Double Excitations and Conical Intersections inDouble Excitations and Conical Intersections in Time-Dependent Density Functional Theory Time-Dependent Density Functional Theory

Chaehyuk KoChaehyuk Ko, Benjamin G. Levine, Richard M. Martin, and Todd J. Martínez, Benjamin G. Levine, Richard M. Martin, and Todd J. MartínezDepartment of Chemistry, University of Illinois at Urbana-ChampaignDepartment of Chemistry, University of Illinois at Urbana-Champaign

Studying PhotochemistryStudying Photochemistry TDDFT in the Frank-Condon regionTDDFT in the Frank-Condon region

Ethylene Absorption SpectrumEthylene Absorption Spectrum Doubly Excited States of ButadieneDoubly Excited States of Butadiene

Energy (eV)

6.5 7.0 7.5 8.0 8.5 9.0

Abs

orpt

ion

(arb

itrar

y un

its)

0.0

0.5

1.0

1.5

2.0

2.5

Energy (eV)

6.5 7.0 7.5 8.0 8.5 9.0

Ab

sorp

tion

(arb

itrar

y u

nits

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

N→R3s

N→V

• Absorption spectra are calculated from simulations of nuclear wavepacket dynamics.

• The PES is calculated ‘on the fly’ at the B3LYP/6-31+G level of theory.

• Results of runs on the valence (N→V) and 3s Rydberg (N→R3s) states are shifted to match experimental excitation energies and summed according to their TDDFT oscillator strengths.

-1

0

1

2

3

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28

Ene

rgy

(eV

)

Bond Alternation

C1

C2

C3

C4

• The dark 21Ag state of butadiene is nearly degenerate to the bright 11Bu state at the Frank-Condon point.

• This dark state contains significant doubly excited character.

TDDFT 21Ag

CASPT2 21Ag

11Bu

TDDFT can accurately reproduce the shape of singly excited potential energy surfaces in the Frank-Condon region.

TDDFT does not capture the doubly excited character of this excited state.

______________________________________________________________________________________ ______________________________________________________________________________________________________________________________________________________________________________________________

TDDFT outside the Frank-Condon regionTDDFT outside the Frank-Condon regionWhat is a Conical Intersection?What is a Conical Intersection?

1

2

( ) ( )

( ) ( )

E R V RH

V R E R

1 2( ) ( )E R E R

( ) 0V R

• A conical intersection is a point of true degeneracy between electronic states.

• Two conditions must be met for degeneracy therefore conical intersections exist not as single points but as N-2 dimensional seams (where N is the number of nuclear degrees of freedom).

• Normally we search for the minimum energy points along these seams (MECIs).

Searching for MECIsSearching for MECIs

• Excited state energy is optimized subject to the constraint that the energy gap is zero.

• Function is smoothed to allow numerical differentiation.

• Function optimized with conjugate gradient method.

• Intersections are optimized at the trusted MS-CASPT2 level and compared with TDDFT results.

( , ) ( ) ( )ExcitedStateg Q E Q E Q

21 ( ) 1( , ) ( ) ( )

2 ( ( ) ) 2ExcitedStateE Q

h Q E Q E QE Q

λ = Lagrange multiplier

α = smoothing parameter

1.58(1.57)[1.58]

1.22(1.21)[1.20]

1.10(1.09)[1.09]

1.39(1.40)[1.41]

1.09(1.09)[1.11]

1.45(1.44)[1.46]

1.34(1.35)[1.36]

1.07(1.07)[1.08]

1.08(1.07)[1.08]

1.07(1.07)[1.08]

B3LYP/6-31G(CAS)

[CASPT2]

FC Transoid CI Me+ CI Me- CI

TDDFT can accurately predict the location and energy of conical intersections.

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Vertical Excitation Energies are Only the BeginningVertical Excitation Energies are Only the Beginning

• We examine important features of the potential energy surface (PES) including excited state minima, conical intersections, and barriers. We also run dynamical simulations.

• States of many different characters are important (singly excited, doubly excited, etc).

• To study these systems we usually utilize multireference ab initio methods such as CASSCF, CASPT2, and MRCI. Accurate treatment of dynamical correlation is important, but very expensive.

TDDFT could provide an inexpensive, highly accurate alternative to multireference ab initio methods.

