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Universita’ dell’Insubria, Como, Italy
Some reflections on nodes and trial wave functions
Is QMC delivering its early promises?
Dario Bressanini
http://scienze-como.uninsubria.it/bressanini
QMCI Sardagna (Trento) 2006
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
3He 4He
m
n
3He
4He
n
0 1 2 3 4
m
Stability Chart
5 6
7 8
9 10 11
0
Bound L=0
1
Unbound
2
3
Unknown
4
L=1 S=1/2
5
L=1 S=1
Terra Incognita
Bound
32
3He 4He
2
2
L=0 S=0
3He 4He
2
4
L=1 S=1
3He 4He
3
8
3He 4He
3
4
L=0 S=1/2
L=1 S=1/2
5
Good for vibrational problems
6
For electronic structure?
Sign Problem
Fixed Nodal error problem
7
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
?
8
He2+: the basis set
The ROHF wave function:
RHF (R)  ( g (1) u (3)   g (3) u (1)) g (2)
1s
E = -4.9905(2) hartree
1s1s’2s3s
E = -4.9943(2) hartree
EN.R.L = -4.9945 hartree
9
He2+: MO’s
Bressanini et al. J. Chem. Phys. 123, 204109 (2005)

E(RHF) =
-4.9943(2) hartree
E(CAS) =
-4.9925(2) hartree
E(CAS-NO) = -4.9916(2) hartree
E(CI-NO) = -4.9917(2) hartree

EN.R.L =



-4.9945 hartree
10
He2+: CSF’s
1s1s’2s3s2p2p’

E(1 csf) = -4.9932(2) hartree
+
2
2
1 u11 ux
 1 u11 uy
E(2 csf) = -4.9946(2) hartree
1s1s’2s3s

E(1 csf) = -4.9943(2) hartree
+
1 1g 1 u1 2 1g
E(2 csf) = -4.9925(2) hartree
11
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 (N.R.L.)
-14.9954
-14.9933(1)
Not all CSF are useful
Only 4 csf are needed to build a statistically exact
nodal surface
12
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
13
Dimers
Bressanini et al. J. Chem. Phys. 123, 204109 (2005)
14
Is QMC competitive ?
15
Carbon Atom: Energy






CSFs
Det.
Energy
1 1s22s2 2p2
1
-37.8303(4)
2 + 1s2 2p4
2
-37.8342(4)
5 + 1s2 2s 2p23d 18
-37.8399(1)
83 1s2 + 4 electrons in 2s 2p 3s 3p 3d shell
422
-37.8387(4)
adding f orbitals
7 (4f2 + 2p34f)
R12-MR-CI
Exact (estimated)
34
-37.8407(1)
-37.845179
-37.8450
16
Ne Atom
Drummond et al.
-128.9237(2) DMC
Drummond et al.
-128.9290(2) DMC backflow
Gdanitz et al.
-128.93701
Exact (estimated)
-128.9376
R12-MR-CI
17
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)
18
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)
19
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)
20
Nitrogen Atom




Simple RHF (1 det)
Simple RHF (1 det)
Simple UHF (1 det)
Simple GVB (4 det)
4
8
11
11
26.0%
42.7%
41.2%
42.3%
91.9%
92.6%
92.3%
92.3%

Clementi-Roetti + J
27
24.5%
93.1%


Param.
E corr. VMC E corr. DMC
Is it worth to continue to add parameters
to the wave function?
21
What to do?

Should we be happy with the “cancellation of
error”, and pursue it?

If so:


Is there the risk, in this case, that QMC becomes Yet
Another Computational Tool, and not particularly
efficient nor reliable?

VMC seems to be much more robust, easy to
“advertise”
If not, and pursue orthodox QMC (no pseudopotentials, no
cancellation of errors, …) , can we avoid the curse of T ?
22
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
23
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
24
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
25
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 )
26
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
27
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)
28
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
29
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 ?
30
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
31
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 )
32
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
33
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
34
A little intermezzo
Why is QMC not
used by chemists?
DMC Top 10 reasons

12. We need forces, dummy!

11. Try getting O2 to bind at the variational level.

10. How many graduate students lives have been lost optimizing
wavefunctions?

9. It is hard to get 0.01 eV accuracy by throwing dice.

8. Most chemical problems have more than 50 electrons.

7. Who thought LDA or HF pseudopotentials would be any good?

6. How many spectra have you seen computed by QMC?

5. QMC is only exact for energies.

4. Multiple determinants. We can't live with them, we can't live without them.

3. After all, electrons are fermions.

2. Electrons move.

1. QMC isn't included in Gaussian 90. Who programs anyway?
http://web.archive.org/web/20021019141714/archive.ncsa.uiuc.e
du/Apps/CMP/topten/topten.html
36
Chemistry and Mathematics
"We are perhaps not far removed from the time,
when we shall be able to submit the bulk of chemical
phenomena to calculation”
Joseph Louis Gay-Lussac - 1808
“The underlying physical laws necessary for the
mathematical theory 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 equations leads to equations much too
complicated to be soluble”
P.A.M. Dirac - 1929
37
Nature and Mathematics
“il Grande libro della Natura e’ scritto
nel linguaggio della matematica, e non
possiamo capirla se prima non ne
capiamo i simboli“
Galileo Galilei
Every attempt to employ mathematical methods in the study of chemical
questions must be considered profoundly irrational and contrary to the
spirit of chemistry… If mathematical analysis should ever hold a prominent
place in chemistry – an aberration which is happily almost impossible – it
would occasion a rapid and widespread degeneration of that science.
Auguste Compte
38
A Quantum Chemistry Chart
Orthodox QMC
J.Pople
The more accurate the calculations became, the more the
concepts tended to vanish into thin air (Robert Mulliken)
39
Chemical concepts

Molecular structure and geometry

Chemical bond

Ionic-Covalent

Singe, Double, Triple
d

NOT DIRECTLY OBSERVABLES
Electronegativity
ILL-DEFINED CONCEPTS

Oxidation number

Atomic charge

Lone pairs

Aromaticity
d
d
d O
C

d
O
40
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 quite good: they “naturally” have these
properties
Checked on small systems: L, Be, He2+. See also Mitas
41
Avoided crossings
Be
e- gas
Stadium
43
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)
44
Avoided nodal crossing

At a nodal crossing,  and  are zero

Avoided nodal crossing is 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
48
Casual similarity ?
1s2s 1S Helium
First unstable antisymmetric stretch orbit of
semiclassical linear helium along with the symmetric
Wannier orbit r1 = r2 and various equipotential lines
49
Casual similarity ?
Superimposed Hylleraas node
50
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)
51
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
52
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
53
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
54
Think Different!
55