Intro Phenomenology Computational Methods
Excited States:
Phenomenology and Computational Aspects
Felix Plasser
Institute for Theoretical Chemistry, University of Vienna
COLUMBUS in China
Tianjin, October 10–14, 2016
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Intro Phenomenology Computational Methods
Electronically Excited States
What are electronically excited states?
Time-Independent Schrödinger Equation
ĤΨ0 (x1 , x2 , . . .) = E0 Ψ0 (x1 , x2 , . . .)
ĤΨI (x1 , x2 , . . .) = EI ΨI (x1 , x2 , . . .)
E0 Ground state energy
Ψ0 Ground state wavefunction
EI > E0 Excited state energy
ΨI Excited state wavefunction
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Electronically Excited States
What are electronically excited states?
Time-Independent Schrödinger Equation
ĤΨ0 (x1 , x2 , . . .) = E0 Ψ0 (x1 , x2 , . . .)
ĤΨI (x1 , x2 , . . .) = EI ΨI (x1 , x2 , . . .)
I
Ψ0 and ΨI are many-electron wavefunctions
→ How can we discuss them?
I
Usually transitions between individual orbitals
→ Excited state phenomenology
I
Computations challenging
→ Many different computational methods exist
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Intro Phenomenology Computational Methods
Excited State Phenomenology
How can we talk about electronically excited states?
I
Intramolecular excitations
I
I
I
Valence states: ππ ∗ , nπ ∗ , πσ ∗
Rydberg states
Intermolecular excitations
I
I
Charge transfer states
Excitonic states
I
One-electron excited states (single excitations)
I
Two-electron excited states (double excitations)
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Valence states
Valence states
Bonding / non-bonding orbital → antibonding orbital
I
Excitation within the valence space
I
Bonding → antibonding orbital: ππ ∗ , πσ ∗ , σσ ∗
I
Lone pair → antibonding orbital: nπ ∗
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Valence states
ππ ∗ states
I
π-orbital: nodal plane in the
molecular plane
I
Typical UV absorbing states
↑
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Valence states
nπ ∗ states
I
↑
Usually "dark"
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Intro Phenomenology Computational Methods
Valence states
πσ ∗ , σσ ∗ states
I Can lead to dissociation of the
molecule
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Rydberg states
Diffuse s-orbital
Rydberg states
I
Excitations into diffuse orbitals
I
Molecular ion circled by an
electron
↑
→ Similar to "Rydberg series" of the
hydrogen atom
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Rydberg states
Diffuse p-orbitals
↑
↑
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Rydberg states
Rydberg states - in practice
↑
↑
↑
Cutoff 0.04
Cutoff 0.03
Cutoff 0.02
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Charge Transfer States
Charge Transfer States
I
Electron transfer between two
chromophores
I
Approximate energy
ECT ≈ IP + EA −
I
↑
1
R
Strong dependence on the
intermolecular separation!
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Excitons
↑ (58%)
Excitons
I
Excited states in larger systems
I
Coupled local excitations
- many orbitals involved
I
↑ (21%)
Alternative viewpoint:
electron-hole pair
- two-body wavefunction
χexc (xh , xe )
↑ (9%)
1 S. A. Mewes, J.-M. Mewes, A. Dreuw, F. Plasser PCCP 2016, 18, 2548.
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Overview
How can we compute electronically excited states?
I
I
Large number of computational methods
Different characteristics:
I
I
I
Single reference vs. multireference
Inclusion of dynamic/nondynamic correlation
Size consistent/extensive?
Configuration interaction vs. response theory
- Ground state at the same level as excited state?
I State specific vs. multistate
- Does the number of states computed affect the outcome?
I Highest possible excitation level
I Specific strengths and weaknesses
I Availability of transition moments / gradients / nonadiabatic couplings
I
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Single reference vs. multireference
Single reference
Multireference
I
Only one reference determinant
Φref
I
Excited states are excitations out
of this determinant
!
X
†
ΨI =
dpq âp âq + . . . Φref
pq
,
,
/
Easy to use
Easy to interpret
I
,
Several reference determinants
General applicability
→ Open-shell ground states
→ Two-electron excitations
→ Conical intersections (S0 /S1 )
/
/
Computationally expensive
More difficult to apply
Limited applicability
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Dynamic/nondynamic correlation
Different types of correlation for excited states
I
Dynamic correlation
→ Different types of states affected differently!
I
Static correlation - open-shell spin states
→ Singlet: 1 Φai = 2−1/2 (Φai + Φāī )
→ Single determinant desciption qualitatively wrong for excited states!
I
Nondynamic correlation - multiconfigurational excited states
→ Excitonic correlation/electron-hole entanglement1
→ Single configurational description often not possible for excited states
All methods discussed later allow for static and nondynamic correlation in the
excited state!
1 F. Plasser J. Chem. Phys. 144, 194107.
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Size consistent/extensive
I
Do the excitation energies change if a non-interacting molecule is added to
the calculation? - size consistency
I
Is the correlation energy and extensive property of the system?
