## Is CASINO QMC realy linear scaling?

General discussion of the Cambridge quantum Monte Carlo code CASINO; how to install and setup; how to use it; what it does; applications.
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### Is CASINO QMC realy linear scaling?

Hello, all.

I found very exciting flag in CASINO QMC namely:
MOLGSCREENING (Logical ) Toggle on and off the use of screening in Gaussian basis set calculations of molecules, i.e., the division of space into boxes and the preparation of lists of which Gaussian basis functions have a significant weight in each box. The use of screening should speed up the calculation of large molecules. The screening information can take up a reasonable amount of memory; hence the existence of this keyword.
If this flag make CASINO QMC linear scaling for sufficiently large molecules? What is the level of error it introduces?

If it's really linear and error is really small it is very demanding option.

For example such option for "gold standard" method https://www.ncbi.nlm.nih.gov/pubmed/28456208 named DLPNO-CCSD(T) is under development, although some parts are ready.
For DLPNO-CCSD(T) linear regime starts from about 50 carbon atoms. Is it the same for MOLGSCREENING=True?

Last edited by Vladimir_Konjkov on Fri Sep 08, 2017 4:55 am, edited 3 times in total.
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Neil Drummond
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### Re: Is CASINO QMC realy linear scaling?

Sadly it doesn't make CASINO scale linearly with problem size.

The cost of evaluating the wave function (or calculating a local energy, etc.) is usually dominated by the cost of evaluating the Slater wave function. For a localised basis set (e.g., blips or Gaussians but not plane waves), each time an electron is moved, a small number of basis functions must be evaluated for every orbital, so the cost is N^2.

If localised orbitals (e.g., Wannier functions, or nonorthogonal localised orbitals) are constructed and truncated to zero at finite range, you can save a factor of N in the scaling, since now you only need to evaluate a small number of basis functions for a small number of orbitals when an electron is moved.

Note that the number of steps required to achieve a fixed error bar increases linearly with N, so the total cost of a "standard" QMC calculation with extended orbitals and localised basis functions is N^3, and the use of localised orbitals could improve this to N^2.

In practice localised orbitals are rarely used because the speedup does not justify the error incurred by truncating the orbitals, and other parts of the wave function such as the two-body Jastrow terms start to become significant. In addition, the cost of evaluating a determinant given the elements of the matrix is N^3, so the standard scaling is really more like N^3 + epsilonN^4, where epsilon is usually negligible over the relevant range of N.

Hope that helps,

Neil.