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Universita’ dell’Insubria, Como, Italy
Some considerations on nodes and
trial wave functions
Dario Bressanini
http://scienze-como.uninsubria.it/bressanini
QMCI Sardagna (Trento) 2008
30+ years of QMC in chemistry
2
The Early promises?

Solve the Schrödinger equation exactly without
approximation (very strong)

Solve the Schrödinger equation with controlled
approximations, and converge to the exact solution
(strong)

Solve the Schrödinger equation with some
approximation, and do better than other methods
(weak)
3
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
4
Good for vibrational problems
5
For electronic structure?
Sign Problem
Fixed Nodal error problem
6
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

The functional form of T(R)
?
7
Improving T

Current Quantum Monte Carlo research focuses on

Optimizing the energy

Adding more determinants (large number of parameters)

Exploring new trial wave function forms (moderately
large number of parameters)
» Pfaffians, Geminals, Backflow ...

Node are improved (but not always) only indirectly
8
Adding more determinants

Use a large Slater basis

Try to reach HF nodes convergence

Orbitals from MCSCF are good

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
9
Li2
CSF (1g2 1u2 omitted)


E (hartree)
2 g2
 3 g2  4 g2  ...  9 g2
-14.9923(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 (N.R.L.)
-14.9954
-14.9914(2)
-14.9933(1)
Not all CSF are useful
Only 4 csf are needed to build a statistically exact nodal
surface Bressanini et al. J. Chem. Phys. 123, 204109 (2005)
10
Dimers
Bressanini et al. J. Chem. Phys. 123, 204109 (2005)
11
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
12
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
13
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 )
14
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. An exponential wall
15
GVB for atoms
18
GVB for atoms
19
GVB for atoms
20
GVB for atoms
21
GVB for molecules

Correct
asymptotic
structure

Is there a nodal
error component
in HF wave
function coming
from incorrect
dissociation?
23
GVB for molecules
Localized orbitals
24
GVB Li2
Wave functions
VMC
DMC
HF 1 det compact
-14.9523(2)
-14.9916(1)
GVB 8 det compact
-14.9688(1)
-14.9915(1)
CI 3 det compact
-14.9632(1)
-14.9931(1)
GVB CI 24 det compact
-14.9782(1)
-14.9936(1)
-14.9933(2)
CI 3 det large basis
CI 5 det large basis
E (N.R.L.)
 1 ux2  1 uy2  2 u2  3 g2
-14.9952(1)
-14.9954
Improvement in the wave function
but irrelevant on the nodes,
25
GVB in QMC

Conclusions

The quality of the wave function improves, giving better
VMC energies …

… but the nodes are not changed, giving the same QMC
energies

Bressanini and Morosi J. Chem. Phys. 129, 054103 (2008)
26
Conventional wisdom on 
Single particle approximations

EVMC(RHF) > EVMC(UHF) > EVMC(GVB)
Consider the N atom

RHF = |1sR 2sR 2px 2py 2pz| |1sR 2sR|

UHF = |1sU 2sU 2px 2py 2pz| |1s’U 2s’U|
EDMC(RHF) > ? < EDMC(UHF)
27
Conventional wisdom on 
We can build a RHF with the same nodes of UHF

UHF = |1sU 2sU 2px 2py 2pz| |1s’U 2s’U|

’RHF = |1sU 2sU 2px 2py 2pz| |1sU 2sU|
EDMC(’RHF) = EDMC(UHF)
EVMC(’RHF) > EVMC(RHF) > EVMC(UHF)
28
Conventional wisdom on 
GVB = |1s 2s 2p3| |1s’ 2s’| -
|1s’ 2s 2p3| |1s 2s’| +
|1s’ 2s’ 2p3| |1s 2s|- |1s 2s’ 2p3| |1s’ 2s|
Same Node
Node equivalent to a UHF |f(r) g(r) 2p3| |1s 2s|
EDMC(GVB) = EDMC(’’RHF)
29
What to do?

Should we be happy with the “cancellation of
error”, and pursue it?

After all, the whole quantum chemistry is built on
it!

