QQ--ChemChem WorkshopWorkshop

APCTCC6, APCTCC6, GyeongGyeong--juju, Korea, Korea

July 10, 2013July 10, 2013

Tim Tim KowalczykKowalczyk

Nagoya UniversityNagoya University

Constrained density functional theoryConstrained density functional theory

in Qin Q--ChemChem

Acknowledgments

Troy Van Voorhis

Ben KadukQin Wu

Brookhaven National Laboratory

MIT Chemistry

MIT Kerberos Consortium

Why Constrained DFT?

Density functional approximations suffer from selfself--interaction errorinteraction error (SIE)

• Mixed-valence compounds:

unphysical charge delocalization

• Highly delocalized excited states:

underestimate excitation energies

• Barrier heights are too low

Strategies to overcome self-interaction error (and other flaws of XC functionals):

1. Make better functionals

2. Abandon DFT, go ab initio

3. Adopt strategies to work around SIE with existing work around SIE with existing functionalsfunctionals

Constrained density functional theory

We identify charge-localized states with ground states of constrained systems

• Introduce charge/spin constraints

on the density

• Constraints force density to conform toconform to

chemical intuitionchemical intuition by design

• Constrained energy is obtained at roughly

the cost of a ground state calculation

Constrained density functional theory

[ ] [ ] ( ) ( )

−+→ ∫∑∑

σ

σσ ρρρkk

k

kNdwVEE rrr

3

wk – weight operators defining the constraint region (Becke weights)

Vk – Lagrange multipliers to enforce the constraint

Nk – value of the desired charge/spin constraint

We identify charge-localized states with ground states of constrained systems

• Introduce charge/spin constraints

on the density

• Constraints force density to conform toconform to

chemical intuitionchemical intuition by design

• Constrained energy is obtained at roughly

the cost of a ground state calculation

Constrained density functional theory

Some rules of thumb:

CDFT delivers diabatic states

• Density-based weight operators (e.g. Becke) are more stable than AO-based

operators (Mulliken, Lowdin)

• Larger fragments give more stable energies and couplings

• CDFT may give unreliable results for very small donor-acceptor distances

States maintain their electronic maintain their electronic

charactercharacter over the reaction

coordinate of interest

CDFT Applications

Sticking curves for O2 incident on the

Al(111) surface

Charge distribution in the

intramolecular charge-transfer

excited state of a fluorescent

sensor dye

Constrained density functional theory –configuration interaction (CDFT-CI)

Electronic couplings between CDFT states

KSKSKS

2

)(ˆ)(ˆ

2

ˆ

B

BBAA

AAB

BA

BAAB

wVwVS

FF

HH

Φ+

Φ−+

≈

ΨΨ=

rr

Constrained density functional theory –configuration interaction (CDFT-CI)

Electronic couplings between CDFT states

KSKSKS

2

)(ˆ)(ˆ

2

ˆ

B

BBAA

AAB

BA

BAAB

wVwVS

FF

HH

Φ+

Φ−+

≈

ΨΨ=

rr

• Need to approximate approximate wavefunctionwavefunction of interacting systemof interacting system with Kohn-Sham

wavefunction – an uncontrolled approximation

• Can build and diagonalizediagonalize HamiltonianHamiltonian in basis of nonorthogonal CDFT states

to get adiabatic ground, excited states

Barrier heights in CDFT-CI

A ― XB AX ― B[A ― X ― B]‡

• Most XC functionals overdelocalizeoverdelocalize the electron densitythe electron density of [A ― X ― B]‡

• 22--state descriptionstate description of the transition state avoids overdelocalization;

reactant and product diabatic states are both well-described by XC functionals

• More realistic charge/spin constraints via promoleculepromolecule densitiesdensities:

CDFT-CI Applications

Qualitatively correct

description of conical intersections

Electronic couplings that show

the correct distance dependence

CASSCF

CDFT-CI

TDDFT

Summary

CDFT is a DFT prescriptionCDFT is a DFT prescription

for for diabaticdiabatic electronic stateselectronic states

•• CDFT energies, gradients, and HessiansCDFT energies, gradients, and Hessians

•• CDFTCDFT--CI energiesCI energies

Availability in QAvailability in Q--ChemChem 4.x:4.x:

Charge and spin constraints are appliedCharge and spin constraints are applied

to a groundto a ground--state Kohnstate Kohn--Sham calculationSham calculation

•• Gradients and Hessians are on the wayGradients and Hessians are on the way……

Improved description of: mixed-valence complexes, spin-frustrated systems, CT excited states, barrier heights…

Thanks!Thanks!