Newton-Raphson procedure, density

A more efficient way of solving the DFT equations is via a Newton-Raphson (NR) procedure as outlined here for a fluid between two surfaces. In this case one starts with an initial guess for the density profile. The self-consistent fields are then calculated and the next guess for density profile is obtained through a single-chain simulation. The difference from the Picard iteration method is that an NR procedure is used to estimate the new guess from the density profile from the old one and the one monitored in the single-chain simulation. This requires the computation of a Jacobian matrix in the course of the simulation, as described below. 

Computationally the super-CI method is more complicated to work with than the Newton-Raphson approach. The major reason is that the matrix d is more complicated than the Hessian matrix c. Some of the matrix elements of d will contain up to fourth order density matrix elements for a general MCSCF wave function. In the CASSCF case only third order term remain, since rotations between the active orbitals can be excluded. Besides, if an unfolded procedure is used, where the Cl problem is solved to convergence in each iteration, the highest order terms cancel out. In this case up to third order density matrix elements will be present in the matrix elements of d in the general case. Thus super-CI does not represent any simplification compared to the Newton-Raphson method. 

In addition to further correlating the ground state of a single molecule, the SCVB procedure can also be used to describe its excited states. However, a minimization procedure based on a first-order approach tends not to give good convergence in such cases. Instead, we have adopted a stabilized Newton-Raphson scheme, as in the usual SC approach, but we use an approximate expression for the second derivative that requires only density matrices up to third order [12]. The resulting procedure has been shown to be quite stable. 

The numerical solution of the system (4.86), by a procedure of the Newton-Raphson type with two variables, requires the calculation of the derivatives dtjds and dt /5Wc . The results we obtained for a square lattice are similar to those by Yonezawa and Odagaki180 for a cubic lattice. The most striking feature is the existence, at low concentration, of a gap in the density of states,179 which isolates the zero energy on which a 3 peak builds up. Thus the HCPA produces a forbidden region of energy for the transport the gap and the 3 peak disappear at a critical concentration, analogous to the percolation threshold of the mean-field of resistances. 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

The super-CI method can be regarded as an approximation in the augmented Hessian variant of the Newton-Raphson (NR) procedure (see for example Ref 43). However, in its original formulation, it does not eonstitute any simplification when compared with the NR method. The matrix elements between the Brillouin states (41) are actually more difficult to compute than the corresponding Hessian matrix elements, since they involve third-order density matrix elements" . 

The polynomial format of the Bender multiparameter EoS yields a simple solution via a Newton-Raphson iteration procedure [9]. We found three iterations provided fluid-phase density values with an associated RCSU of < 2 x 10" % between the third and fourth iteration. We took the value of the 6 -iteration, which gave an RCSU of 0.0% between the 4 - and 6 -iteration. The form of the Bender EoS requires implicit differential expressions [10] to evaluate and propagate the uncertainty to generate the CSU in the amoimt of fluid change from the material balance, leading to u n). Implicit differentials were developed for... 

In this section two iterative schemes for calculating the density given values for the pressure and temperature are described the bisection or interval-halving method, and the Newton-Raphson technique. These methods and others are described in more detail by Burden et al. (1978), who also give algorithms for these procedures. 

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