterative method for correlated many-electron... 

Other examples of optimizing functions that depend quadraticaUy of the parameters include the energy of Hartree-Fock (HF) and configuration interaction (Cl) wave functions. Minimization of the energy with respect to the MO or Cl coefficients leads to a set of linear equations. In the HF case, the Xy coefficients depend on the parameters Ui, and must therefore be solved iteratively. In the Cl case, the number of parameters is typically 10 -10 and a direct solution of the linear equations is therefore prohibitive, and special iterative methods are used instead. The use of iterative techniques for solving the Cl equations is not due to the mathematical nature of the problem, but due to computational efficiency considerations. 

The proof of this last statement uses only the non-negative irreducible and convergent nature of the matrix M. In order to sharpen this last result, as well as introduce the basis for the successive overrelaxation iterative method of Young and Frankel [52 12], we make the following definition. 

It is natural to ask how this Chebyshev semi-iterative method compares... 

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