Size consistent

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

There is a variation on the coupled cluster method known as the symmetry adapted cluster (SAC) method. This is also a size consistent method. For excited states, a Cl out of this space, called a SAC-CI, is done. This improves the accuracy of electronic excited-state energies. 

CC, which is superior to Cl because it is rigorously size consistent... 

The level of theory necessary for computing PES s depends on how those results are to be used. Molecular mechanics calculations are often used for examining possible conformers of a molecule. Semiempiricial calculations can give a qualitative picture of a reaction surface. Ah initio methods must often be used for quantitatively correct reaction surfaces. Note that size consistent methods must be used for the most accurate results. The specific recommendations given in Chapter 18 are equally applicable to PES calculations. 

It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

Size-extensivity is of importance when one wishes to compare several similar systems with different numbers of atoms (i.e., methanol, ethanol, etc.). In all cases, the amount of correlation energy will increase as the number of atoms increases. However, methods that are not size-extensive will give less correlation energy for the larger system when considered in proportion to the number of electrons. Size-extensive methods should be used in order to compare the results of calculations on different-size systems. Methods can be approximately size-extensive. The size-extensivity and size-consistency of various methods are summarized in Table 26.1. 

Size-consistency and size-extensivity are issues that should be considered at the outset of any study involving multiple molecules or dissociated fragments. As always, the choice of a computational method is dependent on the accuracy desired and computational resource requirements. Correction formulas are so simple to use that several of them can readily be tried to see which does best for... 

Abrasion resistance, resistance to C02attack, size consist... 

Size consist, carbon content, resistance to abrasion, impact... 

Fig. 7. Primary functions of coke in a blast furnace showing the various furnace zones where arrows indicate direction of gas flow. The term size consist...

Size consistency the results given for a system of molecules infinitely separated from one another ought to equal the sum of the results obtained for each individual molecule calculated separately. Another way of describing this requirement is that the error in the predictions of any method should scale roughly in proportion to the size of the molecule. When size consistency does not hold, comparing the properties of molecules of different sizes will not result in quantitatively meaningful differences. 

Additional empirical corrections 1 and 2-electron higher-order corrections (size-consistent), spin contamination 2-electron higher-order correction (size-consistent), spin contamination, core correlation for sodium... 

The full Cl method has many of the desirable features of a theoretical model. It is well-defined, size-consistent, and variational. However, it is also very expensive and impractical for all but the very smallest systems. 

A disadvantage of all these limited Cl variants is that they are not size-consistent.The Quadratic Configuration Interaction (QCI) method was developed to correct this deficiency. The QCISD method adds terms to CISD to restore size consistency. QCISD also accounts for some correlation effects to infinite order. QCISD(T) adds triple substitutions to QCISD, providing even greater accuracy. Similarly, QCISD(TQ) adds both triples and quadruples from the full Cl expansion to QCISD. 

The parameter 8,y serves to retain size consistency in the CBS extrapolation for finite values of N. Full Cl pair energies, (N), may be obtained from the... 

Some methods of describing electron correlation are compared from the point of view of requirements for theoretical chemical models. The perturbation approach originally introduced by Mpller and Plesset, terminated at finite order, is found to satisfy most of these requirements. It is size consistent, that is applicable to an ensemble of isolated systems in an additive manner. On the other hand, it does not provide an upper bound for the electronic energy. ... 

The MPn method treats the correlation part of the Hamiltonian as a perturbation on the Hartree-Fock part, and truncates the perturbation expansion at some order, typically n = 4. MP4 theory incorporates the effect of single, double, triple and quadruple substitutions. The method is size-consistent but not variational. It is commonly believed that the series MPl, MP2, MP3,. .. converges very slowly. 

Steife, /. stiffening (starch, glue, size) consistency prop, strut, stay stiffness, rigidity, steifen, v.t. stiffen starch stay, prop. Steifheit,/. stiffness, rigidity. 

In comparison with the more standard Configuration Interaction (Cl) method, the one-particle Green s function approach offers the essential advantages, in the outlook of numerical applications on extended systems, of a stronger and systematic compactness (30) of the configuration spaces in high order approximations and of energy separability (5,31) in the dissociation limit (size-consistency). The latter is a necessary prerequisite ( ) for a correct (i.e. size-... 

Our results fit also with a previous investigation (9) on polyenes based on a version of the 2h-lp Cl scheme restricted to the virtual one-electron states generated by a minimal basis. In our case, however, the fragmentation of lines into satellites is much more pronounced. The reason lies in the size-consistency of the ADC[3] approach (as contrasted with the size-inconsistency of any truncated form of Cl (27d), in the full handling of the virtual space, and (10) in the inclusion of correlation corrections to the reference ground state, leading to (37) a net reduction of the quasi-particle band gap of conjugated polymers. 

For the correlation bands obtained by a convolution of many shake-up lines, the size-consistency and size-intensivity requirements imply a convergence towards some asymptotic profile, when going to the polymer limit. This must ideally be achieved through a balance of the scaling properties of the individual shake-up lines, and the dispersion of the intensity of Ih lines over a rapidly increasing number of excited states with increasing system size. 

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