Self-consistent iteration

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

It does not appear that the ICSEs can be solved by self-consistent iteration, however. In Eq. (68), CSE(2) is expressed in a form that affords the 2-RDM as an explicit functional of the 2-, 3-, and 4-RDMs, but no analogous formulation of ICSE(l) or ICSE(2) is possible, since the 1- and 2-RDMCs appearing in these equations are always acted upon by or g (cf. Eqs. (66) and (67)). Thus the ICSEs are implicit equations for the cumulants. 

The background scattering (e.g. from the sample container and cryostat) can be measured separately but the calculation of multi-phonon contributions from equation (11) might not be straightforward. Alternatively a number of self-consistent iteration techniques have also been used, for details see ref. [21]. In most of our work we have compared the observed (S(Q,o)) =>) g(co) obtained from equation (20) with that calculated from... 

The basis of computational quantum mechanics is the equation posed by Erwin Schrbdinger in 1925 that bears his name. Solving this equation for multielectron systems remains as the central problem of computational quantum mechanics. The difficulty is that because of the interactions, the wave function of each electron in a molecule is affected by, and coupled to, the wave functions of all other electrons, requiring a computationally intense self-consistent iterative calculation. As computational equipment and methods have improved, quantum chemical calculations have become more accurate, and the molecules to which they have been applied more complex, now even including proteins and other biomolecules. 

Now let us examine, in turn, each of the local space matrices arising from the rhs of Eq. (23). The first is [U K,j U]L with K,j = K,j — K,. This matrix can be evaluated by exactly the same methods that were discussed earlier in connection with the HF/LSA method. In fact, using Eqs. (19) for the (non-orthogonal) local space LMOs one is led to exactly the same integral transformation that was discussed in Sect. 2. If this is done by the conventional method, then the entire set (all i > j) of [UK U]L matrices may be determined prior to the self-consistent iterations. Alternatively, the numerical scheme of Sect. 2 can be employed. Then [U K,7 U] ab can be written as a sum of terms each having the form ... 

Gummel, H.K. (1964) A Self-consistent Iterative Scheme for One-dimensional Steady State Transistor Calculations. IEEE Trans. Electron Devices, ED-11, 455-465. Lee, C.M., Lomax, R.J. and Haddad, G.I. (1974) Semiconductor Device Simulation. IEEE Trans. Microw. Theory Techn., MTT-22, 160-177. 

Comparison with Eqs. (1) and (4) shows immediately that in principle this is a four-center integral (basis fimctions at four nuclear sites contribute). The computational cost can be reduced greatly by mapping the problem to a sum of three-center integrals. The scheme is to expand, at each self-consistent iteration, the electron number density Wg in an auxiliary Hennite gaussian basis set Q of the same type (but not necessarily the identical functions) as used to expand the KS functions ... 

However, these benefits come at a price. Both Vgg and Vxc and their contributions to the transformations obviously change at each self-consistent iteration so the net effect is that some very complicated operator products, involving both momentum and direct space representations, must be done at every iteration. What Rosch and co-workers noticed [44] was that the singular part of the Hamiltonian Vxe of course does not change from iteration to iteration, so they attempted an incomplete DKH transformation which retained only V g and incorporated, therefore, the bare electron-electron interactions in the transformed Hamiltonian. 

Fig. 24.3 In-plane arrangement of atomic positions and the initial spin configuration in self-consistent iterations for the [Cu207] molecule...

The method achieves self-consistency within a similar number of self-consistent field iterations as eigensolver-based approaches. However, the replacement of the standard diagonalization at each self-consistent iteration by a polynomial filtering step results in a significant speedup over methods based on standard diagonalization, often by more than an order of magnitude. Algorithmic details of a parallel... 

Given the implicit nature of the expression for induced dipoles several methods have been proposed for their efficient calculation. The most popular of them is the combination of a predictive scheme with the traditional self-consistent iterative procedure for the calculation of non-additive effects [76] and variations around [91]... 

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