Conical IntersectionExcited State

Minima

Barriers

Product Minima

y

x

H O Hx

y

DFT

TDDFT

Linear Water Intersection - TDDFT

H O Hx

y

-0.02

-0.01

0.00

0.01

0.02

-2.878

-2.876

-2.874

-2.872

-2.870

0.0000.001

0.0020.003

0.004

Ene

rgy

(eV

)

H2x

H2y

x

y

Linear Water Intersection – CASSCF – Everything an intersection should be

• Degeneracy is lifted in two independent directions.

• State characters mix as molecule moves ‘around’ intersection (follows red arrow).

• Energy gap varies linearly in region surrounding intersection.

• Degeneracy is lifted only in one direction. (V(R)=0 for all geometries because Brillouin’s theorem applies to the coupling between the DFT ground state and TDDFT excited states.)

• State characters cannot mix.

• Energy gap changes dramatically and nonlinearly in region surrounding intersection.

TDDFT fails to produce correctly shaped PESs in the region surrounding intersections involving DFT ground states.

ConclusionsConclusions

• TDDFT accurately predicts the shape of the PES of singly excited states in the Frank-Condon region.

• TDDFT fails to accurately describe states with significant doubly excited character.

• TDDFT does predict the existence of intersections between states where they exist according to high level ab initio calculations.

• The dimensionality and shape of intersections between the DFT ground state and TDDFT excited states are pathological.

Future WorkFuture Work

• Investigate possible extensions and alternatives to TDDFT as accurate and low cost electronic structure method for the study of photochemistry.

Picture from Michl, J. and Bonačič, V. Electronic Aspects of Organic Photochemistry. New York: John Wiley & Sons, Inc., 1990, p 53

0 i ti i tC e dt

ˆ; expi i ii t iHt

R

Torsional Coordinate Driving Curve for the model chromophore of Photoactive Yellow Protein (PYP)

TDDFT agrees very well with multi-reference perturbation theory if the states are not of double excitation character

• The isomerizable double bond dihedral angle φ was driven while therest of geometrical parameters were optimized with respect to S1 energyat state-averaged CASSCF(6-31G*) level of therory. CASPT2 was doneat these geometries.• After crossing the isomerization barrier on S1, S2 has significant doubleexcitation character.

TDDFT for Large Molecules/Condensed Phases:Pseudospectral Implementation of Configuration Interaction Singles

► Tamm-Dancoff approximation (TDA) TDDFT and Configuration Interaction Singles (CIS)

TDDFT working equation,

In TDA/TDDFT, B matrix is ignored,

▪ If the exact exchange type integrals in CISare replaced with exchange-correlation integrals,TDA/TDDFT can be implemented.▪ Pseudospectral implementation of CIS pavesthe way for pseudospectral TDA/TDDFT.

Pseudospectral approach

▪ Potential operators are diagonal in physical space, but purely numerical solution requires too many grid points.▪ In pseudospectral methods, both physical space basis (grid) and spectral (analytical) basis are used.▪ Explicit calculation of two electron integrals is avoided (Anm: one electron integrals)▪ Reduction of scaling from N4 to N3 (N: # of basis functions. ) is obtained.

Pseudospectral CIS (PS-CIS) on pCA-n(H2O) cluster (0≤n≤50)

▪ GAMESS program package was used for Spectral CIS (SP-CIS).

▪ Neutral pCA was surrounded by the specified number of water molecules.

▪ 1.2GHz AMD Athlon XP 1800+ used for timing.

▪ The coarse grid (~80 points / atom) was used in CIS iterations [6-31G*].

▪ Projected Computation Time for the first singlet excited state of the entire PYP with 6-31G* basis set.

0

1

2

3

4

5

6

0 2 4 6 8 10

CASPT2/ TDDFT at SA5-CAS(6/5) Geometries (Neutral)

SA5-CAS(6/5)-PT2

TDDFT (B3LYP)

φ

Mass-Weighted Distance / amu1/2 Angstrom

Ene

rgy

/ e

V

[ Neutral p CA ]

• AcknowledgementAcknowledgement •

This work is supported in part by the National Science Foundation under Award Number DMR-03 25939 ITR, via the Materials Computation Center at the University of

Illinois at Urbana-Champaign.

y = 0.0005x2.3908

y = 0.0004x2.6482

0

2000

4000

6000

8000

10000

12000

14000

16000

0 200 400 600 800 1000 1200 1400

number of basis functions

tim

e /

se

c

Pseudospectral

Spectral