Excited-StateDynamics Based on
TDDFT
Yoshiyuki Miyamoto
Research Center for Computational Design ofAdvanced Functional Materials, AIST, Tsukuba

outline
1. Time-Dependent Density Functional Theory
i. Frequency domain vs. Real-time
propagation
2. Review of Excited state MD Simulations
i. Combination with classical MD (EhrenfestDynamics)
ii. Energy conservation rule
3. Application on Quantum Beats in Moleculesi. Two antiphase pulses to major coherent
time in moleculesii. Size dependence of coherence1. Time-Dependent Density Functional Theory
DFT: unique relation between 𝑉𝑒𝑥𝑡 (𝒓) and 𝜌(𝒓) under the ground state−1For Kohn-Sham equation 𝜌 𝒓 = 𝑂𝑐𝑐𝑢𝑝 𝜓𝑛 (𝒓) 2 and 𝑇 = 𝑂𝑐𝑐𝑝 𝜓𝑛∗ 𝒓Δ𝜓𝑛 𝒓 𝑑𝒓
2

Minimizes 𝐸𝑡𝑜𝑡 under condition of0=𝑂𝑐𝑐𝑝

𝜓𝑛 (𝒓) 2 𝑑𝒓 = 𝑁: 𝐶𝑜𝑛𝑠𝑡.

𝛿(𝐸𝑡𝑜𝑡 −𝜖𝑛 𝑁)−1=
Δ + 𝑉𝑒𝑥𝑡 𝒓 + 𝑉𝑒−𝑒 𝜌 𝒓 − 𝜖𝑛 𝜓𝑛 (𝒓)𝛿𝜓𝑛∗ (𝒓)2Kohn-Sham eq.

TDDFT: unique relation between 𝑉𝑒𝑥𝑡 (𝒓, 𝑡) and 𝜌(𝒓, 𝑡) under the well-described
𝑉𝑒𝑥𝑡 (𝒓, 𝑡 = 0)) and 𝜌(𝒓, 𝑡 = 0).
For Kohn-Sham equation 𝜌 𝒓, 𝑡 = 𝑂𝑐𝑐𝑢𝑝 𝜓𝑛 (𝒓, 𝑡) 2
Minimizes Action 𝐴 = 𝑖0=𝑂𝑐𝑐

𝜓𝑛∗ 𝒓, 𝑡𝜕𝜓 𝒓, 𝑡 𝑑𝑟 − 𝐸𝑡𝑜𝑡 (𝑡)𝜕𝑡 𝑛𝛿𝐴𝜕−1=𝑖𝜓𝒓,𝑡−Δ + 𝑉𝑒𝑥𝑡 𝒓, 𝑡 + 𝑉𝑒−𝑒 𝜌 𝒓, 𝑡𝛿𝜓𝑛∗ (𝒓, 𝑡)
𝜕𝑡 𝑛2

𝜓𝑛 (𝒓, 𝑡)

Time-dependent Kohn-Sham eq.

Frequency domain with Linear Response(We won’t go details)
𝜕−1𝑖 𝜓𝑛 𝒓, 𝑡 =Δ + 𝑉𝑒𝑥𝑡 𝒓, 𝑡 + 𝑉𝐻𝑋𝐶 𝜌 𝒓, 𝑡𝜕𝑡2
Linear response theory𝜌 𝒓, 𝑡 = 𝜌0 𝒓 + 𝛿𝜌 𝒓, 𝑡 , 𝑉𝐻𝑋𝐶 𝜌 𝑟, 𝑡𝛿𝜌 𝒓, 𝑡 =𝑑𝑡′= 𝑉𝐻𝑋𝐶 𝜌0 𝑟 + 𝛿𝑉𝐻𝑋𝐶 (𝑟, 𝑡)𝑑𝒓′𝜒(𝒓, 𝒓′ , 𝑡 − 𝑡 ′ )𝑉𝑒𝑥𝑡 (𝒓′ , 𝑡 ′ )=𝛿𝑉𝐻𝑋𝐶 (𝒓, 𝑡) =𝜓𝑛 (𝒓, 𝑡)

𝑑𝑡′𝑑𝒓′𝜒𝐾𝑆 (𝒓, 𝒓′ , 𝑡 − 𝑡 ′ ) 𝑉𝑒𝑥𝑡 𝒓′ , 𝑡 ′ + 𝛿𝑉𝐻𝑋𝐶 (𝒓′ , 𝑡 ′ )𝐾𝐻𝑋𝐶 𝒓, 𝒓′ , 𝑡 − 𝑡′ 𝛿𝜌 𝒓′ , 𝑡′ 𝑑𝒓′ 𝑑𝑡′

