Putting a spin on time dependent electronic structure ... · of Electronic Excited State Methods...
Transcript of Putting a spin on time dependent electronic structure ... · of Electronic Excited State Methods...
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Putting a spin on time dependent electronic structure theory!
!Joshua Goings!
General Exam!!
Thursday, May 14, 10:00am CHB 339
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Thank you to General Exam Committee
Xiaosong Li (chair)!Jim Pfaendtner (GSR)!Matthew Bush!David Masiello!Stefan Stoll!!
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How can we predict and understand the electronic and magnetic responses
of molecules and nano-materials?
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Research Directions !(2014—2015)
Quantum !Dots
Excited !States
Magnetic !Materials
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Research Directions (2014—2015)
Excited !States
Quantum !Dots
Magnetic !Materials
Other
1. Goings, J. J., Caricato, M., Frisch, M. J., & Li, X. (2014). JCP, 141(16), 164116.
2. Goings, J. J., Schimpf, A. M., May, J. W., Johns, R. W., ! ! Gamelin, D. R., & Li, X. (2014). JPC C, 118(46), 26584.
3. Ding, F., Goings, J. J., Frisch, M. J., & Li, X. (2014). JCP, 141(21), 214111.4. Goings, J. J., Ding, F., Frisch, M. J., & Li, X. (2015). JCP, 142(15), 154109.
5. Goings, J. J., Ohlsen, S. M., Blaisdell, K. M., & Schofield, D. P. (2014). ! JPC A, 118(35), 7411.6. Goings, J. J., Ding, F., & Li, X. (2014). Proceedings of MEST 2012: ! Electronic Structure Methods with Applications to Experimental ! Chemistry, 68, 77.
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Research Directions (2014—2015)
Excited !States
Quantum !Dots
Magnetic !Materials
Other
1. Goings, J. J., Caricato, M., Frisch, M. J., & Li, X. (2014). JCP, 141(16), 164116.
2. Goings, J. J., Schimpf, A. M., May, J. W., Johns, R. W., ! ! Gamelin, D. R., & Li, X. (2014). JPC C, 118(46), 26584.
3. Ding, F., Goings, J. J., Frisch, M. J., & Li, X. (2014). JCP, 141(21), 214111.4. Goings, J. J., Ding, F., Frisch, M. J., & Li, X. (2015). JCP, 142(15), 154109.
5. Goings, J. J., Ohlsen, S. M., Blaisdell, K. M., & Schofield, D. P. (2014). ! JPC A, 118(35), 7411.6. Goings, J. J., Ding, F., & Li, X. (2014). Proceedings of MEST 2012: ! Electronic Structure Methods with Applications to Experimental ! Chemistry, 68, 77.
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Research Directions !(2014—2015)
Quantum !Dots
Excited !States
Magnetic !Materials
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Research Directions !(2014—2015)
Quantum !Dots
Excited !States
Magnetic !Materials
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Excited !States
Goings, J. J., Caricato, M., Frisch, M. J., & Li, X. (2014). Assessment of low-scaling approximations to the equation of motion coupled-cluster singles and doubles equations. JCP, 141(16), 164116.
Balancing Cost + Accuracy !of Electronic Excited State Methods
• O(N4) !
• Unpredictable !
• No excited state exchange kernel
!• Limited to single
electron phenomena
• O(N6) !
• Systematically improvable
!• Excited state
electron correlation
!• Multi-electron
phenomena
Balance cost + accuracy!
!(P)-EOM-MBPT2,!
CC2, etc.!
EOM-CCSD!LR-TDDFT!
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Excited !States
Goings, J. J., Caricato, M., Frisch, M. J., & Li, X. (2014). Assessment of low-scaling approximations to the equation of motion coupled-cluster singles and doubles equations. JCP, 141(16), 164116.
Balancing Cost + Accuracy !of Electronic Excited State Methods
• Applied perturbation theory to coupled cluster (CC) equations!
!• Reduced computational time
of CC equations by an order of magnitude!
!• Accuracy generally
outperforms density functional theory
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Research Directions !(2014—2015)
Quantum !Dots
Excited !States
Magnetic !Materials
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Quantum !Dots
Goings, J. J., Schimpf, A. M., May, J. W., Johns, R. W., Gamelin, D. R., & Li, X. (2014). Theoretical Characterization of Conduction-Band Electrons in Photodoped and Aluminum-Doped Zinc Oxide (AZO) Quantum Dots. JPC C, 118(46), 26584-26590.
