Is QMC delivering its early promises?

1
Universita dellInsubria, 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
2
30 years of QMC in chemistry
3
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)

4
Good for Helium studies

• Thousands of theoretical and experimental papers

have been published on Helium, in its various
forms
Small Clusters
Droplets
Bulk
Atom
5
3Hem4Hen Stability Chart
0 1 2 3 4 5 6 7 8 9
10 11
0 1 2 3 4 5
Terra Incognita
32
3He34He8 L0 S1/2
3He24He2 L0 S0
3He34He4 L1 S1/2
3He24He4 L1 S1
6
Good for vibrational problems
7
For electronic structure?
Sign Problem Fixed Nodal error problem
8
The influence on the nodes of YT

• QMC currently relies on YT(R) and its nodes
  (indirectly)
• How are the nodes YT(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

?
9
He2 the basis set
The ROHF wave function
1s E -4.9905(2) hartree
1s1s2s3s E -4.9943(2) hartree
EN.R.L -4.9945 hartree
10
He2 MOs
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

11
He2 CSFs

• 1s1s2s3s2p2p
• E(1 csf) -4.9932(2) hartree

• 1s1s2s3s
• E(1 csf) -4.9943(2) hartree

12
Li2
E (hartree)
CSF
(1sg2 1su2 omitted)
-14.9923(2)
-14.9914(2)
-14.9933(2)
-14.9933(1)
-14.9939(2)
-14.9952(1)
-14.9954
E (N.R.L.)

• Not all CSF are useful
• Only 4 csf are needed to build a statistically
  exact nodal surface

13
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

14
Dimers
Bressanini et al. J. Chem. Phys. 123, 204109
(2005)
15
Is QMC competitive ?
16
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) 34 -37.8407(1)
• R12-MR-CI -37.845179
• Exact (estimated) -37.8450

17
Ne Atom
Drummond et al. -128.9237(2) DMC
Drummond et al. -128.9290(2)
DMC backflow
Gdanitz et al. -128.93701
R12-MR-CI
Exact (estimated) -128.9376
18
Conventional wisdom on Y
Single particle approximations

• EVMC(YRHF) gt EVMC(YUHF) gt EVMC(YGVB)

Consider the N atom

• YRHF 1sR 2sR 2px 2py 2pz 1sR 2sR
• YUHF 1sU 2sU 2px 2py 2pz 1sU 2sU

EDMC(YRHF) gt ? lt EDMC(YUHF)
19
Conventional wisdom on Y
We can build a YRHF with the same nodes of YUHF

• YUHF 1sU 2sU 2px 2py 2pz 1sU 2sU
• YRHF 1sU 2sU 2px 2py 2pz 1sU 2sU

EDMC(YRHF) EDMC(YUHF)
EVMC(YRHF) gt EVMC(YRHF) gt EVMC(YUHF)
20
Conventional wisdom on Y
YGVB 1s 2s 2p3 1s 2s - 1s 2s 2p3 1s
2s 1s 2s 2p3 1s 2s- 1s
2s 2p3 1s 2s
Node equivalent to a YUHF f(r) g(r) 2p3 1s 2s
EDMC(YGVB) EDMC(YRHF)
21
Nitrogen Atom

• Y Param. E corr. VMC E corr. DMC
• Simple RHF (1 det) 4 26.0 91.9
• Simple RHF (1 det) 8 42.7 92.6
• Simple UHF (1 det) 11 41.2 92.3
• Simple GVB (4 det) 11 42.3 92.3
• Clementi-Roetti J 27 24.5 93.1

Is it worth to continue to add parametersto the
wave function?
22
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 YT ?

23
The curse of the YT

• QMC currently relies on YT(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

For M109 and p3 ? N6
p parameters per variable M total parameters
needed
The Exponential Wall
24
Convergence to the exact Y

• We must include the correct analytical structure

Cusps
QMC OK
QMC OK
3-body coalescence and logarithmic terms
Often neglected
Tails
25
Asymptotic behavior of Y

• Example with 2-e atoms

is the solution of the 1 electron problem
26
Asymptotic behavior of Y

• The usual form

does not satisfy the asymptotic conditions
A closed shell determinant has the wrong structure
27
Asymptotic behavior of Y

• In general

Recursively, fixing the cusps, and setting the
right symmetry
Each electron has its own orbital,
Multideterminant (GVB) Structure!
2N determinants. Again an exponential wall
28
PsH Positronium Hydride

• A wave function with the correct asymptotic
  conditions

Bressanini and Morosi JCP 119, 7037 (2003)
29
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
30
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 ?

31
The other half of Quantum mechanics
The Schrodinger equation in the momentum
representation
Some QMC (GFMC) should be possible, given the
iterative form
Or write the imaginary time propagator in
momentum space
32
Better?

• For coulomb systems

• There are NO cusps in momentum space. Y
  convergence should be faster
• Hydrogenic orbitals are simple rational functions

33
Another (failed so far) example

• Different dimensionality Hypernodes
• Given HY (R) EY (R) build

• The hope was that it could be better than Fixed
  Node

34
Hypernodes

• The energy is still an upper bound
• Unfortunately, it seems to recover exactly the
  FN energy

35
Why is QMC not used by chemists?

• A little intermezzo

36
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.edu/Apps/CMP/topten/topten.html
37
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
38
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
39
A Quantum Chemistry Chart
J.Pople
The more accurate the calculations became, the
more the concepts tended to vanish into thin air
(Robert Mulliken)
40
Chemical concepts

• Molecular structure and geometry
• Chemical bond
• Ionic-Covalent
• Singe, Double, Triple
• Electronegativity
• Oxidation number
• Atomic charge
• Lone pairs
• Aromaticity

41
Nodes
Should we concentrate on nodes?

• Conjectures on nodes
• have higher symmetry than Y 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
42
Be Nodal Topology
43
Avoided crossings
Be
e- gas
Stadium
44
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)

45
Nodal Regions
Nodal Regions
46
Avoided nodal crossing

• At a nodal crossing, Y and ?Y are zero
• Avoided nodal crossing is the rule, not the
  exception
• Not (yet) a proof...

47
He atom with noninteracting electrons
48
(No Transcript)
49
Casual similarity ?
First unstable antisymmetric stretch orbit of
semiclassical linear helium along with the
symmetric Wannier orbit r1 r2 and various
equipotential lines
50
Casual similarity ?
Superimposed Hylleraas node
51
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

52
Coordinate transformation

• Take a wave function with the correct nodal
  topology

• Change the nodes with a coordinate transformation
  (Linear? Feynmans backflow ?) preserving the
  topology

Miller-Good transformations
53
Feynman on simulating nature

• Nature isnt classical, dammit, and if you want
  to make a simulation of Nature, youd better make
  it quantum mechanical, and by golly its a
  wonderful problem, because it doesnt look so
  easy
• Richard Feynman 1981

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
55
Think Different!