Dyson likeequation for 𝜒

At particular snapshot of ions’ positions, we
can compute excitation energy with thisformula. (Born-Oppenheimer Approximation)

Potential

Frequency Domain: Changing χ(𝒓, 𝒓′, 𝑡) as χ(𝒓, 𝒓′, 𝜔)→ Casida’s equation for excitation energy

Reaction Coordinates

Real-time propagation𝑖𝜕
−1𝜓𝑛 𝒓, 𝑡 =Δ + 𝑉𝑒𝑥𝑡 𝒓, 𝑡 + 𝑉𝐻𝑋𝐶 𝜌 𝒓, 𝑡
𝜕𝑡2Δ𝑡𝑒 −𝑖𝐻 𝜌 𝒓,𝑡+𝐾𝑆2 Δ𝑡 𝜓𝑛(𝒓, 𝑡)𝜌(𝒓, 𝑡)𝜓𝑛𝐾𝑆 (𝒓, 𝑡)𝑡 + Δ𝑡𝑡𝐻 𝜌∗ (𝒓, 𝑡 + Δ𝑡

𝐻 𝜌(𝒓, 𝑡)

Guessed 𝜌∗22)

𝜓𝑛 (𝒓, 𝑡)𝜌(𝒓, 𝑡 + Δ𝑡)
𝜓𝑛𝐾𝑆 (𝒓, 𝑡 + Δ𝑡)𝑡 + Δ𝑡time

Compare guessed𝜌∗ (𝒓, 𝑡 + Δ𝑡 2) with𝜌𝑛𝑒𝑤 (𝒓, 𝑡 + Δ𝑡 2)
which is interpolatedfrom 𝜌(𝒕 + Δ𝑡) and𝜌(𝒓, 𝑡)→SCF

2.-i Review of Excited-State MD Simulation

MD: Newton’s equation for ions
MD under electronic excited state(Born-Oppenheimer Approximation)

Potential

Potential

MD under electronic ground state(Born-Oppenheimer Approximation)

PES

More complicatedissue: Nonadiabatic
transition among PESs.

Reaction coordinate

Potential

Reaction coordinate

Reaction coordinate

Why do we need to concern this case?
Potential

Because we do classical MD forions. (Need of force field, needof PES)

Reaction coordinate

Maybe it doesn’t matter when wetreat electrons and ions quantum
mechanically, but this approach is notpractical except for cases of few H
atoms (in low dimension).

If we don’t care and just perform classical MD for ions withquantum mechanical dynamics for electrons together, whathappens?

Ehrenfest dynamics
Potential

A classical MD
trajectory just go
along neither PESs,
but along with
mixture of them.

Reaction coordinate
In practical: The amount of
mixture is dependent onsystems, my lecture in 2015
CMD27

Trans
S1

S0

S2Cis

2.-ii Energy Conservation Throughout theEhrenfest Dynamics under Time-Dependent
External Field

Total Energy within DFT
Without an external field we know
𝐸𝐹𝑇𝐸𝑡𝑜𝑡𝑡𝑂𝐶𝐶𝑃𝜓𝑛𝐾𝑆∗ 𝒓, 𝑡

=1+2𝑛

−1 𝜕 2 𝐾𝑆𝜓 𝒓, 𝑡 𝑑𝒓 +2 𝜕𝒓2 𝑛

𝜌 𝒓′ , 𝑡 𝜌(𝒓, 𝑡) ′𝑑𝒓 𝑑𝒓 +𝒓′ − 𝒓𝜓𝑛𝐾𝑆∗ 𝒓′ , 𝑡 𝑉𝑛𝑙 𝒓′ , 𝒓, 𝑡 𝜓𝑛𝐾𝑆 𝒓, 𝑡 𝑑𝒓′ 𝑑𝒓𝜌(𝒓, 𝑡)𝜀𝑋𝐶 𝜌(𝒓, 𝑡) 𝑑𝒓 +Let optical field 𝑬(𝒓, 𝑡) as obtained by𝑍𝐼𝐼

𝜌 𝒓, 𝑡𝑑𝒓𝒓 − 𝑹𝐼 (𝑡)𝑍𝐼 𝑍𝐽+𝑹𝐼 (𝑡) − 𝑹𝐽 (𝑡)𝐼≠𝐽𝜕𝑉𝑒𝑥𝑡 𝒓, 𝑡−= 𝑬(𝒓, 𝑡)
𝜕𝒓And assume fictitious charge 𝜌𝑒𝑥𝑡 𝒓, 𝑡 which gives

