Extensivity, size

By carefully adjustmg the variational wavefiinction used, it is possible to circumvent size-extensivity problems for selected species. For example, the Cl calculation on Bc2 using all 2 CSFs fomied by placing the four valence electrons into the 2a, 2a, 30g, 3a, In, and iTt orbitals can yield an energy equal to twice that of the Be atom described by CSFs in which the two valence electrons of the Be atom are placed into the... 

D) MOST PERTURBATION AND CC METHODS ARE SIZE-EXTENSIVE, BUT DO NOT PROVIDE UPPER BOUNDS AND THEY ASSUME THAT ONE CSF DOMINATES... 

In contrast to variational metliods, perturbation tlieory and CC methods achieve their energies by projecting the Scln-ddinger equation against a reference fiinction (transition formula (expectation value ( j/ It can be shown that this difference allows non-variational teclmiques to yield size-extensive energies. 

Method Variational/size extensive Computational scaling... 

A. Variational Methods Such as MCSCF, SCF, and Cl Produce Energies that are Upper Bounds, but These Energies are not Size-Extensive... 

In general, one finds that if the monomer uses CSFs that are K-fold excited relative to its dominant CSF to achieve an accurate description of its electron correlation, a size-extensive variational calculation on the dimer will require the inclusion of CSFs that are 2K-fold excited relative to the dimer s dominant CSF. To perform a size-extensive... 

B. Non-Variational Methods Sueh as MPPT/MBPT and CC do not Produee Upper Bounds, but Yield Size-Extensive Energies... 

P I H I P >. It ean be shown (H. P. Kelly, Phys. Rev. 131, 684 (1963)) that this differenee allows non-variational teehniques to yield size-extensive energies. This ean be seen in the MPPT/MBPT ease by eonsidering the energy of two non-interaeting Be atoms. The referenee CSF is = Isa 2sa Isb 2sb the Slater-Condon rules limit the CSFs in P whieh ean eontribute to... 

This expression, onee the SC rules are used to reduee it to one- and two- eleetron integrals, is of the additive form required of any size-extensive method ... 

The additivity of E and the separability of the equations determining the Cj eoeffieients make the MPPT/MBPT energy size-extensive. This property ean also be demonstrated for the Coupled-Cluster energy (see the referenees given above in Chapter 19.1.4). However, size-extensive methods have at least one serious weakness their energies do not provide upper bounds to the true energies of the system (beeause their energy funetional is not of the expeetation-value form for whieh the upper bound property has been proven). 

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. 

The accuracy of these two methods is very similar. The advantage of doing coupled cluster calculations is that it is a size extensive method (see chapter 26). Often, coupled-cluster results are a bit more accurate than the equivalent... 

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. 

It is possible to make a method approximately size-extensive by adding a correction to the final energy. This has been most widely used for correcting CISD energies. This is a valuable technique because a simple energy correction formula is easier to work with than full Cl calculations, which require an immense amount of computational resources. The most widely used correction is the Davidson correction ... 

Another method for making a method size-extensive is called the self-consistent dressing of the determinant energies. This is a technique for modifying the Cl superdeterminant in order to make a size-extensive limited Cl. The accuracy of this technique is generally comparable to the Davidson correction. It performs better than the Davidson correction for calculations in which the HF wave function has a low weight in the Cl expansion. 

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... 

The principal deficiency of CISD is the lack of the TI term, which is the main reason for CISD not being size extensive. Furthermore, this term becomes more and more important as the number of electrons increases, and CISD therefore recovers a smaller and smaller percentage of the correlation energy as the system increases. There are various approximate corrections for this lack of size extensivity which can be added to standard CISD. The most widely known of these is the Davidson correction, sometimes denoted CISD - - Q(Davidson), where the quadruples contribution is approximated as... 

Another commonly used method is Quadratic CISD (QCISD). It was originally derived from CISD by including enough higher-order terms to make it size extensive. 

It has later been shown that the resulting equations are identical to CCSD where some of the terms have been omitted. The omitted terms are computationally inexpensive, and there appears to be no reason for using the less complete QCISD over CCSD (or QCISD(T) in place of CCSD(T)), although in practice they normally give very similar results. There are a few other methods which may be considered either as CISD with addition of extra terms to make them approximately size extensive, or as approximate versions of CCSD. Some of the methods falling into this category are Averaged... 

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

The use of Cl methods has been declining in recent years, to the profit of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl or MCSCF based methods have been the prefen ed methods here. More recently propagator or equation of motion (Section 10.9) methods have been developed for coupled cluster wave functions, which allows calculation of exited state properties. 

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