Transcript No Slide Title

Universita’ dell’Insubria, Como, Italy
Nodal structure and the construction
of trial wave functions for Quantum
Monte Carlo
Dario Bressanini
http://scienze-como.uninsubria.it/bressanini
UNAM, Mexico City, 19/09/2007
Monte Carlo Methods

How to solve a deterministic problem using a
Monte Carlo method?

Rephrase the problem using a probability
distribution
A   P(R) f (R)dR

R  N
“Measure” A by sampling the probability distribution
1
A 
N
N
 f (R )
i 1
i
R i ~ P(R )
2
VMC: Variational Monte Carlo

Start from the Variational Principle
H

(R ) H(R )dR


E
  (R)dR
0
2
Translate it into Monte Carlo language
H   P(R ) EL (R )dR
H (R )
EL (R ) 
 (R )
P(R ) 
 (R )
2
2

 (R)dR
3
VMC: Variational Monte Carlo
E  H   P(R ) EL (R )dR

E is a statistical average of the local energy over P(R)
1
E H 
N

N
E
i 1
L
(R i )
R i ~ P(R )
Recipe:
 take an appropriate trial wave function
 distribute N points according to P(R)
 compute the average of the local energy
4
5
VMC for PsH – Positronium Hydride

A wave function with the correct asymptotic conditions:
(1,2, p)  (1  Pˆ12 )( H  ) f (rp )( Ps) g (r1 p )
Bressanini and Morosi: JCP 119, 7037 (2003)
6
Diffusion Monte Carlo

VMC is a “classical” simulation method
Nature is not classical, dammit, and if you
want to make a simulation of nature, you'd
better make it quantum mechanical, and by
golly it's a wonderful problem, because it
doesn't look so easy.
Richard P. Feynman

Suggested by Fermi in 1945, but implemented
only in the 70’s
7
Diffusion equation analogy
 The time dependent
Schrödinger equation
is similar to a diffusion
equation
 The diffusion
equation can be
“solved” by directly
simulating the system

2 2
i

   V
t
2m
C
2
 D C  kC
t
Time
evolution
Diffusion
Branch
Can we simulate the Schrödinger equation?
8
Imaginary Time Sch. Equation

The analogy is only formal

 is a complex quantity, while C is real and positive
 (R, t )  e iEnt /   n (R)

If we let the time t be imaginary, then  can be real!

 D 2   V

Imaginary time Schrödinger equation
9
 as a concentration

 is interpreted as a concentration of fictitious
particles, called walkers

The schrödinger equation
is simulated by a process
of diffusion, growth and
disappearance of walkers

2
 D   V

 (R, )   ai  i (R)e  ( Ei  ER )
i
(R,  )   0 (R)e ( E0  ER )
Ground State
10
Diffusion Monte Carlo
SIMULATION: discretize time
•Diffusion process

 D 2 

(R,  )  e
 ( R  R 0 ) 2 / 4 D
•Kinetic process (branching)

 (V (R )  ER )

(R,  )  e
 (V ( R )  ER ) 
(R,0)
11
The DMC algorithm
12
Good for Helium studies

Thousands of theoretical and experimental papers
Hˆ n (R)  En n (R)
have been published on Helium, in its various forms:
Atom
Small Clusters
Droplets
Bulk
13
The Fermion Problem

Wave functions for fermions have nodes.

Diffusion equation analogy is lost. Need to introduce
positive and negative walkers.
The (In)famous Sign Problem


If we knew the exact nodes of , we could exactly simulate
the system by QMC methods, restricting random walk to a
positive region bounded by nodes.
Unfortunately, the exact nodes
are unknown. Use approximate
nodes from a trial . Kill the
walkers if they cross a node.
+
-
14
For electronic structure?
Sign Problem
Fixed Nodal error problem
15
The Helium triplet

First 3S state of He is one of very few systems
where we know the exact node

For S states we can write
  (r1 , r2 , r12 )
•For the Pauli Principle
(r1 , r2 , r12 )  (r2 , r1 , r12 )

Which means that the node is
r1  r2
or
r1  r2  0
16
The Helium triplet node

Independent of r12

The node is “more
symmetric” than the
wave function itself


It is a polynomial in r1
and r2
r1
r12
r2
r1  r2    0
r1
r2
Present in all 3S states of
two-electron atoms
r1  r2    0
17
The influence on the nodes of T