𝑉𝑒𝑥𝑡 𝒓, 𝑡 =Note thatAnd let

𝜌𝑒𝑥𝑡 (𝒓′ )𝑑𝒓′𝒓′ − 𝒓𝜌𝑒𝑥𝑡 𝒓, 𝑡 𝑑𝒓 = 0

𝜌 𝒓, 𝑡 → 𝜌 𝒓, 𝑡 + 𝜌𝑒𝑥𝑡 (𝒓, 𝑡)
except the term including exchange-correlation

Total Energy within DFT(continued)
Then, with an external field we know
𝐸𝐹𝑇
𝐸𝑡𝑜𝑡𝑡
𝑂𝐶𝐶𝑃

=𝑛

1+
22−1
𝜕
𝜓𝑛𝐾𝑆∗ 𝒓, 𝑡𝜓𝑛𝐾𝑆 𝒓, 𝑡 𝑑𝒓 +22 𝜕𝒓𝜓𝑛𝐾𝑆∗ 𝒓′ , 𝑡 𝑉𝑛𝑙 𝒓′ , 𝒓, 𝑡 𝜓𝑛𝐾𝑆 𝒓, 𝑡 𝑑𝒓′ 𝑑𝒓

𝜌 𝒓′ , 𝑡 + 𝜌𝑒𝑥𝑡 (𝒓′, 𝑡) 𝜌(𝒓, 𝑡) + 𝜌𝑒𝑥𝑡 (𝒓, 𝑡)
𝑑𝒓′ 𝑑𝒓 +′𝒓 −𝒓𝜌(𝒓, 𝑡)𝜀𝑋𝐶 𝜌(𝒓, 𝑡) 𝑑𝒓

Electron-Light interaction+𝑍𝐼𝐼

Ion-Light interaction

𝜌 𝒓, 𝑡 + 𝜌𝑒𝑥𝑡 (𝒓, 𝑡)
𝑑𝒓 +𝒓 − 𝑹𝐼 (𝑡)𝑍𝐼 𝑍𝐽𝐼≠𝐽𝑹𝐼 (𝑡) − 𝑹𝐽 (𝑡)The energy conservation rule
𝐷𝐹𝑇𝑑𝐸𝑡𝑜𝑡
(𝑡)

𝑑𝑡

𝑂𝐶𝐶𝑃

=𝑛+𝐷𝐹𝑇𝐷𝐹𝑇
𝜕𝜓𝑛𝐾𝑆∗ (𝒓, 𝑡) 𝛿𝐸𝑡𝑜𝑡(𝑡)𝜕𝜓𝑛𝐾𝑆 (𝒓, 𝑡) 𝛿𝐸𝑡𝑜𝑡(𝑡)++𝜕𝑡𝜕𝑡𝛿𝜓𝑛𝐾𝑆∗ (𝒓, 𝑡)𝛿𝜓𝑛𝐾𝑆 (𝒓, 𝑡)𝜕𝜌𝑒𝑥𝑡 (𝒓, 𝑡) 𝜌 𝒓′ , 𝑡 + 𝜌𝑒𝑥𝑡 (𝒓′ , 𝑡)
𝑑𝒓′ 𝑑𝒓 +′𝜕𝑡𝒓 −𝒓𝐼𝐼𝐷𝐹𝑇𝜕𝑹𝐼 (𝑡) 𝛿𝐸𝑡𝑜𝑡(𝑡)∙𝜕𝑡𝛿𝑹𝐼 (𝑡)𝜕𝜌𝑒𝑥𝑡 (𝒓′ , 𝑡)
𝑍𝐼𝑑𝒓′𝜕𝑡𝒓′ − 𝑅𝐼 (𝑡)We know
𝐷𝐹𝑇𝛿𝐸𝑡𝑜𝑡−1 𝜕 2 𝐾𝑆=𝜓 𝒓, 𝑡 +2 𝜕𝒓2 𝑛
𝛿𝜓𝑛𝐾𝑆∗∗ (𝒓, 𝑡)𝛿𝜌(𝒓, 𝑡)+ 𝐾𝑆∗
𝛿𝜓𝑛 (𝒓, 𝑡)

𝑉𝑛𝑙 𝒓′ , 𝒓, 𝑡 𝜓𝑛𝐾𝑆 𝒓′ , 𝑡 𝑑𝒓′

𝜌 𝒓′ , 𝑡 + 𝜌𝑒𝑥𝑡 (𝒓′ , 𝑡)
𝑑𝒓′ + 𝜇𝑋𝐶 𝜌(𝒓, 𝑡) +𝒓′ − 𝒓= 𝐻 𝐾𝑆 𝒓, 𝑡 + 𝑉𝑒𝑥𝑡 (𝒓, 𝑡) 𝜓𝑛𝐾𝑆 (𝒓, 𝑡)

