New Variational Approaches to Excited and (Nearly) Degenerate States in Density Functional Theory

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Helsinki Finland June 22-27 2009
Chair Manuel Yanez
1520 Wednesday June 24 2009
New Variational Approaches to Excited and
(Nearly) Degenerate States in Density Functional
Theory
Tom Ziegler Department of Chemistry University
of Calgary Calgary Alberta Canada
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Excited States
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Basic Ground State KS-theory
Ziegler, et al. J. Chem. Phys. 2009, 130,154102
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Ziegler, et al. J. Chem. Phys. 2009, 130,154102
Basic Ground State KS-theory
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Ziegler, et al. J. Chem. Phys. 2009, 130,154102
Basic Ground State KS-theory
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Electronic Ground-State Hessian
Ziegler, et al. J. Chem. Phys. 2009, 130,154102
Here Uai are our variational parameters
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Electronic Ground-State Hessian
Corresponding to the determinantal wave-function
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Electronic Ground-State Hessian
Ziegler, et al. J. Chem. Phys. 2009, 130,154102
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Stationary Points and Excitation Energies
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Stationary Points and Excitation Energies
Ziegler, et al. J. Chem. Phys. 2009, 130,154102
In the expression for the new density
Only the last two terms give rise to a change in
the energy if ro(1,1) is optimized
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Stationary Points and Excitation Energies
Only the last two terms give rise to a change in
the energy if ro(1,1) is optimized
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Stationary Points and Excitation Energies
Ziegler, et al. J. Chem. Phys. 2009, 130,154102
We have a transfer of charge
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Ziegler, et al. J. Chem. Phys. 2009, 130,154102
Stationary Points and Excitation Energies
and l is a Lagrange multiplier
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Stationary Points and Excitation Energies
Thus the requirement that dL/dUai leads to
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Stationary Points and Excitation Energies
This equation is identical to that obtained from
TD-DFT/TD within the adiabatic approximation
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Ziegler, et al. J. Chem. Phys. 2009, 130,154102
Derivation of transition moments
In wave function mechanics we have from
Raleigh-Schrödinger perturbation theory for the
polarizability tensor
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Derivation of transition moments
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Derivation of transition moments
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Derivation of transition moments
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Charge Transfer
Ziegler, et al. J. Chem. Phys. 2008, 129, 184114
Time-dependent density functional theory (TD-DFT)
at the generalized gradient (GGA) level of
approximation has shown systematic errors in the
calculated excitation energies. This is
especially the case for energies representing
electron transitions between two separated
regions of space or between orbitals of different
spatial extent.
Ground state
Triplet excited state
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Charge Transfer
Ziegler, et al. J. Chem. Phys. 2008, 129, 184114
Ziegler et al. Theochem. 2009
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Ground state
Triplet excited state
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Charge Transfer
Ziegler, et al. J. Chem. Phys. 2008, 129, 184114
Ziegler et al. Theochem. 2009
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Ground state
Triplet excited state
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Charge Transfer in steps
Ziegler et al. Theochem. 2009
Ground state
Triplet excited state
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Charge Transfer in steps
Ziegler et al. Theochem. 2009
II. Ionization
Frozen orbitals
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Charge Transfer in steps
Ziegler et al. Theochem. 2009
III. Electron attachment
Frozen orbitals
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Ziegler et al. Theochem. 2009
Charge Transfer
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Ground state
Triplet excited state
IV. Recombining
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Ziegler et al. Theochem. 2009
Charge Transfer
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Ground state
Triplet excited state
Combining all terms
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Coupling of a Single Occupied/Virtual Orbital
Pair to all Orders.
Ziegler, et al. J. Chem. Phys. 2008, 129, 184114
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KS-Case
We have

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A Revised Electronic Hessian for Approximate
CV-DFT
Ziegler, et al. J. Chem. Phys. 2008, 129, 184114
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A Revised Electronic Hessian for CV-DFT
We now obtain the revised Hessian
where
Ziegler, et al. J. Chem. Phys. 2008, 129, 184114
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A Revised Electronic Hessian for CV-DFT
Revised KS-Case
Triplet excited state
Ground state
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Numerical example
Orbital optimization
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Numerical example
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Numerical example
CV-DFT
TD-DFT
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Excited States
 92nd Canadian Chemistry Conference and
Exhibition in Hamilton, ON, May 30 - June 3,
2009
DFT-methods
2 TD-DFT (Gross,Cassida 1995)
1600 Tuesday June 2 2009 Heritage-Sheridan
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Excited States
 92nd Canadian Chemistry Conference and
Exhibition in Hamilton, ON, May 30 - June 3,
2009
DFT-methods
2 TD-DFT (Gross,Cassida 1995)
Slater Sum rules Ziegler 1976
Spin-Restricted Open-Shell-KS ROKS Filatov 2000
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Excited States
 92nd Canadian Chemistry Conference and
Exhibition in Hamilton, ON, May 30 - June 3,
2009
DFT-methods
2 TD-DFT (Gross,Cassida 1995)
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Excited States
 92nd Canadian Chemistry Conference and
Exhibition in Hamilton, ON, May 30 - June 3,
2009
DFT-methods
III. Density matrix functional theory Baerends
(2009)
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H2 Dissociation
Missing orbital optimization
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Dr. Mike Seth
Dr. Mykhylo Krykunov
Prof. Jochen Autschbach
Prof. Fan Wang
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Concluding Remarks
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Basic Time Dependent Density Functionl Theory
Basic Equation
M.E.Casida
TD-approximation
Gross,E.K. Kohn W.
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The Ground State Hessian
Energy change due to change in density away from
ground state
Change in density away from ground state
Ground state density
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The Ground State Hessian for HF
HF
Change in density away from ground state
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The Ground State Hessian for HF
KS
Change in density away from ground state
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Basic Ground State KS-theory
Normal Restrictions
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Similar Constraint in Wave Function Mechanics
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Similar Constraint in Wave Function Mechanics
To second order