QMC currently relies on T(R) and its nodes (indirectly)

How are the nodes T(R) of influenced by:

The single particle basis set

The generation of the orbitals (HF, CAS, MCSCF, NO, …)

The number and type of configurations in the multidet.
expansion
?
18
Nodes and Configurations
A better  does not necessarily mean better
nodes
Why? What can we do about it?
20
Beryllium Atom
RHF  1s( r1 )2 s( r2 )  1s( r3 )2s( r4 ) 
 HF predicts 4 nodal regions
Bressanini et al. JCP 97, 9200 (1992)
 Node: (r1-r2)(r3-r4) = 0
  factors into two determinants
each one “describing” a triplet
Be+2. The node is the union of the
two independent nodes.
The HF node is wrong
•DMC energy -14.6576(4)
•Exact energy -14.6673
Plot cuts of (r1-r2) vs (r3-r424
)
Be Nodal Topology
r1+r2
r1+r2
r3-r4
r3-r4
r1-r2
HF  0
r1-r2
  1s 2 2 s 2  c 1s 2 2 p 2
CI  0
25
Be nodal topology
Node is

Now there are only two
nodal regions

It can be proved that the
exact Be wave function has
exactly two regions
( r1  r2 )( r3  r4 )  c( r132  r142  r232  r242 )  ...  0
See Bressanini, Ceperley and Reynolds
http://scienze-como.uninsubria.it/bressanini/
26
Be energy (hartree)

HF node
-14.6565(2) 1s2 2s2

Bressanini et al.
-14.66729(2) +1s2 2p2 small basis

“
no improvement
+1s2 ns ms

“
no improvement
+1s2 np mp
-14.66734(4) +1s2 2p2 large basis


Exact
-14.6673555
Why some configurations are not important for QMC ?
27
Be Node: considerations

they give the same contribution to the node expansion

ex: 1s22s2 and 1s23s2 have the same node ( r1  r2 )( r3  r4 )  0

ex: 2px2, 2px3px and 3px2 have the same structure
x1 x2 f1 (r1 ) f 2 (r2 )

It seems that the nodes of "useful" CSFs belong to
higher and different symmetry groups than the exact

2
2
2
2
(r1  r2 )(r3  r4 )
 
r12  r34
iˆ34 (1s 2 s )  1s 2 s
iˆ34 (1s 2 2 p 2 )  1s 2 2 p 2
28
Li2
CSF (1g2 1u2 omitted)
2 g2
 3 g2  4 g2  ...  9 g2


E (hartree)
-14.9923(2)
-14.9914(2)
 1 ux2  1 uy2
 4 n ux2  n uy2
-14.9933(2)
 1 ux2  1 uy2  2 u2
-14.9939(2)
 1 ux2  1 uy2  2 u2  3 g2
-14.9952(1)
E (exact N.R.L.)
-14.9954
-14.9933(1)
Not all functions are useful
Only 5 determinants are needed to build a
statistically exact nodal surface and exact energy
32
A tentative recipe

Use a large Slater basis

But not too large

Try to reach HF nodes convergence

Orbitals from CAS seem better than HF, or NO

Not worth optimizing MOs, if the basis is large enough

Only few configurations seem to improve the FN energy

Use the right determinants...


...different Angular Momentum CSFs
...and not the bad ones

...types already included
iˆ34 (1s 2 2s 2 )  1s 2 2s 2
iˆ34 (1s 2 2 p 2 )  1s 2 2 p 2
iˆ34 (1s 2 3s 2 )  1s 2 3s 2
33
Dimers
Bressanini et al. J. Chem. Phys. 123, 204109 (2005)
34
The curse of the T

QMC currently relies on T(R)

Walter Kohn in its Nobel lecture (R.M.P. 71, 1253 (1999))
“discredited” the wave function as a non legitimate
concept when N (number of electrons) is large
M  p3N
3  p  10
p = parameters per variable
For M=109 and p=3  N=6
M = total parameters needed
The Exponential Wall
36
Playing directly with nodes?