Understanding Dopant Influence !on Excitations in n-type Quantum Dots
Al3+:ZnO e-:ZnOextra conduction
band electronextra conduction
band electrontough to oxidize easy to oxidize
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Quantum !Dots
Goings, J. J., Schimpf, A. M., May, J. W., Johns, R. W., Gamelin, D. R., & Li, X. (2014). Theoretical Characterization of Conduction-Band Electrons in Photodoped and Aluminum-Doped Zinc Oxide (AZO) Quantum Dots. JPC C, 118(46), 26584-26590.
Understanding Dopant Influence !on Excitations in n-type Quantum Dots
• n-type ZnO QDs!!• UV-Vis spectra from
aluminum doped and photodoped QDs!
!• Rationalized theoretical/
experimental results in terms of particle-in-a-sphere
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Quantum !Dots
Goings, J. J., Schimpf, A. M., May, J. W., Johns, R. W., Gamelin, D. R., & Li, X. (2014). Theoretical Characterization of Conduction-Band Electrons in Photodoped and Aluminum-Doped Zinc Oxide (AZO) Quantum Dots. JPC C, 118(46), 26584-26590.
Understanding Dopant Influence !on Excitations in n-type Quantum Dots
• n-type ZnO QDs!!• UV-Vis spectra from
aluminum doped and photodoped QDs!
!• Rationalized theoretical/
experimental results in terms of particle-in-a-sphere
S P D
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Research Directions !(2014—2015)
Quantum !Dots
Excited !States
Magnetic !Materials
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Magnetic !Materials
Goings, J. J., Ding, F., Frisch, M. J., & Li, X. (2015). Stability of the complex generalized Hartree-Fock equations. The Journal of chemical physics, 142(15), 154109.
Exploring spin-frustrated molecules !with the generalized Hartree Fock (GHF) method
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Magnetic !Materials
Take a three site lattice
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Magnetic !Materials
Now add two electrons!(assume antiferromagnetism is favored)
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Magnetic !Materials
?
Now add the third electron.!No orientation simultaneously favors !
all anti-ferromagnetic interactions.
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Magnetic !Materials
Collinear Non-collinearConventional methods force you to pick a collinear configuration.!
Few methods can give you non-collinear state.
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Magnetic !Materials
S Lounis. “Non-collinear magnetism induced by frustration in transition-metal nanostructures deposited on surfaces”. JPCM 26 (2014) 273201.
via STMCr3
Ag fcc(111) surface
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Magnetic !Materials
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Magnetic !Materials
Can we describe this with Hartree Fock (HF)?
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Magnetic !Materials
Hartree-Fock (HF): minimize the energy of a single Slater Determinant
E h�|H|�ih�|�i
The solution is variational;! an upper bound to the exact energy.
This is an independent particle model (IPM).!It is the quantitative basis of molecular orbital theory.
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Magnetic !Materials Types of Hartree-Fock
Restricted (RHF) Unrestricted (UHF) General (GHF)
orbital spatial
yes no no
yes yes no
system closed shell open shell spin frustrated
�(r)↵(!) + �(r)�(!)
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Magnetic !Materials Generalized Hartree Fock (GHF)
• GHF allows you to change spin smoothly!!• Lowest energy HF solution possible!!• Largely insensitive to guess multiplicity!!• Only HF method to treat spin frustration
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Magnetic !Materials Generalized Hartree Fock (GHF)
• GHF allows you to change spin smoothly!!• Lowest energy HF solution possible!!• Largely insensitive to guess multiplicity!!• Only HF method to treat spin frustration
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Magnetic !Materials Generalized Hartree Fock (GHF)
• GHF allows you to change spin smoothly!!• Lowest energy HF solution possible!!• Largely insensitive to guess multiplicity!!• Only HF method to treat spin frustration
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Magnetic !Materials Generalized Hartree Fock (GHF)
• GHF allows you to change spin smoothly!!• Lowest energy HF solution possible!!• Largely insensitive to guess multiplicity!!• Only HF method to treat spin frustration
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Magnetic !Materials
Problem: !Just because you can get the lowest energy
solution, doesn’t mean you will.
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Magnetic !Materials How to obtain GHF local minima?
