Divide and Conquer QM Development

• Muller, Ward, Goddard• Linear Scaling• Massive Parallelizability Form

H

Diagonalize

Did Change?

Guess

No

Done

Yes

O(N4)PS/Jaguar O(N2)

O(N3)Difficult to reduce: Krylof space, Conjugate gradient

Currently only important if N > 2000

Divide and Conquer Methods

• Break a big problem into small problems• Physically based in quantum chemistry

Divide and Conquer

F F F F F F F F F

F F F F F F F F

F F F F F F F F F

Cl Cl Cl Cl Cl Cl Cl Cl

ActiveZone

BufferZone

BufferZone

H

Hamiltonian:Divided into fragments andbuffer zones

nbf

Divide and Conquer Results

• Linear Alkanes

– 14-98 atoms

– 170-817 bfns

H3C

CH2

H2C

CH3

n

• Linear scaling AND parallel!

Divide, Conquer, and Recombine• Buffer zones don’t scale

– Metallic Systems– Same problem n times

• Hierarchically recombine– E-value decomp of off-diagonal sections– Can also approximate.

Diagonal Blocks:

ˆ H A =ˆ t +ˆ v A + ˆ v Bexternal

field

{

ˆ v i = ˆ V aN + ˆ V a

J + ˆ V aXC

[ ]atomsa

fragmenti

∑

Off -Diagonal Blocks:

H AB =t +vA +vBignoreas

Rab increases

1 2 4 4 3 4 4

Recombine:

A C

C B

⎛

⎝ ⎜

⎞

⎠ ⎟ =

A λ1u1v1T

λ1u1v1T B

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

=A −λ1u1u1

T 0

0 B−λ1v1v1T

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

+λ1u1v1

⎡

⎣ ⎢

⎤

⎦ ⎥ u1

T v1T

[ ]