Quantum Monte Carlo for
“Difficult” Systems in Materials
Chemistry
Ainsley A. Gibson
Howard University
Washington, DC 20059
Black Box Computing
• Pros:
– Widespread adoption of techniques
– Relative ease of use
– Always gets a number as output
• Cons:
– Often promotes misconceptions
– Usually no error estimation
– Always gets a number as output
State-of-the-Art Computing
• Pros:
– End results are well-analyzed
– Results are frequently great!
– Near-complete explanation
• Cons:
– Expensive (human, not CPU) cost
– Not for everyone
– Potentially highly selective
“Golden Box” Computing
• Lies somewhere between black box and state
of the art.
• Use of high level techniques in a generalized
form.
• Tradeoff between high accuracy/high
expertise and variable accuracy/low
expertise.
Method
Electron Correlation,
in Principle
Electron Correlation,
in Practice
Density Functional
Theory
Characteristic density and
exact density functional
recover system’s properties
Exact functional unknown,
functionals generated by fit
to experiment or theory
Traditional ab initio
post-HF methods
Infinite excitations from
reference state(s) provide
approximation from oneelectron basis
Truncated number of
excitation types; selected
reference state(s) used
Quantum Monte
Carlo (QMC)
Random sampling of
wavefunction-based
probability in real 3dimensional space
Explicit inter-particle
interaction added to
independent-particle trial
functions
Method
Pros
Cons
Density Functional
Theory
Inexpensive, and
functionals exist that are
well-tuned to specific
chemistries
No hierarchy of functionals,
low to medium accuracy
Traditional ab initio
post-HF methods
High to very high accuracy;
has a hierarchy of methods
Moderately expensive to
extremely expensive, may
fail regardless
Quantum Monte
Carlo (QMC)
Massively parallel, very
high accuracy, simple error
estimation, and simple
excited state energies
Expensive to very
expensive, small energy
differences challenging
This Work…
• There is a significant degree of “art” in QMC
calculations, due to the lack of strict
restriction on trial function form.
• We wish to determine the degree of
necessary “art” in trial function form.
• We also wish to retain the ability to accurately
describe “difficult” systems.
Difficult? This isn’t rocket science…
• Typical “difficult” systems have:
– Ionized or excited states
– Radical or metallic character
– Significant delocalization or resonance
• More broadly, “difficult” systems require
use of an atypical variant of the
technique that need not be used for
95% of chemical systems.
Applications
• Atomic Excited States
• Beryllium Dimer
• Nanoscale Ternary Compounds (HU-CREST)
• Transition Metal Energetics (AHPCRC)
• Atmospherically Interesting
Atomic Excited States
• Simple test of ability to describe electronic
structure
• Some reactions require accurate description
of excited states
• “Proof of capability” study for future
applications to molecular systems
Atomic Excited States
Beryllium Dimer
• Poorly described by simpler traditional basis
set ab initio techniques.
• Multi-reference character due to 2s-2p near
degeneracy.
• Motivated by prior success with atomic
excited states.
• A few-electron system amenable to allelectron fixed-node DMC.
Dissociation Energies, in cm-1
Method
MRSDCI1
R12MRCI, MRACPF2
estimated FCI/cc-pV5Z3
“extensive” ab initio3
VMC/CASSCF(4,8) 90%
VMC/CASSCF(4,8) 95%
DMC/CASSCF(4,8) 90%
DMC/CASSCF(4,8) 95%
Experiment4
De, cm-1
1049
898(8)
803.61 - 822.71
944(25)
-867(71)
-7422(152)
1293(52)
829(64)
839(10)
Nanoscale Ternary Compounds
• Formation of novel compounds at the
•
•
nanoscale have been proposed.
The reactions use carbon and oxygen in the
presence of a nitrogen plasma.
We propose to predict some basic properties
of proposed reactions and compounds using
QMC techniques.
Higher Excited States
•
•
•
•
Reactions proposed may proceed through
excited and/or ionized states.
QMC offers the allure of unprecedented
accuracy for ionized and excited species.
QMC is generalized for any electronic state.
The higher states of nitrogen are first in a
series of excited state calculations.
Nitrogen Excited States
4P
CISD/
cc-pVTZ
VMC
DMC
Exp’t
2D
2.9227
2.50(3)
2.43(4)
2.3835
2P
3.1458
3.34(4)
3.45(4)
3.5756
(2s2p4) 10.6570
10.95(6) 10.84(4) 10.9239
Transition Metals
• When carefully chosen, there are methods
able to describe selected metallic systems.
• Satisfaction with price, performance and
general applicability is elusive.
• QMC shows promise for metallic systems,
and has three features in its favor:
– System-independent methodology
– Consistent error estimates
– Ideal for HPC environments
X-alpha, LSDA, and PL density functionals
B- functionals: B971, BLYP and BPW91
B1- functionals: B1B95 and B1LYP
B3- functionals: B3P86, B3PW91, B3LYP
Selected post-HF Ionization Potentials
QMC/HF Ionization Potentials
2.67,
VMC
1.52,
DMC
QMC/NO Ionization Potentials
1.71,
VMC
1.57,
DMC
Ozone Dissociation Energy
• Traditional ab initio has difficulties:
– Resonance character of ozone
– Low-lying excited state contributions
• Estimates of the dissociation limit are
relatively small (1.02 – 1.13 eV).
• Various excited states lie above and
below the dissociation limit.
Results to Date, in eV
MRCI
0.943
MRCI+Q
1.049
VMC
0.70(4)
DMC
1.06(16)
Exp.
1.0625(4)
Exp.
1.132(1)
HU-CREST Current Work:
Novel Nanoscale Compounds
• Characterization of excited states:
– CN, CO, NO, N2, C2, O2
– ONC, OCN
• Energetic profile of proposed reactions
• Large-scale network compounds
AHPCRC Current Work:
Transition Metals
•
•
•
•
•
Electron Affinity
Proton Affinity
Small Clusters, Mx, x = 2,…,10
Surfaces and solids
Silver nanoparticle stability (collaborative with
CREST)
Future/Current Work:
Atmospherically Interesting
• Ozone dissociation and excited state
characterization
• S4 inter-conversion energetics
• Excited and ionized states of binary (O, N, C)
compounds as atmospheric species
Acknowlegements
•
•
•
•
•
•
•
•
John A. W. Harkless*
William Lester, Jr.
James Mitchell
William Hercules
Floyd Fayton
Gordon Taylor
José González
Mike Towler
• NSF CREST Center for
Nanomaterials
Characterization and
Design
• Army High Performance
Computing Research
Center
• Computer Learning and
Design Center
• TTI