Optimized cluster theory

Our starting point is Eqs. (68) and (69) for si and g in terms of graphs containing ho and p bonds. We define the renormalized potential as 

See Fig. 8 for a pictorial representation of this equation. In the graphs in this series, each field point has only two bonds attached, at least one of which must be a bond. Moreover, each root has only one bond attached. This is the same function that appears as part of the contribution to h in the y-ordered expansion in Eq. (83). An explicit expression for for monatomic fluids is given in Eqs. (84) and (85). (The generalization to multicomponent fluids is given by Andersen and Chandler. ) 

Performing the topological reduction on the series for si is somewhat less straightforward, since graphs that are rings of p bonds and ho bonds do not satisfy the conditions of the topological reduction theorem and must hence be 

Finally it is convenient to reexpress Eq. (87) for g by noting the fact that for every graph in the series that has an ho bond between the roots there is another graph in the series that is identical to it except that it does not have an ho between the roots (and vice versa). Adding these two graphs together, we obtain 

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Lennard-Jones potential. The new radial distribution function is estimated using the exponential approximation to the optimized cluster theory (31)... 

Fig. 8. Graphical series for the renormalized potential in optimized cluster theory. The dashed lines are ho bonds and the solid lines are tp bonds.

Chandler D and Andersen H C 1972 Optimized cluster expansions for classical fluids II. Theory of molecular liquids J. Chem. Phys. 57 1930... 

D. Chandler and H. C. Andersen, /. Chem. Phys., 57, 1930 (1972). Optimized Cluster Expansions for Classical Fluids. II. Theory of Molecular Liquids. 

Certain Schrodinger equation based methods, such as coupled cluster theory, are not based on a variational principle. They fall outside schemes that use the energy expectation value as a optimization function for simulated annealing, although these methods could be implemented within a simulated annealing molecular dynamics scheme with alternative optimization function. 

Analysis and control of ultrafast processes in atomic clusters in the size regime in which each atom counts are of particular importance from a conceptual point of view and for opening new perspectives for many applications in the future. Simultaneously, this research area calls for the challenging development of theoretical and computational methods from different directions, including quantum chemistry, molecular dynamics, and optimal control theory, removing borders between them. Moreover, it provides stimulation for new experiments. 

There are numerous alternative methods that introduce electron correlation in the molecular calculations at a more precise level that can be profitably used. We mention here the MC-SCF approach (the acronym means that this is a variant of HF (or SCF) procedure starting fi om the optimization no more of a single antisymmetric orbital product, but of many different products, or configurations), the Coupled-Cluster theory, etc., all methods based on a MO description of single-electron functions. 

Andersen, H. C. and D. Chandler. 1972. Optimized cluster expansions for classical fluids. 1. General theory and variational formulation of the mean spherical model and hard sphere Percus-Yevick equations. Journal of Chemical Physics. 57, 1918. 

At first, aU these methods were developed for closed-shell systems only. Later research in this area was directed towards local methods for open-shell systems and excited states, local triples corrections beyond (T) (triples included in coupled cluster iterations), [138], local energy gradients for geometry optimizations of large molecules [139], combination of the local correlation method with explicitly correlated wavefunctions. It is evident from the discussion that these local 0 N) methods open the applications of coupled-cluster theory to entirely new classes of molecules, which were far ont-of-scope for such an accurate treatment before. Possible applications lie, for example, in the determination of the thermochemistry of reactions involving... 

Benchmark Studies on Small Molecules Configuration Interaction Coupled-cluster Theory Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Geometry Optimization 1 Geometry Optimization 2 ... 

Basis Sets Correlation Consistent Sets Complete Active Space Self-consistent Field (CASSCF) Second-order Perturbation Theory (CASPT2) Configuration Interaction Coupled-cluster Theory Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field G2 Theory Geometry Optimization 1 Gradient Theory Inter-molecular Interactions by Perturbation Theory Molecular Magnetic Properties NMR Chemical Shift Computation Ab Initio NMR Chemical Shift Computation Structural Applications Self-consistent Reaction Field Methods Spin Contamination. 

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