𝐼𝑍𝐼𝒓 − 𝑹𝐼 (𝑡)The energy conservation rule (continued)
Thus
𝐷𝐹𝑇𝑑𝐸𝑡𝑜𝑡(𝑡)

𝑑𝑡

𝑂𝐶𝐶𝑃

=𝑛+𝑈 𝑡𝐷𝐹𝑇𝐷𝐹𝑇𝜕𝜓𝑛𝐾𝑆∗ (𝒓, 𝑡) 𝛿𝐸𝑡𝑜𝑡(𝑡)𝜕𝜓𝑛𝐾𝑆 (𝒓, 𝑡) 𝛿𝐸𝑡𝑜𝑡(𝑡)++𝜕𝑡𝜕𝑡𝛿𝜓𝑛𝐾𝑆∗ (𝒓, 𝑡)𝛿𝜓𝑛𝐾𝑆 (𝒓, 𝑡)𝜕𝜌𝑒𝑥𝑡 (𝒓, 𝑡) 𝜌 𝒓′ , 𝑡 + 𝜌𝑒𝑥𝑡 (𝒓′ , 𝑡)
𝑑𝒓′ 𝑑𝒓 +′𝜕𝑡𝒓 −𝒓

𝐷𝐹𝑇= 𝐸𝑡𝑜𝑡
𝑡+𝐼

𝐷𝐹𝑇𝑑𝑈(𝑡) 𝑑𝐸𝑡𝑜𝑡
(𝑡)
=+𝑑𝑡
𝑑𝑡

=𝐼𝑀𝐼 𝜕𝑹𝐼 (𝑡)2𝜕𝑡2𝐼𝐼𝐷𝐹𝑇𝜕𝑹𝐼 (𝑡) 𝛿𝐸𝑡𝑜𝑡(𝑡)∙𝜕𝑡𝛿𝑹𝐼 (𝑡)𝜕𝜌𝑒𝑥𝑡 (𝒓′ , 𝑡)𝑍𝐼𝑑𝒓′
𝜕𝑡𝒓′ − 𝑅𝐼 (𝑡)𝜕𝑹𝐼 (𝑡)𝜕 2 𝑹𝐼 (𝑡)∙ 𝑀𝐼𝜕𝑡𝜕𝑡 2

𝜕𝜌𝑒𝑥𝑡 (𝒓, 𝑡) 𝜌 𝒓′ , 𝑡 + 𝜌𝑒𝑥𝑡 (𝒓′ , 𝑡)
𝑑𝒓′ 𝑑𝒓 +′𝜕𝑡𝒓 −𝒓𝑑𝑊𝑒𝑥𝑡 (𝑡)=𝑑𝑡𝐼𝜕𝜌𝑒𝑥𝑡 (𝒓′ , 𝑡)
𝑍𝐼𝑑𝒓′𝜕𝑡𝒓′ − 𝑅𝐼 (𝑡)𝑈 𝑡 − 𝑊𝑒𝑥𝑡 (𝑡) = 𝐶𝑜𝑛𝑠𝑡Summary of excited state MD
1. Coupling of quantum mechanicalelectron dynamics and classical iondynamics
2. Non-adiabatic transition among adiabatic
potential energy surface
3. Ehrenfest dynamics (Mixing of thepotential energy surfaces)
4. Energy conservation rule 𝑈 𝑡 − 𝑊𝑒𝑥𝑡 (𝑡) = 𝐶𝑜𝑛𝑠𝑡

3. Application on quantum beat inmolecules

Lesson: Two-level system
Two-level system with transition probability 𝑉𝑔𝑒 = 𝑒 𝑉 𝑔 as pure
imaginary number 𝑖𝑉, and excitation energy 𝜖.

