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N ice ,N o ve m ber19 2005 Some mathematical questions related to quantum Monte Carlo approaches for molecules Michel Caffarel Laboratoire de Physique Quantique, IRSAMC Universite Paul Sabatier, Toulouse – p.1/28

  • ro-vibrational spectroscopy

  • con- tinuous space

  • discrete systems

  • solid-state physics

  • high tc superconductivity

  • he3 very rich

  • very large

  • theoretical condensed-matter physics


Publié le : mardi 19 juin 2012
Lecture(s) : 21
Source : math.unice.fr
Nombre de pages : 28
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Nice, November 19 2005
Some mathematical questions related
to quantum Monte Carlo approaches
for molecules
Michel Caffarel
´
Laboratoire de Physique Quantique, IRSAMC Universite Paul Sabatier, Toulouse
– p.1/28QMC ?
Quantum Monte Carlo (QMC) = generic name for a variety
of methods “solving stochastically” the stationary
Schrödinger equation.
More precisely: Finding the lowest eigenvalueE of:
0
1 2 d
Continuous systems: H = r +V (x) x2R d = 3N
2
Discrete systems: H= very large sparse matrix (linear size
10
much greater than 10 )
– p.2/286
QMC community
Four pertinent parameters:
Temperature: T = 0 orT = 0
Statistics: F (Fermions), B (Bosons), Bz (Boltzmannons:
distinguishable particles)
Nature of configuration space: D (discrete), C
(continuous)
Extension of the system: infinite (solids,liquids, the
thermodynamical limit has to be taken: N !1, finite
(molecules, cluster). M(macro) ou m(micro)
) 2x3x2x2= potentially: 24 communities...
– p.3/28Physics:
(0,C,B,M), (T,C,B,M): Bosonic liquids He , Superfluidity
4
(0,C,F,M),(T,C,B,M): Fermi liquids He Very rich phase
3
diagram
(0,D,F,M),(0,D,F,M): Theoretical condensed-matter physics
Hubbard model, High-T superconductivity
c
(0,C,F,M): Solid-state physics Silicium
(0,C,F,m): Nuclear physics Tritium nucleus
Chemistry:
(0,C,Bz,m): Ro-vibrational spectroscopy Water IR spectrum
(0,C,F,m): Electronic Structure of molecules H ?, rather Li...
2
– p.4/28Warning!
Numerous QMC algorithms with various acronyms:
VMC, DMC, PDMC, GFMC, PIMC, projector MC, Worldline
MC, etc....
Here: Electronic Sructure for quantum Chemistry: T=0 , con-
tinuous space, fermions, finite system.
– p.5/28– p.6/28Fundamental idea
Quantum Monte Carlo (QMC) = “stochastic” power method
Letju > be an arbitrary vector:
0
n
Power method: H ju >
0
) Extraction of the eigenvector associated with the largest
n
eigenvalue [ (E H) ju >: lowest eigenvalue]
0
P
N
Fundamental step: Multiplication of H by |u>; H u
ij j
j=1
Full multiplication = usual deterministic power method
Importance sampled multiplication = QMC (only the most
significant contributions to the sum are computed and
accumulated)
– p.7/28In practice
G(H): G(H)ju >
0
Lattice: G(H) = 1 (H E )
T
Continuous: G(H) = exp[ (H E )]
T
Matrix elements ofG(H) are written as a product of a
probability transition (stochastic matrix) times a weight:
G =p(i!j)w
ij ij
Example:
2
<x; exp[ (H E );y> <x; exp(=2r ;y>
T small
exp[ (V (x) E )]
T
2
d
(x y)
1
withp(x!y) = ( ) exp( )
2 2
– p.8/28Feynman-Kac type formulae
p(i!j) allow to generate series of “states” (walkers,
particles,...) in time (iteration numbern)
! notion of “trajectories”
Quantum averages can be written as average over the set
of trajectories (path integral formalism)
! Feynman-Kac point of view
Example:
+1V (x(s))ds
>
0
(x)<e
0
<>= average over the set of all free brownian
trajectories starting at x; (this formula is just the rewritting of
H n
[e ] applied to(x y)
– p.9/28Importance sampling
As in any Monte Carlo scheme, importance sampling has to
be introduced.
Here, importance sampling = introduction of some good
T
guess of the unknown eigenvector,ju > or
T
T T T
p (i!j) =u =u p(i!j)
j i
Continuous systems:
2
(y x b(x))
d
1
T
2
p (x!y) = ( ) e
2
b(x) =r = drift vector
T T
! “drifted” random walks instead of free random walks
– p.10/28

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