If not, and pursue orthodox QMC (no pseudopotentials, no
cancellation of errors, …) , can we avoid the curse of T ?
30
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
31
Numbers and insight

There is no shortage of accurate calculations for
few-electron systems


−2.90372437703411959831115924519440444669690537 a.u.
Helium atom (Nakashima and Nakatsuji JCP 2007)
However…
“The more accurate the calculations became, the
more the concepts tended to vanish into thin air “
(Robert Mulliken)
32
A little intermezzo (for the students)
We need new,
and different, ideas
We need new, and different, ideas

Different representations

Different dimensions

Different equations

Different potential

Radically different algorithms

Different something
Research is the process of going up alleys to see if they are
blind.
Marston Bates
35
Just an example

Try a different representation

Is some QMC in the momentum representation

Possible ? And if so, is it:

Practical ?

Useful/Advantageus ?

Eventually better than plain vanilla QMC ?

Better suited for some problems/systems ?

Less plagued by the usual problems ?
36
The other half of Quantum mechanics
 ( p)  Fˆ ( (r ))
The Schrodinger equation in the momentum representation
2
p
(E 
) ( p)  (2 ) 1/ 2  Vˆ ( p  p) ( p)dp
2m
Some QMC (GFMC) should be possible, given the iterative form
Or write the imaginary time propagator in momentum space
37
Better?

For coulomb systems:
1
2
ˆ
ˆ
V ( p)  F ( ) 
rij
pi  p j
2

There are NO cusps in momentum space.  convergence
should be faster

Hydrogenic orbitals are simple rational functions
(8Z 5 )1/ 2
1s ( p) 
2
2 2
(p  Z )
38
Another (failed so far) example

Different dimensionality: Hypernodes

Given H (R) = E (R) build
H   H (R1 )  H (R 2 )
6 N dimensions
T  B (R1 )F (R 2 )  F (R1 )B (R 2 )
•Use the Hypernode of T
• The hope was that it could be better than Fixed Node
39
Hypernodes
The intuitive idea was that the system could correct the
wrong fixed nodes, by exploring regions where T ( R)  0
Fixed Node
Trial node
Fixed HyperNode
Trial node
Exact node
Exact node

The energy is still an upper bound

Unfortunately, it seems to recover exactly the FN energy
40
Feynman on simulating nature

Nature isn’t 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 Feynman 1981
41
Nodes
Should we concentrate on nodes?

Conjectures on nodes

have higher symmetry than  itself

resemble simple functions

the ground state has only 2 nodal volumes

HF nodes are often a god starting point
42
How to directly improve nodes?

Fit to a functional form and optimize the
parameters (maybe for small systems)

IF the topology is correct, use a coordinate
transformation
R  T (R)
43
He2+: “expanding” the node
Node (1 ) : c  0
r1A  r1B  r3 A  r3 B
Node (2 ) : c  1
z1  z3  0

It is a one
parameter  !!
Exact
44
“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???????
45
Coordinate transformation

Take a wave function with the correct nodal topology
  HF  

Change the nodes with a coordinate transformation
(Linear? Feynman’s backflow ?) preserving the topology
R  T (R)
Miller-Good transformations
46
The need for the correct topology

Using Backflow alone, on a single determinant 
is not sufficient, since the topology is still wrong

More determinants are necessary (only two?)
BF  T (R )  T (R ) 
T (R )  T (R )   0 at least two nodes
47
Be Nodal Topology
r1+r2
r1+r2
r3-r4
r3-r4
r1-r2
HF  0
r1-r2
CI  0
  1s 2 2s 2  c 1s 2 2 p 2
48
Avoided crossings
Be
e- gas
Stadium
49
Nodal topology

The conjecture (which I believe 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)
50
Avoided nodal crossing

At a nodal crossing,  and  are zero

Avoided nodal crossing is the rule, not the exception

Not (yet) a proof... (any help is appreciated)
  0

  0
1 eq. 
3N  1 with 3N variables
3N eqs.
If HF has 4 nodes HF   has 2 nodes, with a proper 
52
He atom with noninteracting electrons
1
3s5s S
53
54
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
57
Think Different
Take a look at your nodes!
58