Progress in Calculating the Potential Energy Surface of H 3 +
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Progress in Calculating the Potential Energy Surface of H3
+
Ludwik Adamowicz and Michele Pavanello, Department of Chemistry and Biochemistry
February 9th, 2012
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UA SUPERCOMPUTERSShared Memory - SGI Altix 4700
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New Cluster - SGI Altix ICE 84002112-core, 6-core INTEL Xeon
Soon To Be InstalledSGI Altix UVIBM: DataPlex HTC
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Goal: Accurate PES of H3+
MotivationInterstellar chemistry: (Hn)+
Spectroscopy: H3+
What has been done in the past?
Why Molecules with Hydrogen?
Method Single Point (cm-1) PES (cm-1)CI, (CC), (MPn) 200 – 10 200 – 10
R12 – CI, (CC, MPn) <10 <10
ECGs* < 10-3 0.02Our work ? ?
*Cencek JCP, 108, 2831 (1998) ; Explicitly Correlated Gaussians (ECGs)
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What are ECGs?
Expansion in terms of basis functions
The basis set is made of explicitly correlated Gaussians with floating centers
rrrr MMM gcgcgc 2211
...exp
...exp,...,,2
23232
1212
222
21121
rr
rrrrrgkk
kknek
...exp
...exp,...,,2
23232
1212
2
222
2
11121
rr
SrSrrrrgkk
kkkknek
linear and non-linear parameters
The case of H3+
ner
rr
2
1
r
Atoms
Molecules
jiij rrr
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The cusp condition1. Electron-Nucleous cusp2. Electron-Electron cusp
Zr
Rr21
11
Zr
FRr
211)(
11
r
Kato’s condition*
*T. Kato (1957)
Cusp function:
The derivative of Ψ in this point counts!
Is the nucleous really a point charge?
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Optimization of ФM
EnergyE
GradientG
Determine the step size
and move
PPP
gPPHggPHPggPHgP
gPHgccE
lklklk
M
i
M
jlklk
ˆˆˆ
;ˆˆˆˆˆˆˆˆˆ
;ˆˆ1 1
M
i
M
jlk
ilki
i
jikijnei
ki
gPHgx
ccxG
x
1 1
ˆˆ
,
The step size is determined as a function of G and E.
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• Variational PrincipleMM
MM HE
|rel.non
Does our approach work?Single point calculation
Basis size CPU Time (days)
Energy1
(au)Energy2
(au)150 8 -1.343 835 599 83 -1.343 835 540300 40 -1.343 835 623 08 -1.343 835 615600 8 -1.343 835 624 94 -1.343 835 624
1000 0.1 -1.343 835 625 02 NA
1) Pavanello et. al. J. Chem. Phys., 130, 034104 (2009) 2) Cencek et al. Chem. Phys. Lett. 246, 417-420 (1995)
1. Non-linearity: M*7 parameters2. Encounter linear dependencies
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5 or 6 electrons maximumI. Antisymmetrize electrons: ne! II. Basis set size: M2
Schrödinger EquationI. Born-Oppenheimer approximationII. RelativityIII. Coulomb HamiltonianImplementation – Parallelization – Numerical InstabilityI. Encounter linear dependenciesII. Memory constraints
Limits
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What if we move the geometry?Can we carry out PES calculations?
1. Re-optimize from scratch the basis set for each PES grid point. a. Takes a long time to optimize the basis setb. Hundreds, sometimes thousands of geometries need to be considered
2. Guess the basis set from nearby geometriesa. How?b. Is it precise?c. Is the precision maintained for each grid point?
We need a benchmark!
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Test of the spring method Benchmark PES of H3
+ Generated a 377-point PESThe wavefunction at a certain
geometry was generated from one of a nearby geometry
Pavanello et al. J. Chem. Phys. 130, 001033 (2009)
The spring model
Convergence dictated by the value of the analytical gradient ( GTG < 10-11 a.u. ) and not of
the energy
M=9006300 parameters
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1st benchmark: D3h symmetry
-6 -4 -2 0 2 4 6
-1.36
-1.34
-1.32
-1.3
-1.28
-1.26
-1.24
-1.22
-1.2
-1.18 -6 -4 -2 0 2 4 6
-10
-9.5
-9
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
Total Energy (a.u.) ΔE(a.u. x 10-8)
We notice:Our energies are always 0.01 cm-1 below the
best in the literature.Stretched geometries seem to show better
improvement
The more negative the better!
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2nd benchmark: C2v & asymmetric
-5 -4 -3 -2 -1 0 1 2 3
-10
-9.5
-9
-8.5
-8
-7.5
-7-4 -3 -2 -1 0 1 2 3
-10-9.5
-9-8.5
-8-7.5
-7-6.5
-6-5.5
-5
asymmetric
C2v 3C2v 4
0 1 2 3 4 5 6 7 8
-12
-10
-8
-6
-4
-2
0
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The challenge: a complete PES of H3+
toward sub 0.01 cm-1 accuracy
0 2 4 6 8 10 12
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
11A1
1E1
Tota
l Ene
rgy
(a.u
.)
ρ (a.u.)
R12
R 13
R23
Viegas, Alijah and Varandas, JCP (2007)Johnson, JCP (1980)Whitten and Smith (1968)
sinsin13
34sinsin1
3
34sinsin1
3
22
13
2223
22
12
R
R
R
3hD0sin
Alijah at al. usedMR-CI with cc-pV5Z
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1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
MR-CI vs "exact"1 2 3 4 5 6 7 8
-1.4
-1.35
-1.3
-1.25
-1.2
-1.15
-1.1
-1.05
-1Rij
Ener
gy (a
.u.)
1 2 3 4 5 6 7 8
-1.2E-03
-1.0E-03
-8.0E-04
-6.0E-04
-4.0E-04
-2.0E-04
0.0E+00
Alija
h’s
mos
t diff
use
func
tion
H3+ [H H H]+ 2H+H+
20 c
m-1
Alijah et al.
Our work (ECGs)
Ener
gy D
iffer
ence
(a.u
.)
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Vibrational Wave function Plots
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Conclusions on H3+
We developed:• ECG with analytical gradients, tested on single
point calculations• Spring method to calculate PESs, tested on a 69
point PES portion of H3+
• The code is applicable to any (ne<7) molecular system
We achieved:• Most accurate variational energies to date• Most accurate (≈ 0.01cm-1) and extensive PES
(42000 grid points) of H3+
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Conclusions on H3+
To be developed:• Leading relativistic corrections• Non-adiabatic corrections• Leading QED corrections
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Equivalent treatment of nuclei and electrons in H3
+
The total laboratory-frame nonrelativistic Hamiltonian:
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Separating out the center of mass motion
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The internal Hamiltonian Molecular atom.
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Explicitly Correlated Gaussian Functions for non-BO calculations of H3
+
diatomics
H3+
or
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Expectation values of the ground state non-BO energies, virial coefficients (η) , and internuclear distances for some isotopologues of H3
+ . All values are calculated for an optimized 50 term explicitly correlated Gaussian basis set and are in atomic units.
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Acknowledgements Coworkers:
Pawel Kozlowski Donald Kinghorn Mauricio Cafiero Sergiy Bubin Michele Pavanello Wei-Cheng Tung
Collaborators: Alexander Alijah Nikolai Zobov Irina I. Mizus Oleg Polyansky Jonathan Tennyson
Tamás Szidarovzsky Attila Császár Max Berg Annemieke Petrignani Andreas Wolf
Support: NSF