Goings, J. J., Ding, F., Frisch, M. J., & Li, X. (2015). Stability of the complex generalized Hartree-Fock equations. The Journal of chemical physics, 142(15), 154109.
stableunstable
E
coefficient
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Magnetic !Materials How to obtain GHF local minima?
We want stable electronic solutions to the GHF model.
Local minima:!!(a) First variation equal
to zero!(b) Second variation
greater than zero
Goings, J. J., Ding, F., Frisch, M. J., & Li, X. (2015). Stability of the complex generalized Hartree-Fock equations. The Journal of chemical physics, 142(15), 154109.
stableunstable
E
coefficient
(a) = 0!(b) < 0
(a) = 0!(b) > 0
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Magnetic !Materials
@E
@qai=
✓h�a
i |H|0ih0|H|�a
i i
◆=
✓faifia
◆
Differentiate E with respect to! wave function coefficients
This must equal zero, known as Brillouin’s theorem.
(Trivially satisfied as long as HF converges to something)
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Magnetic !Materials
@2E
@qai @qbj
=
h�a
i |H|�bji h�ab
ij |H|0ih0|H|�ab
ij i h�bj |H|�a
i i
!
Second derivative of energy !with respect to wave function coefficients
The Hessian, which must be positive definite if our solution is to be a minimum.
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✓A BB⇤ A⇤
◆
Aia,jb = (✏a � ✏i)�ia,jb + haj||ibi, Bia,jb = hab||iji
Magnetic !Materials
Hessian =
Compute eigenpairs of Hessian matrix.
If all eigenvalues are positive, we are at minima
If any eigenvalues are negative, !lower energy solution exists.
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Magnetic !Materials What if we run into instability?
1. Take the most negative eigenvalue and eigenvector!2. Step wave function in direction of eigenvector!3. Re-optimize GHF solution.
Steepest Descent Method
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Magnetic !Materials
Full details and working equations in the paper!
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Magnetic !Materials
Looking for lowest energy solutions !in spin frustrated rings
0.00 kcal mol-1
-0.49 kcal mol-1
-7.52 kcal mol-1
Cr3 : high-spin solutions unstable.!Lower energy non-collinear solution exists.
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Magnetic !Materials
GHF UHF
Hydrogen rings
for odd-member! rings, GHF
lowest energy solution
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Magnetic !Materials In summary:
GHF allows for the lowest-energy HF solution !!Can determine multiplicity without user input!!Can handle spin frustrated systems!!Loses all good spin quantum numbers
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Future outlook
GHF can handle spin transitions, !but lacks any spin operators !
(only Coulomb exchange interaction)
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Ding, F., Goings, J. J., Frisch, M. J., & Li, X. (2014). Ab initio non-relativistic spin dynamics. The Journal of chemical physics, 141(21), 214111.
Including Arbitrary Magnetic Fields !into Ab Initio Electron Dynamics
Initial Magnetization Time Evolution (ps)
• Arbitrary magnetic field!!• Magnetic moments
precess with field!!• No spin coupling!
Neutral Li3 trimer, 20T field perpendicular to plane
RT-TD-GHF
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Observation: TD-GHF allows smooth evolution of spin states, but lacks spin coupling operators
Can we extend this description by adding explicit spin operators to the TD-GHF description?
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Spin is a non-classical effect. !It arises from relativistic quantum mechanics.
V c� ·⇧
c� ·⇧ V� 2c2
� L
S
�= E
L
S
�
One electron, Dirac equation:
This replaces the one electron operators in GHF
L =
L↵
L�
� S =
S↵
S�
�Note four
component:
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V12 =q1q2r12
� q1q22
↵1 ·↵2
r12+
(r12 ·↵1) (r12 ·↵2)
r312
�
For multiple electrons, we have the approximate interaction operator—the Breit operator
This replaces the two electron Coulomb operator
↵ =
✓02 �� 02
◆
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V12 =q1q2r12
� q1q22
↵1 ·↵2
r12+
(r12 ·↵1) (r12 ·↵2)
r312
�
For multiple electrons, we have the approximate interaction operator—the Breit operator
This replaces the two electron Coulomb operator
↵ =
✓02 �� 02
◆
(Coulomb) (Breit correction, note spin-dependence)
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We can reduce to two component form
HLL HLS
HSL HSS
� L
S
�
U
HLL HLS
HSL HSS
�U�1 !
