Size consistency/inconsistency

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. 

Very similarly, higher-order processes can be shown to yield a size-consistent redistribution of the intensity of shake-up states among themselves, via multiple 2h-lp/2h-lp interactions. Any restriction on this balance will therefore yield a size-inconsistent description of correlation bands, which will tend to vanish with increasing system size (11). A nice example is provided here, with the necessary introduction of a lower limit on pole strengths in the block-Davidson diagonalization procedure. 

The above description is one way to realize that the SD-CI method is not size-consistent. Another way is to look in detail at what happens when this method is used on the composite and on the separated systems. It is clear that if the energy of A and B should be additive the corresponding wavefunction for (A+B) should be equal to the wavefunction of A times the wavefunction of B. This means that since in the calculations on the separated systems there are local double excitations on both A and B, the product wavefunction will contain certain quadruple excitations. In the SD-CI calculation on the composite system these quadruple excitations are clearly missing and this is the reason for the size-inconsistency. It is also clear that for the SD-CI method E(A)+E(B) must be lower in energy than E(A+B). 

The method of single- and double-excitation configuration mixing is not size-consistent. The method remains size-inconsistent even if a multi-configurational reference function is used. For a configuration mixing calculation, with... 

This size inconsistency occurs because the two open-shell electrons on the atoms must be singlet-coupled to produce the correct dissociation limit, and a supermolecule, two-determinant approach is therefore required. This difficulty also applies to coupled cluster or perturbation-based wavefunctions that use the RHF determinant as a reference these methods cannot be size consistent for a given molecular system unless the reference wavefunction is size consistent. 

Because is an extensive properly (i.e., size consistent), we should expect possible size inconsistency from the last term in Ej if the (second-order) factor in front of Ej is also extensive would then contain terms that are proportional to the square of the system s size (for identical noninler-acting subsystems). Let us now look at this situation more closely. By introducing the spectral representation of the projector Q given in Eq. (3.21), Ej- can be written as... 

If one wishes to use RSPT to perform ab initio quantum-chemical calculations that yield size-consistent energies, then care must be taken in computing the factors that contribute to any given E For example, if were calculated as in Eq. (3.28), limitations of numerical precision might not give rise to the exact cancellation of size-inconsistent terms, which we know should occur. This would certainly be the case for an extended system (for which the size-inconsistent terms would dominate). In addition, it is unpleasant to have a formalism in which such improper terms arise in the first place. It is therefore natural to attempt to develop approaches to implementing RSPT in which the size-inconsistent factors are never even computed. Such an approach has been developed and is commonly referred to as many-body perturbation theory (MBPT). The method of implementing MBPT is discussed once we have completed the present treatment of RSPT. 

To obtain more insight into the structure of the size-consistent and -inconsistent terms, we derive an explicit expression for the quantity A appearing in Eq. (3.29) within MPPT. Realizing that only the doubly excited determinants k > contribute in Eq. (3.29) because of the BT, we obtain... 

Having now completed our treatment of RSPT, let us return to the problem of finding a mechanism for explicitly computing, in any given order, only those terms in or /" > that arc size consistent. Recall that RSPT, as normally expressed, contains size-inconsistent terms that cancel when grouped together properly, but that nevertheless appear in the formal RSPT expression. Recall also that the size-inconsistent terms could be characterized by a factorization into products of two or more terms that did not share common summation indices. In MBPT, the formal cancellation of size-inconsistent terms in RSPT is carried out explicitly (Brueckner, 1955a,b Bartlett and Silver, 1975 Kelly, 1969 Lowdin, 1968 Brandow, 1977). Formally, we may thus write as... 

The question for a more systematic inclusion of electronic correlation brings us back to the realm of molecular quantum chemistry [51,182]. Recall that (see Section 2.11.3) the exact solution (configuration interaction. Cl) is found on the basis of the self-consistent Hartree-Fock wave function, namely by the excitation of the electrons into the virtual, unoccupied molecular orbitals. Unfortunately, the ultimate goal oi full Cl is obtainable for very small systems only, and restricted Cl is size-inconsistent the amount of electron correlation depends on the size of the system (Section 2.11.3). Thus, size-consistent but perturbative approaches (Moller-Plesset theory) are often used, and the simplest practical procedure (of second order, thus dubbed MP2 [129]) already scales with the fifth order of the system s size N, in contrast to Hartree-Fock theory ( N ). The accuracy of these methods may be systematically improved by going up to higher orders but this makes the calculations even more expensive and slow (MP3 N, MP4 N ). Fortunately, restricted Cl can be mathematically rephrased in the form of the so-called coupled clus-... 

However, application of nucleation theory to extended and chain-folded crystals did not reveal consistent results in that the ere values as determined by equation (3a) are very different and the ratio of surface free energy of folded to extended chain crystals was about 12. This ratio size was inconsistent with other values determined for the melting points of the various crystals. Nucleation theory when applied to the growth rates of the extended chain crystals implied that multiple surface nucleation, i.e., Kyi of equation (3a) was involved, but when applied to the growth rates of folded chain crystals at high supercooling, uni-molecular nucleation, i.e., Ky was involved,... 

For a dilute gas of molecules treated as a supermolecule, the overlap between any wavefunction involving only a finite level of excitation and the exact wavefunction must vanish exponentially with the number of molecules. This includes the SCF wavefunction, a SD-CI wavefunction, or an MPl wave-function. Thus SD-Cl energies and wavefunctions are size inconsistent in the sense that the error grows with N in a nonlinear manner. MPl wavefunctions are also size inconsistent, but the second-order perturbation energy can be obtained from them in a size-consistent manner by special tricks that avoid computing the average energy of the wavefunction. 

There are two major routes to improve the Cl methods. The first is by including size-consistent corrections for dynamical EC leading to the CIS(D) and CIS-MP2 methods " (which are closely related to the coupled-cluster methods described in a later section). The second is by improving the reference wave functions, which leads to multireference Cl (MRCI) methods, that, however, remain size-inconsistent. 

There are some approximate methods in quantum chemistry which suffer from the so-called size inconsistency error The eigenvalue of the dissociated Hamiltonian of Eq. (16.9) is not equal to the sum of the solutions of the two independent Hamiltonians. A typical wave function of this type is that of a truncated Cl calculation. Size consistent wave functions are preferred for the study of intermolecular interactions. 

The level of confidence in source assignment can be Increased if other observables such as time variability, size dependence, etc, are consistent with prior knowledge of a source s emission characteristics. Confidence can also be enhanced by showing that the characteristics of other possible sources are inconsistent with the 2imbient observables. It follows that the highest level of confidence... 

I well remember my shock when I recognized that COSMO-RS was not thermodynamically consistent, just after having quit my safe position at Bayer AG and had started my own company. But within an hour I had located the origin of the inconsistency in the solvent-size correction, and within a day I found an expression for the size correction that ensures Gibbs-Duhem consistency, while leaving the limits of pure solvents unchanged compared with our original COSMO-RS expression ... 

The prototype facility on which Modesto was based uses manual tire feed.6 Modesto personnel felt it necessary to automate the tire feed system. The initial system, however, did not deliver a consistent feed of tires to the furnaces. The one weigh station, located near the tire pile, could not make allowances for the variability in size and type of tire entering the conveyor apparatus.6 Inconsistent power generation resulted. 

---