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Reduced Explicitly Correlated Hartree-‐Fock in the Nuclear Electronic Orbital framework applied to positronic atoms
Kurt Brorsen PI: Sharon Hammes-‐Schiffer Blue Waters Symposium
June 13th, 2016
Positrons
• AnOparOcle of electron, e+ • Experimentally observed in 1932 by Carl David Anderson
• ProducOon by emission by β+ decay
!1223Mg→ 11
23Na+e+ + ve
Positrons
• AnnihilaOon of electron-‐positron pair – At low energies normally results in the producOon of 2 γ rays of 0.511 MeV
• Positronium – Bound atom like electron-‐positron pair
Positrons
• Positron emission tomography – Images of metabolic processes – Fluorodeoxyglucose (18F)
• Positron emission spectroscopy – Observe voids and defects in solids – AnnihilaOon rate of Ps is environment dependent
CalculaOons on positronic and positronium-‐ atoms
• Many different methods depending on the atom – SVM, SVMFC, CI∞FC, MBPT – Mean-‐field and DFT approaches have not predicted binding
• Two channels to calculate binding energies for positronic atoms
• Theory is currently ahead of experiment for atomic systems
Cheng, Babikov, Schrader. PRA, 2011
!
e+A→ e+ + Ae+A→ Ps + A+
Experimental Positron Binding • Current
disconnect between experiments and computaOon for systems with a positron – Experiments are very difficult
• Emphasis on annihilaOon rates that are enhanced by vibraOonal Feshbach resonances
Gribakin, Young, Surko. Reviews of Modern Physics 2010.
Class Molecule Binding Energy (meV) Small inorganics NH3 >0
CH3F >0 Methyl halides CH3Br 40
C2H6 >0 Alkanes C3H8 10
C6H4 80 C12H26 220 CH3OH >0 C2H5OH 45
Alcohols C6H6 150 C10H8 300
Previous calculaOons on positronic molecules
• High level calculaOons for diatomics and triatomics
• Ab ini4o calculaOons with mean-‐field reference for amino acids and DNA base pairs
• Molecules with a dipole moment greater than 1.625 debye are predicted to bind a positron
• No explicit electron-‐positron correlaOon for larger molecules – Failure of ab ini4o methods for alkane molecules Koyanagi, et al. PCCP, 2013
Charry, et al. PRA, 2014
Nuclear Electronic Orbital Reduced Explicitly Correlated Hartree-‐Fock (NEO-‐RXCHF)
• N electrons, 1 QM nucleus/positron, Nr regular electrons, Ns special electrons
Sirjoosingh, Pak, Swalina, Hammes-‐Schiffer, JCP 2013
ΨRXCHF x1e ,…x N
e ,x p( ) = ΨRXCHF x1r ,…x Nr
r ,x1s ,…x Ns
s ,x p( )= Φe,r x1
r ,…x Nr
r( )Φe,s x1s ,…x Ns
s( )χ p x p( )G r1s ,…rNs
s ,r p( ),
G r1
s ,…rNs
s ,r p( ) = g ris ,r p( )
i=1
Ns
∑ g ri
s ,r p( ) = bke−γ k ri
s−r p 2
k=1
Ngem
∑
F – H – F !
N ≡ C – H !
!
NEO RXCHF • Apply variational conditions to wavefunction ansatz
approximate exchange, if needed
ERXCHF-ne =ΨRXCHF H ΨRXCHF
ΨRXCHF ΨRXCHF
= ERHF + EXCHF + E int
ERXCHF-ae =ERXCHF-ne +Eex ≡ EHF+ EXCHF+ Eint−ae
FrCr = SeCrEr
FsCs = SeCsEs
F pC p = S pC pE p
HNEO = − 12
∇i2
i
Ne
∑ −ZA
| rie − rA
c |A
Nc
∑i
Ne
∑ + 1| ri
e − rje |i> j
Ne
∑
− 12mp
∇ ′i2
′i
Np
∑ +ZA
| r ′ip − rA
c |+ 1
| r ′ip − r ′j
p |′i > ′j
Np
∑A
Nc
∑′i
Np
∑
− 1| ri
e − r ′ip |i
Ne
∑′i
Np
∑
RXCHF • High computaOonal expense
– Two-‐parOcle integrals for Hatree-‐Fock
– Three-‐, four-‐, and five-‐ parOcle integrals for RXCHF
– HCN, 6-‐31G(d,p) basis, # of non-‐permutaOonal unique integrals
• RXCHF: ~1.5*1012 integrals • HF: ~1.5*106 integrals • Previously all integrals stored in memory (> 12 TB)
!!!
Ω3 p,1,2,3( ) = g r1s ,rp( ) hs r3s( )+Vep r3s ,r p( )+Vee r1s ,r3s( )+Vee r2s ,r3s( )⎡⎣
⎤⎦ g r2
s ,rp( )+g r1
s ,rp( )Vee r2s ,r3s( )g r1s ,rp( )
!!µν 1
r12λσ
! µνχδ Ω3 λστυ = ρµλρνσ ρχτ ρδυΩ3∫
• Last year: HCN, electronic pc-‐0 basis, 11s proton basis – 5 hours on 128 nodes.
• Faster integral code – ~105-‐106 speed up – HCN
• Last year’s calculaOon now in 6 seconds on 16 nodes • Direct algorithm
– Compute integrals on-‐the-‐fly instead of storing them in memory
• Memory required now similar to a Hatree-‐Fock calculaOon • HCN, 6-‐31G(d,p) electronic basis, 5s2p2d proton basis
– 39 minutes on 32 nodes
Recent RXCHF Improvements
RXCHF Positronic Atom CalculaOons
• Alkali and alkaline earth metal atoms – Couple 1 spin orbital for alkali – Couple 2 spin orbitals for alkaline
• Standard electronic basis sets • S-‐type basis sets for positron
– Diffuse – Even-‐tempered
• Different ansatz for correlaOon factor – G for alkali – 1+G for alkaline
RXCHF Positronic Atom Binding Energy Elec Basis Positron Basis Binding Energy (eV) Best Computed Value (eV)
Single Coupled G ansatz
e+Li -‐0.0675 aug-‐pc-‐seg-‐1 8s -‐0.191 aug-‐pc-‐seg-‐2 8s -‐0.157 aug-‐pc-‐seg-‐3 8s -‐0.133
e+Na aug-‐pc-‐seg-‐0 7s -‐0.041 -‐0.0129 aug-‐pc-‐seg-‐2 10s -‐0.115
Double Coupled 1+G ansatz
e+Be -‐0.086 aug-‐pc-‐seg-‐0 6s -‐0.102 aug-‐pc-‐seg-‐1 6s -‐0.082
e+Mg -‐0.464 aug-‐pc-‐seg-‐0 8s -‐0.349 aug-‐pc-‐seg-‐1 8s -‐0.341
RXCHF Future
• Positronium Atoms – Larger basis sets than positronic atoms
• Molecules – Alkanes, amino acids, DNA base pairs
• Add electron-‐electron correlaOon – CI type expansion for explicitly correlated electrons and positrons
– MulO-‐component DFT
Acknowledgements
• UIUC – Sharon Hammes-‐Schiffer
– Mike Pak • Stanford
– Todd MarOnez – Andrew Komornicki
• NSF • Blue Waters