HLL 00 HSS
�
Usually approximate
Off diagonal terms zero to some order
Order 1/c gives Breit-Pauli!Order V gives Douglas-Kroll-Hess L =
L↵
L�
�
S =
S↵
S�
�
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Uncertain best way to include these !operators into the TDSE
Four Component Breit-Pauli !(two component, 1/c)
Douglas-Kroll-Hess!
(two component,V)
Naive cost?
Variational? No (yes, in practice) No Yes
Interpretation?Must account for
positron-like component
Clear relation to spin-Hamiltonian Spin-terms mixed
2(2N)42(4N)4 2(2N)4
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Future Roadmap
Time !Domain
DKH
4c
BreitProperty!Transf.
spin-spin
cheaper,!harder to code
expensive,!simpler
picture-!change
Spin-dependent electronic dynamics!Heavy-element response properties
retardation effects
response properties,!physical observables
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How can we predict and understand the electronic and magnetic responses
of molecules and nano-materials?
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Maybe settle for atoms at the moment?
TD-DKHSodium D-lines (daug-cc-pVTZ)
Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2014). NIST Atomic Spectra Database (ver. 5.2), [Online]
1c-TD-DKH 2c-TD-DKH Exp
2S1/2 !2 P3/2
2S1/2 !2 P1/2
Splitting
2.10232.1044
1.97331.9733
1.97301.9736
0.0000 0.0006 0.0021
[eV]
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/Nadoub.gif
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Maybe settle for atoms at the moment?
TD-DKHSodium D-lines (daug-cc-pVTZ)
Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2014). NIST Atomic Spectra Database (ver. 5.2), [Online]
1c-TD-DKH 2c-TD-DKH Exp
2S1/2 !2 P3/2
2S1/2 !2 P1/2
Splitting
2.10232.1044
1.97331.9733
1.97301.9736
0.0000 0.0006 0.0021
[eV]
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/Nadoub.gif
A lot more more needs to be done!
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Excited !States
Quantum !Dots
Magnetic !Materials
Spin-dependent dynamics and response theory
Future work
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Spin-dependent dynamics and response theory
Future work
Spin in Time-Dependent
Theory
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Thank you!!!
Li Group!Exam Committee!Ernest Davidson
Excited !States
Quantum !Dots
Magnetic !Materials
Spin in Time-Dependent
Theory
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Coupling between charge carriers and Mn2+ in ZnO QDs
holes in p-type dopants couple with Mn2+ dimer !
May, Joseph W., Ryan J. McMorris, and Xiaosong Li. JPCL 3.10 (2012): 1374-1380.
exciton couples to Mn2+ in ZnO QDs!Norberg, Nick S., et al. JACS 126.30 (2004): 9387-9398.
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It seems reasonable that excess CB electrons would couple to Mn
Al3+:ZnO QDs; no evidence excess electrons couple with Mn
Inte
grat
ed M
CD
Inte
nsity
6543210Field (T)
0.2
0.1
0.0
Abso
rban
ce
200
150
100
50
0
(mde
g)
20001800160014001200Wavelength (nm)
6 T
295 K
1.8 K
100
80
60
40
20
0
(mde
g)
20001800160014001200Wavelength (nm)
3 T1.8 K5 K10 K20 K
Inte
grat
ed M
CD
Inte
nsity
6543210Field (T)
0.2
0.1
0.0
Abso
rban
ce
200
150
100
50
0
(mde
g)
20001800160014001200Wavelength (nm)
6 T
295 K
1.8 K
100
80
60
40
20
0
(mde
g)
20001800160014001200Wavelength (nm)
3 T1.8 K5 K10 K20 K
Data courtesy !Dr. Alina Schimpf
Inte
grat
ed M
CD
Inte
nsity
6543210Field (T)
0.2
0.1
0.0
Abso
rban
ce
200
150
100
50
0
(mde
g)
20001800160014001200Wavelength (nm)
6 T
295 K
1.8 K
100
80
60
40
20
0
(mde
g)20001800160014001200
Wavelength (nm)
3 T1.8 K5 K10 K20 K
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Of course, similar story with photodoped e-:ZnO
All systems studied so far are heavily doped
Schimpf, A. M., Thakkar, N., Gunthardt, C. E., Masiello, D. J., & Gamelin, D. R.
(2013). ACS Nano, 8(1), 1065-1072.
Currently, Gamelin group looking at low-carrier
concentrations
Can we help explain this with spin dependent electronic structure theory?