Документ взят из кэша поисковой машины. Адрес оригинального документа : http://classic.chem.msu.su/gran/gamess/gencon.pdf Дата изменения: Wed Nov 22 14:24:38 2006 Дата индексирования: Mon Oct 1 19:38:21 2012 Кодировка:

General Contraction Basis Sets: fast direct Fock Matrix formation
Alexander A. Granovsky

Laboratory of Chemical Cybernetics, M.V. Lomonosov Moscow State University, Moscow, Russia
May 04, 2005


Fock Matrix
F2(D) = J(D) - K(D)
Jpq = rs(pq|rs)Drs Kpr = qs(pq|rs)Dqs where (pq|rs) are 2-e integrals in AO basis

Note: required not only for HF/DFT
CIS, TDHF, TDDFT, CI, MCSCF, ... Orbital response equations for properties Molecular gradients: orbital response equations for all non fully orbital-variational methods Molecular hessians: orbital response equations for all methods


GTO Basis Sets
Segmented Contraction (SC)
i=j cijxkylzmexp(-ijr2) Examples: 6-31*, etc...

General Contraction (GC)
i=j cijxkylzmexp(-jr2)
common set of exponents within each shell type (very) long expansions

Examples: Roos' ANO, Dunning's cc-pVXZ (to some degree)


Standard Direct Fock matrix formation code
loop over all shells (P index)
loop over Q P, Q in P domain
loop over all shells (R index)
· loop over S R, S in R domain Check (PQ|RS) using selection criterion Calculate non-negligible (PQ|RS) if any Add their contributions to the Fock matrix end loop end loop

end loop

end loop


Simplified code calculating fourcenter integrals over shell quartet
loop over all primitives in P shell
loop over all primitives in Q shell
loop over all primitives in R shell
· loop over all primitives in S shell Calculate contribution due to combination of four primitive functions to (PQ|RS) end loop end loop

end loop

end loop


Preliminary considerations
Standard code is very bad for GC basis sets:
does not common does not high L. does not reuse already calculated integrals over primitives like long contractions, especially for use integral screening efficiently


Modified Fock matrix formation code, most common way
loop over all shell subsets (P index)
loop over Q subsets P, Q in P domain
loop over all shell subsets (R index)
· loop over S subsets R, S in R domain · Calculate all non-negligible (pq|rs) combinations · Perform small four-index transformation to get all (PQ|RS) · Save on disk or add contributions to the Fock matrix end loop end loop

end loop

end loop


Common way problems:
Requires non-trivial extra logic in loops Complex algorithms Non-trivial implementation of:
parallel mode algorithms integral screening use of molecular symmetry

Can be a memory hog
Example: g-functions, 10 primitives:
154*104 500 000 000 words of memory are required only to store all intermediate (pq|rs)


Alternative way to direct Fock Matrix construction:
We do not need 2-e integrals themselves, only their contributions to the Fock matrix Let us consider transformation from (pq|rs) to (PQ|RS) as four-index transformation using specific transformation matrix:
(PQ|RS) = p q r s CPpCQqCRrCSs(pq|rs)
Matrix C is completely defined by the basis set contraction coefficients Matrix C is block-diagonal (this will not be very important)


Alternative way to direct Fock Matrix construction, contd.:
(PQ|RS) = p q r s CPpCQqCRrCSs(pq|rs) JPQ = RS(PQ|RS)DRS JPQ = RS p q r s CPpCQqCRrCSs(pq|rs) DRS JPQ = p q CPpCQqr s (pq|rs) RS CRrCSsDRS Let us define:
Drs=RS CRrCSsDRS <=> D = C+DC Jpq = r s (pq|rs) Drs

Then:
JPQ = p q CPpCQq Jpq <=> J = CJC+


How it works:
Having initial GC basis set and density matrix:
Define intermediate (fully or partially uncontracted) basis Define constant transformation matrix C Transform density to the intermediate basis set Calculate F and/or J & K matrix (matrices) using standard direct SCF code

Finally, transform F (J, K) back to the initial basis set


Example: FeO molecule/Roos Augmented Triple Zeta ANO
Fe:
(21s,15p,10d,6f,4g) -> [8s,7p,5d,3f,2g]

O:
(14s,9p,4d,3f) -> [5s,4p,3d,2f]

FeO:
39 shells, 174 Cartesian/139 spherical basis functions


Example, contd.
PC GAMESS direct RHF calculations, single-CPU Opteron 144
Using GAMESS (US) 2-e code:
First iteration: 175 min., last iteration: 175 min.

Using PC GAMESS 2-e code:
First iteration: 83 min., last iteration: 83 min.

Using PC GAMESS GENCON code:
First iteration: 51 sec., last iteration: 33 sec.


Thank you for your attention!