The document summarizes research using quantum Monte Carlo methods to study the phase of hydrogen at high pressures. Key points: 1) Quantum Monte Carlo allows for highly accurate ab initio simulations beyond density functional theory. The method is used to optimize trial wave functions and perform molecular dynamics simulations of hydrogen. 2) Results on small molecules show good agreement with experiment. Simulations of extended systems use periodic boundary conditions without Ewald sums. 3) Simulations of hydrogen at high pressures find evidence of a possible resonant valence bond phase with off-diagonal long-range order at low temperatures. This could indicate superconductivity. 4) Larger simulations will be needed to fully characterize size effects and dynamic properties,

1. RVB phase of
hydrogen at high
pressure towards
the first ab-initio
Molecular Dynamics
by Quantum
Monte Carlo
Claudio Attaccalite

2. The electronic structure problem
P.A.M. Dirac:The fundamental laws necessary for the
mathematical treatment of a large part of physics and the
whole of chemistry are thus completely known, and the
difficulty lies only in the fact that application of these laws
leads to equations that are too complex to be solved.

3. Why QMC?
Accuracy
Simple empirical potentialsSimple empirical potentials
­ Force fields, pair potentials…
Intermediate methodsIntermediate methods
­ Tight­binding,
­ many­body potentials: EAM
ab­initioab­initio techniques techniques
­ Hartree­Fock
­ Density functional theory (DFT)
Beyond DFT:Beyond DFT:
­ Quantum Monte Carlo
­ Quantum Chemical: CI
Accuracy

4. Outline
• Quantum Monte Carlo
• The trial wave­function
• Optimization Methods
• Results on Molecules
• Molecular Dynamics using QMC forces
• Results on high pressure hydrogen
• New phase in high pressure hydrogen?

5. Variational Monte Carlo
Monte Carlo integration is necessary because the wave­function contains
explicit particle correlations that
leads to non­factoring multi­dimension integrals.

6. The trial wave­function
The trial­function completely determines
quality of the approximation for the physical
observables
r1, r2,... ,rn=AGP J1 J2 J3
The JAGP wave­function
Pairing DeterminantPairing Determinant
3­body Jastrow3­body Jastrow
2­body Jastrow2­body Jastrow
1­body Jastrow1­body Jastrow

7. The AGP wave­function
where
i,j are spin up and down
electrons
a,b are different atoms
l,m different orbitals
The major advantage of the AGP is that it implicitly
contains many
Slater determinants but can be evaluate with the cost of
single
The
matrix
has to be symmetric in order to have a spin
singlet
a ,b
l,m

8. H2
molecule and AGP

9. The one­body Jastrow factor
It's difficult satisfy nuclear cusp conditions
with the pairing determinant
Nuclear cusp:
where:

10. The two­body Jastrow
As for the nuclear­cusp
where
In this functional form the cusp condition for anti­parallel
spin electrons is satisfied, whereas the one for parallel
spins is neglected in order to avoid the spin contamination

11. The Three­body Jastrow
The three­body Jastrow factor:
• describes Van Der Waals forces
• suppress superconductivity
• describes Mott insulators
• keeps charge constant on atoms or molecules (not fixed by AGP)
The three­body Jastrow factor must:
• preserve nuclear and electronic cusp conditions
• whenever the atomic distances are large it factorizes into a product of
independent contributions located near each atom

12. Geminal expansion and orbitals
r=C1e
−Z1
∣r∣
Yl,m
r=C1e
−Z1
∣r∣2
Yl,m

13. Stochastic Reconfiguration 1
In the Stochastic Reconfiguration one adjusts the
parameters
of the original trial-function to be closer as much as possible
to the projected one.

14. Stochastic Reconfiguration 2
we require that a set of mixed average correlation functions,
corresponding to the two wave­functions are equal:
this is equivalent to
substituting the first equation in the others we obtain

15. Stochastic Reconfiguration 3
in analogy with the steepest descent
where
at each interaction the wave function parameters
are iteratively updated according to
the energy variation for a small change of the parameters is

16. Noise and Optimization
Stochastic Reconfiguration is similar to a Langevin
dynamics in the parameters space.
T
eff
2
=〈
2
〉where and 〈
2
〉~1/number of VMCsteps
to reduce statistical fluctuations we averages parameters values
after the equilibration
and increase the number of steps in the VMC simulations
In the limit of infinite bin length the
Euler conditions are satisfied exactly
thermal fluctuations

17. Resonance in Benzene
+
Kekule’
+ +
Dewar Claus­Armstrong
Baeyer

18. Forces with finite variance
the bare force
in order to overcome this problem we follow the idea
of Assaraf and Caffarel we added another operator
with zero expectation value and finite variance
the simplest choice to cancel the divergence in the force is:

19. Structure Optimization
using the stochastic reconfiguration
with

20. Li2
bonding length

21. Benzene Structure

22. Structure optimization results

23. Results on molecules 1

24. Results on molecules 2

25. Simulation of Extended Systems
Hamiltonian with periodic boundary conditions
and Ewald 's sums

26. Periodic Wave­Function
Periodic coordinatesPeriodic coordinates
no need of Ewald's sumno need of Ewald's sum
k=0

27. Forces in Periodic Systems
zero­variance principle in periodic systems
renormalized force

28. Comparison with other results

29. Noise in QMC

30. Finite Temperature Simulation
It is possible to use the forces noise to produce
a finite temperature using
a Generalized Langevin Equation

31. Integration of the Langevin Dynamics
The Impulse integrator (R.D. Skeel, J. A. Izaguirre 2002)
major advantage is stable for large friction
generalization to non diagonal frictiongeneralization to non diagonal friction

32. Temperature and Pressure
Pressure
In the Generalized Langevin Dynamics
the temperature can be estimated at posteriori by equality

33. SR optimization
with Hessian acceleration
Using Hessian matrix information
to accelerate the convergence
in order to have a stable optimization
the variation of the WF has to be small

34. Time­Step Error 1

35. Phase Diagram of Hydrogen
LogT
[K]
4
3
2
Electron Proton Plasma
Electron
Proton
Liquid
Molecular
Liquid
Molecular Solid Proton Solid
Clustered liquid
1 2 3 LogP[GPa]

36. High pressure hydrogen

37. Comparison with other result

38. Proton­proton g(r) in molecular fluid
Rs=2.1 T=4530k

39. Proton­proton g(r) in molecular fluid
Rs=1.31 T=3000k

40. Condensation Energy

41. Off­Diagonal Long Range Order
two­body density matrix
ODLRO
projected two­body density matrix
the estimator we used

42. Eigenvalues of pair function

43. Superconductivity in high pressure hydrogen?
x

44. Condensation energy VS Size

45. ODLRO for 54 hydrogen atoms at 300K
x

46. Conclusions
A new highly correlated wave­function
JOURNAL OF CHEMICAL PHYSICS 121 (15): 7110­7126 OCT 15 2004
COMPUTER PHYSICS COMMUNICATIONS 169 (1­3): 386­393 JUL 1 2005
A new technique to simulate finite
temperature systems
ARTILE IN PREPARATION
Full optimization during the ion dynamics
RVB phase in High Pressure Hydrogen?
(ARTICLE?)

47. Future Work
(a lot things to do!)
Reduction of the number of parameters
Comparison with other methods (CEIMC ...)
Improving the wave­function optimization
Size effects and TABC
Dynamic properties?
GLQ technique:GLQ technique:
Other:Other:
Simulation of Molecular Hydrogen
QM/MM with GLQ technique to simulate
larger systems (biological)

48. Acknowledgement
Sandro Sorella
Michele Casula

49. Time­Step Error 2