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Configuration interaction vs. response theory
How are the working equations derived?
Configuration interaction
I
I
,
Problem expanded in terms of
configurations
Response methods, etc.
Extended physical model
I
Diagonalization
Ground and excited states treated
at the same level
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Time-dependent perturbation
theory
- Linear response, quadr. resp.
I
Equaton of motion
I
Polarization propagator
/
Distinct reference state
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State specific vs. multistate
State specific
I
/
/
Multistate
Independent equations for each
excited state
Incorrect intersection topology
No transition moments or
couplings
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I
/
States treated simultaneously
Number of states can affect the
results of the lower states
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Excitation level
I
Can two-electron excitations be described with the same accuracy as
one-electron excitations?
I
Problem for single reference methods
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TDDFT
Time-dependent density functional theory
I
Single reference - Kohn-Sham determinant
I
Linear response theory
I
Dynamic correlation (implicit) + “trivial” nondynamic correlation
I
One-electron excitations
/
/
Problems for charge transfer and Rydberg states
Strong dependence on the functional chosen
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TDDFT
Response equation
A
B
B
X
1
=ω
A
Y
0
0
−1
X
Y
ω Excitation energy
X Excitation vector
Y De-excitation vector
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Coupled cluster
Coupled cluster
I
So far only single reference CC in common use
I
Different ways to construct excited states
I
Linear response theory: CC2, CC3
I
Equaton-of-motion: EOM-EE-CCSD, EOM-EE-CCSD(T)
,
,
Size-consistent
Tuning between computational efficiency and treatment of electron
correlation possible
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ADC
Algebraic diagrammatic construction (ADC) method for the polarization
propagator
I
,
,
I
Different derivation but similar properties to CC
Size-consistent
Hermitian eigenvalue problem
ADC(2) similar to CC2
- Excited state analogues to MP2
I
ADC(3) cheaper than CC3, but similar quality(?)
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Spin flip methods
Spin flip methods
I
Idea: start from a triplet determinant
→ Flip the spin to construct a singlet
I
,
,
/
/
Available for CC, ADC, ...
Ground and excited states resulting from the same calculation
Multiconfigurational ground states and two-electron excitations
accessible in a single reference treatment
Spin-contamination
Only HOMO and LUMO
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MCSCF
Multiconfiguration self-consistent field (MCSCF)
I
Select a set of configurations
→ Optimize CI coefficients and orbital coefficients at the same time
I
Nondynamic correlation, but no dynamic correlation
MCSCF
ΨMCSCF (C, d) =
X
di Φi (C)
i
di CI coefficient
d CI vector
C orbital coefficient matrix
Φi Slater determinant or configuration state function
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MCSCF
How to choose the active space?
- Many options available!
Complete active space (CAS)
I
I
I
CAS(m, n) - distribute m electrons over n orbitals
Consider all possibilities (for a given spin and spatial symmetry)
I
Restricted active space (RAS)
I
Perfect pairing, generalized valence bond
→ Choice according to chemical intuition and experience
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Intro Phenomenology Computational Methods
MCSCF
State specific
I
/
State averaging
Optimize each state independently
States and orbitals are not
orthogonal
I
,
,
/
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Orbitals minimize the average
energy of several states
Transition moments and nonad.
couplings
Consistent intersection topology
Number of states (and their
weights) affects the outcome for
all states!
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MCSCF
I
/
,
The MCSCF energy E(C, d) is a nonlinear function
Local minima in parameter space!
Local minima identified by orbital rotations
→ Orbitals can be manually rearranged
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MR-CI
Multireference configuration interaction (MR-CI)
I
,
/
Add dynamic correlation to an MCSCF calculation
Dynamic and nondynamic correlation
Not size-consistent (but corrected versions exist)
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MR-CI
Uncontracted
I
,
I
Internally contracted
Excite every reference
configuration individually
I
Gradients and nonadiabatic
couplings
COLUMBUS
,
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Excite the MCSCF wavefunction
as a whole
Ψ
MRCI =
P
†
pq dpq âp âq + . . . ΨMCSCF
Smaller wavefunction expansion
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Perturbation theory
Multireference perturbation theory
I
CASPT2
- Single state vs. multistate
- Internally contracted vs. uncontracted
,
/
I
(Approximately) size-consistent
For practical applications: level shifts, IPEA shift, ...
Other “flavors”: NEVPT2, XMCQDPT2, ...
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x
x
x
x
x
(x)
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(x)
(x)
()
(x)
x
x
(x)
Indep. of num. states
x
x
Nonad. coupl.
(x)
(x)
x
Gradients
2-electron exci.
x
x
Con. int. S1 /S0
()
Con. int. Sn /Sn−1
SR
SR
SR
MR
MR
MR
MR
MR
Size-consistent
DFT
CC/ADC
Spin-flip
SS-MCSCF
SA-MCSCF
MR-CI
SS-CASPT2
MS-CASPT2
Dyn. Corr.
SR/MR
Summary
x
()
x
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