Scaling, linear

The essential property of any true many-body theory of electronic structure is a linear scaling of the energy components, E, with the number of electrons, N, in the system [39,55,63,64], that is, 

Any terms which scale non-linearly are unphysical and, therefore, should be discarded. Equally, any theory which contains such unphysical terms is not acceptable as a valid many-body method. Either the theory is abandoned or corrections are made in an attempt to restore linear scaling, such as that of Davidson [65], which is used in limited configuration interaction studies. 

Brillouin-Wigner perturbation theory was, however, used as a step in the development of an acceptable many-body perturbation theory most notably by Brandow [67] in his pioneering work on multi-reference formalisms for the many-body problem. In a review entitled Linked-Cluster Expansions for the Nuclear Many-Body Problem and published in 1967, B.H. Brandow writes  

As we have seen, in their 1967 text on many-body methods for quantum systems, March, Young and Sampanthar [55] dismiss Brillouin-Wigner perturbation theory as a valid many-body technique. They write (p. 71)  

In a volume entitled Atomic Many-Body Theory published in 1982, Lindgren and Morrison [63] state that  

It is worth a pause, however, to consider how such models should best be used. Part of the motivation for developing linear scaling models has been to permit QM calculations to be carried out on biomolecules, e.g., proteins or polyiiucleic acids. However, one may legitimately ask whether there is any point in such a calculation, beyond demonstrating that it can be done. Because of the relatively poor fashion with which semiempirical models handle non-bonded interactions, there is every reason to expect that such models would be disastrously bad at predicting biomolecular geometries - or at the very least inferior to the far more efficient force fields developed and optimized for this exact purpose. 

---

Fig. XI-6. Polymer segment volume fraction profiles for N = 10, = 0-5, and Xi = 1, on a semilogarithinic plot against distance from the surface scaled on the polymer radius of gyration showing contributions from loops and tails. The inset shows the overall profile on a linear scale, from Ref. 65.

Strain M C, Scuseria G E and Frisch M J 1996 Linear scaling for the electronic quantum coulomb problem Science 271 51-3... 

The localized nature of the atomic basis set makes it possible to implement a linear-scaling TB algoritluu, i.e. a TB method that scales linearly with the number of electrons simulated [17]. (For more infonnation on linear scaling methods, see section B3.2.3.3.)... 

Bowler D R, Aoki M, Goringe C M, Florsfield A P and Pettifor D G 1997 A comparison of linear scaling tight-binding methods Modeiiing Simuiation Mater. Sc/. 5 199... 

Artacho E, Sanchez-Portal D, Orde]6on P, Garcia A and Soler J M 1999 Linear-scaling ah initio calculations for large and complex systems Phys. Status Soiidi B 215 809... 

Goedecker S 1999 Linear scaling electronic structure methods Rev. Mod. Phys. 71 1085... 

Gain G 2000 Large-scale electronic structure calculations using linear scaling methods Status Solidi B 217 231... 

Watson S C and Carter E A 2000 Linear-scaling parallel algorithms for the first principles treatment of metals Comp. Pbys. Common. 128 67-92... 

There is no simple linear scaling of the variance of the momentum with the number of degrees of freedom. 

The problem with most quantum mechanical methods is that they scale badly. This means that, for instance, a calculation for twice as large a molecule does not require twice as much computer time and resources (this would be linear scaling), but rather 2" times as much, where n varies between about 3 for DFT calculations to 4 for Hartree-Fock and very large numbers for ab-initio techniques with explicit treatment of electron correlation. Thus, the size of the molecules that we can treat with conventional methods is limited. Linear scaling methods have been developed for ab-initio, DFT and semi-empirical methods, but only the latter are currently able to treat complete enzymes. There are two different approaches available. 

One recent development in DFT is the advent of linear scaling algorithms. These algorithms replace the Coulomb terms for distant regions of the molecule with multipole expansions. This results in a method with a time complexity of N for sufficiently large molecules. The most common linear scaling techniques are the fast multipole method (FMM) and the continuous fast multipole method (CFMM). 

DFT N With linear scaling algorithms (very large molecules)... 

One of the major selling points of Q-Chem is its use of a continuous fast multipole method (CFMM) for linear scaling DFT calculations. Our tests comparing Gaussian FMM and Q-Chem CFMM indicated some calculations where Gaussian used less CPU time by as much as 6% and other cases where Q-Chem ran faster by as much as 43%. Q-Chem also required more memory to run. Both direct and semidirect integral evaluation routines are available in Q-Chem. 

Fig. 20. Primary x-ray line and Bremsstrahlung background excited by bombardment with 15 keV electrons, (a) Linear scale plot, (b) Logarithmic scale...

Rotameters The rotameter, an example of which is shown in Fig. 10-21, has become one of the most popular flowmeters in the chemical-process industries. It consists essentially of a plummet, or float, which is free to move up or down in a vertical, slightly tapered tube having its small end down. The fluid enters the lower end of the tube and causes the float to rise until the annular area between the float and the wall of the tube is such that the pressure drop across this constriction is just sufficient to support the float. Typically, the tapered tube is of glass and carries etched upon it a nearly linear scale on which the position of the float may be visually noted as an indication of the flow. 

The above drying curves have been generated via testing on a plate-diyer simulator. The test unit duphcates the physical setup of the production diyer, therefore linear scale-up from the test data can be made to the full-scale diyer. Because of the thin product layer on each plate, diying in the unit closely follows the norm diying cui ve... 

Forward Rates as Analog distances on linear scale... 

Quite often isochronous data is presented on log-log scales. One of the reasons for this is that on linear scales any slight, but possibly important, non-linearity between stress and strain may go unnoticed whereas the use of log-log scales will usually give a straight-line graph, the slope of which is an indication of the linearity of the material. If it is perfectly linear the slope will be 45°. If the material is non-linear the slope will be less than this. 

Many of these fitting schemes were derived before linear scaling techniques (Section 3.8.6) were fully developed, and it is not clear whether they have any advantages. For calculation of energy derivatives, they acmally seem counterproductive, since the fitting procedures seriously complicate the computational expressions. ... 

---