It would be useful to be able to optimize only those
parameters that alter the nodal structure

A first “exploration” using a simple test system
He2+

Can we “expand” the
nodes on a basis?
  (1  c)1  c2
37
He2+: “expanding” the node
Node (1 ) : c  0
r1 A  r1B  r3 A  r3 B
Node (2 ) : c  1
z1  z3  0

It is a one
parameter !!
Exact
38
“expanding” nodes

This was only a kind of “proof of concept”

It remains to be seen if it can be applied to larger
systems

Writing “simple” (algebraic?) trial nodes is not
difficult ….

The goal is to have only few linear parameters
to optimize

Will it work ??
39
Nodal Topology Conjecture
WARNING: Conjecture Ahead...
The HF ground state of Atomic
and Molecular systems has 4
Nodal Regions, while the Exact
ground state has only 2
40
Nodal topology

The conjecture (which likely is true) claims that there
are only two nodal volumes in the fermion
ground state

See, among others:

Ceperley J.Stat.Phys 63, 1237 (1991)

Bressanini and coworkers. JCP 97, 9200 (1992)

Bressanini, Ceperley, Reynolds, “What do we know about wave
function nodes?”, in Recent Advances in Quantum Monte Carlo
Methods II, ed. S. Rothstein, World Scientfic (2001)

Mitas and coworkers PRB 72, 075131 (2005)

Mitas PRL 96, 240402 (2006)
41
How to directly improve nodes?

Fit to a functional form and optimize the
parameters (maybe for small systems)

“Expand” the exact (unknown) nodes

IF the topology is correct, use a coordinate
transformation preserving the topology
R  T (R)
42
Conclusions

The wave function can be improved by
incorporating the known analytical
structure…

… but the nodes do not seem to improve
sistematically

It seems more promising to directly
“manipulate” the nodes.
43
A QMC song...
He deals the cards to find the answers
the sacred geometry of chance
the hidden law of a probable outcome
the numbers lead a dance
Sting: Shape of my heart
44
... and a suggestion
Take a look at your nodes
45
Avoided nodal crossing

At a nodal crossing,  and  are zero

Avoided nodal crossing should be the rule, not the
exception

Not (yet) a proof...
  0

  0

3N  1 with 3N variables
3N eqs.
1 eq.
If HF has 4 nodes HF   has 2 nodes, with a proper 
46
He atom with noninteracting electrons
1
3s5s S
47
Convergence to the exact 

We must include the correct analytical structure
Cusps:
r12
 (r12  0)  1 
2
(r  0)  1  Zr
QMC OK
3-body coalescence and logarithmic terms:
Tails:
QMC OK
Often neglected
48
Asymptotic behavior of 

Example with 2-e atoms
1 2
1 1
1
2
H  (1   2 )  Z (  ) 
2
r1 r2 r12
1 2
Z Z 1
2
H  (1   2 )  
2
r1
r2
r2 
r2 
  0 (r1 )r2
0 (r1 )
( Z 1) /  1  r2
e
  2 EI
is the solution of the 1 electron problem
49
Asymptotic behavior of 

The usual form
   (r1 ) (r2 ) J (r12 )
does not satisfy the asymptotic conditions
 (r2  ) 0 (r1 ) (r2 )

 (r1  )  (r1 ) 0 (r2 )
A closed shell determinant has the wrong structure
  ( (r1 ) (r2 )   (r2 ) (r1 )) J (r12 )
50
Asymptotic behavior of 

r1 
In general  N  r a1 (1  c r 1  O(r 2 ))e r1 / b1Y m1 (r ) N 1 (2,...N )
0
1
1 1
1
l1
1
0
Recursively, fixing the cusps, and setting the right symmetry…
U
ˆ
  A( f1 (1) f 2 (2)... f N ( N ) N )e
Each electron has its own orbital, Multideterminant (GVB) Structure!
Take 2N coupled electrons
2 N  (1 2  1 2 )( 3  4  3 4 )...
2N determinants. Again an exponential wall
51
Basis

In order to build compact wave functions we used basis
functions where the cusp and the asymptotic behavior is
decoupled
ar
1s  e
2 px  x e

ar br2
1 r
ar br2
1 r
e
e
r 0
e
r 
br
ar br2
1 cr
Use one function per electron plus a simple Jastrow
52
GVB for atoms
53
GVB for atoms
54
GVB for atoms
55
GVB for atoms
56
GVB for atoms
57