Quantum Monte Carlo

James B. AndersonDepartments of Chemistry and Physics

Pennsylvania State University

“The underlying physical laws necessary for the mathematical

treatment of a large part of physics and the whole of chemistry

are thus completely known, and the difficulty is only that the

exact application of these laws leads to equations much too

complicated to be soluble.”

P.A.M. Dirac, Proc. R. Soc. London Ser. A 123, 714 (1929)

“It therefore becomes desirable that approximate practical

methods of applying quantum mechanics should be

developed, which can lead to an explanation of the main

features of complex atomic systems without too much

computation.”

P.A.M. Dirac, Proc. R. Soc. London Ser. A 123, 714 (1929)

“The dreaded

Sign Problem

In Quantum

Monte Carlo”

“Exact”

Accuracy of EQMC calculation:

±0.000014 or 3 cm-1

Clementi et al., Chem. Rev. 91, 679 (1991), estimated that

an analytic variational calculation for H-H-H accurate to

within 10 cm-1 of the exact value

“would call for somewhere around 3500 years”

on a computer similar to the one we used.

EQMC CI

3 months vs. 2 millennia

Experiment ahead of theory?

Theory ahead of experiment?

Forever?

BACK TO FIXED-NODE METHOD (not exact)

Some examples of larger systems from other labs:

Guanine

GuanineCytosine

Cytosine

Why is the electron correlation

important in biomolecules ?

DNA Base Pair Stacking

HF or DFT cannot

describe the weak

attraction due to van der

Waals (dispersion)

interaction appropriately.

weakly

bound

(K. Fukuzawa et al., J. Comput. Chem. 27 (2006) 948.)

As for FMO-MP2 description for vdW interaction, see

Fragment Molecular Orbital Method

I

J

IK K

J

I

Divide a molecule into fragments

N pieces of fragments (monomers)

[N(N-1)/2] pieces of fragment pairs

(dimers)

( )<

+=JI

JIIJ

I

IEEEEE

FMO Total Energy Environmental

electrostatic potential

Environmental

electrostatic potential

Examples of FMO calculation

Carbon Fixation Enzyme “Rubisco”

FMO-HF/STO-3G

Enzyme (467 amino acid residues)

+ Substrate

7269 atoms, 22057 AOs

Calculation time: only 5 hours

Opteron (2.0GHz) x 32 CPUs

(MP2/6-31G within 1 week)

(H. Watanabe et al., Biochem. Biophys. Res. Commun.

316 (2007) 367.)

Test Calculation

Glycine trimer

Fragment 1

Fragment 2 Fragment 3

HF-SCF calculation: ABINIT-MP

VMC calculation: CASINO code

(the Cambridge quantum Monte Carlo code)

Basis set: STO-3G

Structure Optimization

Gaussian98 (HF/STO-3G)

(R. Maezono et al., J. Phys. Soc. Jpn. 76 (2007) 064301.)

DNA Base Pair

Stacking

3 4 5

−10

0

10STO−3G3−21G

6−31G6−31G*

FMO−HF

FMO−MP2

Distance (A)

Sta

ckin

g E

nerg

y (k

cal/m

ol)

Study underway

Guanine

GuanineCytosine

Cytosine

DNA Base Pair Stacking