Systematic Improvability

In molecular orbital theory, there is a clear and well defined path to the exact solution of the Schrodinger equation. All we need do is express our wave function as a Unear combination of all possible configurations (full CI) and choose a basis set that is infinite in size, and we have arrived. While such a goal is essentially never practicable, at least the path to it can be followed unambiguously until computational resources fail. 

All that being said, experience dictates that, across a surprisingly wide variety of systems, DFT tends to be remarkably robust. Thus, unless a problem falls into one of a few classes of well characterized problems for DFT, there is good reason to be optimistic about any particular calculation. 

Finally, it seems clear that routes to further improve DFT must be associated with better defining hole functions in arbitrary systems. In particular, the current generation of functionals has reached a point where finding efficient algorithms for correction of the classical self-interaction error are likely to have the largest qualitative (and quantitative) impact. 

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By integrating over the hard cores in the SL expansion and collecting tenns it is easily shown this expansion may be viewed as a correction to the MS approximation which still lacks the complete second virial coefficient. Since the MS approximation has a simple analytic fomi within an accuracy comparable to the Pade (SL6(P)) approximation it may be more convenient to consider the union of the MS approximation with Mayer theory. Systematic improvements to the MS approxunation for the free energy were used to detemiine... 

A first step towards a systematic improvement over DFT in a local region is the method of Aberenkov et al [189]. who calculated a correlated wavefiinction embedded in a DFT host. However, this is achieved using an analytic embedding potential fiinction fitted to DFT results on an indented crystal. One must be cautious using a bare indented crystal to represent the surroundings, since the density at the surface of the indented crystal will have inappropriate Friedel oscillations inside and decay behaviour at the indented surface not present in the real crystal. 

Most of the techniques described in this Chapter are of the ab initio type. This means that they attempt to compute electronic state energies and other physical properties, as functions of the positions of the nuclei, from first principles without the use or knowledge of experimental input. Although perturbation theory or the variational method may be used to generate the working equations of a particular method, and although finite atomic orbital basis sets are nearly always utilized, these approximations do not involve fitting to known experimental data. They represent approximations that can be systematically improved as the level of treatment is enhanced. 

There is no systematic way in which the exchange correlation functional Vxc[F] can be systematically improved in standard HF-LCAO theory, we can improve on the model by increasing the accuracy of the basis set, doing configuration interaction or MPn calculations. What we have to do in density functional theory is to start from a model for which there is an exact solution, and this model is the uniform electron gas. Parr and Yang (1989) write... 

The geometry from step 2 is now used in a number of single-point calculations at higher levels of theory, starting with MP4/6-31IG. This energy is now systematically improved, and the improvements are assumed to be additive. 

Numerical Representation The theory should be systematically improvable with respect to basis sets or integration schemes. 

Calculating the exact response of a semiconductor heterostructure to an ultrafast laser pulse poses a daunting challenge. Fortunately, several approximate methods have been developed that encompass most of the dominant physical effects. In this work a model Hamiltonian approach is adopted to make contact with previous advances in quantum control theory. This method can be systematically improved to obtain agreement with existing experimental results. One of the main goals of this research is to evaluate the validity of the model, and to discover the conditions under which it can be reliably applied. 

Quantum-mechanical approximation methods can be classified into three generic types (1) variational, (2) perturbative, and (3) density functional. The first two can be systematically improved toward exactness, but a systematic correction procedure is generally lacking in the third case. 

From the above, one may be left with the impression that the MD technique has major problems. It is important to realise that there are relatively straightforward ways to systematically improve the method. In the future, the force fields will become more accurate, the computer power will increase and allow larger box sizes and longer (real-time) simulation times. Even today, MD simulations are the closest to this ideal situation as compared with other methods. 

At nonzero temperatures the mass gap decreases as a function of the chemical potential already in the phase with broken chiral symmetry. Hence the model here gives unphysical low-density excitations of quasi-free quarks. A systematic improvement of this situation should be obtained by including the phase transition construction to hadronic matter. However, in the present work we circumvent the confinement problem by considering the quark matter phase only for densities above the nuclear saturation density no, i.e. ub > 0.5 no. 

So far, CG approaches offer the most viable route to the molecular modeling of self-organization phenomena in hydrated ionomer membranes. Admittedly, the coarse-grained treatment implies simplifications in structural representation and in interactions, which can be systematically improved with advanced force-matching procedures however, it allows simulating systems with sufficient size and sufficient statishcal sampling. Structural correlations, thermodynamic properties, and transport parameters can be studied. 

The CCS, CC2, CCSD, CC3 hierarchy has been designed specially for the calculation of frequency-dependent properties. In this hierarchy, a systematic improvement in the description of the dynamic electron correlation is obtained at each level. For example, comparing CCS, CC2, CCSD, CC3 with FCI singlet and triplet excitation energies showed that the errors decreased by about a factor 3 at each level in the coupled cluster hierarchy [18]. The CC3 error was as small as 0.016 eV and the accuracy of the CC3 excitation energies was comparable to the one of the CCSDT model [18]. 

The results presented in this chapter are complementary to the ones of Ref. [4], where the Cauchy moments were calculated for the Ne atom using the CCS, CC2, CCSD hierarchy. A systematic improvement in the quality of the Cauchy... 

Adoption of a many-body theory that is systematically improvable (at least in principle) and testable for its numerical convergence ... 

Realization with basis sets for Bloch orbital expansions that are physically, analytically and/or practically motivated, and also systematically improvable and testable ... 

Possibility of a practical, yet again systematically improvable introduction of temperature via phonons, so central to condensed matter properties. 

Criterion 1 seems to exclude the DET-based methods, because systematic improvements are elusive, and we cannot state, in any precise manner, which many-body effects have been included. Criterion 2 suggests only limited use of... 

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