𝜖|𝑒|𝑔Hamiltonian

𝐻 𝑡 = 𝐻0 − 𝑖𝑉𝑒 −𝑖𝜔𝑡 |𝑒 𝑔| + 𝑖𝑉𝑒 𝑖𝜔𝑡 |𝑔 𝑒|

Wavefunction
𝜓 𝑡 = 𝐶𝑔 𝑡 |𝑔 𝑒 −𝑖𝜖𝑔 𝑡 + 𝐶𝑒 (𝑡)|𝑒 𝑒 −𝑖𝜖𝑒𝑡Schrödinger equation 𝑖 𝜓(𝑡) = 𝐻(𝑡)𝜓(𝑡) gives
𝑖 𝐶𝑔 𝑡 = 𝐶𝑒 𝑡 𝑖𝑉𝑒 𝑖 −𝜖+𝜔 𝑡
At a special condition 𝜖 = 𝜔,we know solution of a beat.
𝑖 𝐶𝑒 𝑡 = −𝐶𝑔 𝑡 𝑖𝑉𝑒 𝑖 𝜖−𝜔 𝑡𝐶𝑔 𝑡 = cos 𝑉𝑡 C𝐶𝑒 𝑡 = sin(𝑉𝑡)tDynamics of dipole of two-level system𝜓 𝑡 = 𝑐𝑜𝑠 𝑉𝑡 |𝑔 𝑒 −𝑖𝜖𝑔 𝑡 + 𝑠𝑖𝑛 (𝑉𝑡)|𝑒 𝑒 −𝑖𝜖𝑒𝑡
𝑃 𝑡 = 𝜓(𝑡) 𝑟 𝜓(𝑡) ∝ sin 2𝑉𝑡 cos(𝜔𝑡)Usually 𝜔 ≫ 2𝑉𝑃(𝑡)Optical frequency𝑡

So far, this is an argument of pure two-level system

Realization of pure two-level systems
Use of ion trap technique

Blinov et al., Nature, 428,153 (2004)

Trapped Cd ion

Isolation to other materials
Simple electronic structure…so onI won’t explain details of this but justsay coherent control of electron
dynamics was madeTesting quantum beat of trapped molecules𝜕−1𝑖 𝜓𝑛 𝒓, 𝑡 =Δ + 𝑉𝑒𝑥𝑡 𝒓, 𝑡 + 𝑉𝐻𝑋𝐶 𝜌 𝒓, 𝑡𝜕𝑡2
𝜕 2 𝑹𝐼 (𝑡)𝑀𝐼= 𝑭𝐼 (𝑡)2𝜕𝑡𝜓𝑛 (𝒓, 𝑡)Real example : polycyclic benzene:
Much more than two-levels

Optical absorption spectrum

Computational experiment of twoshort anti-phase pulse1)

2) anti-phase of 1)

𝐸 𝑡 = 𝐸0 sin(𝜔𝑡)Dipole oscillation
Total Energy Increase

Dipole suppression
Total Energy Decrease𝑡−𝑡1 2𝑡−𝑡2 2−−𝜏𝜏𝑒−𝑒Computational conditions
1. Periodic boxes of (11.04 Å)3 for benzene,(11.04 Å)2x12.82 Å for naphthalene, and(11.04 Å)2x15.28 Å for anthracene
2. Plane-waves basis set with Ecut=62 Ry
3. Normconserving Pseudopotentials
4. Small ∆t(0.03 a.u. =0.726 attosec)
5. Scaler potential for E-field (0.1 V/Å)6. MD simulation within Ehrenfest
approximation

Benzene: Single-Pulse vs Double Antiphase Pulsesℏ𝜔 = 6.818 𝑒𝑉Naphthalene:
Single-Pulse vs Double Antiphase Pulsesℏ𝜔 = 5.61 𝑒𝑉

Anthracene:
Single-Pulse vs Double Antiphase Pulsesℏ𝜔 = 4.843 𝑒𝑉

Increase interval of two-pulses:
Naphthalene cases

ECoherence of quantum beat has finite timeSize dependenceSize dependence I:
Naphthalene vs. Anthracene
Naphthaleneℏ𝜔 = 5.61 𝑒𝑉EAnthraceneℏ𝜔 = 4.843 𝑒𝑉E

Why the larger size has shorter
coherent time?

Energy level

Small

Large

Larger systems have many
orbital levels with smaller
interval of orbital-energies.

Many decay channels ofexcited states

Size dependence II:monomer vs. dimer
Example: naphthalene dimer

Naphthalene
monomerNaphthalenedimerSize dependence: monomer vs. dimerEEInfluence of ion dynamics

Ehrenfest vs. frozen ions

Ion motion seems to help quantum beatSnapshot of velocity at 12.4 fs
18

19

620

E

54

171

15

1021

7816

32142413912 1123

TDDFT-MD Ehrenfest
dynamics induced
symmetric distributionof velocities

22

This is NOT the thermallyequilibrium case.

Electron-ion interaction usually destroys
quantum coherence
Applies randomized initial velocities on ions ofnaphthalene at the beginning of the TDDFT-MD simulation

Long time
Short timeSummary of quantum beat1. Molecules (benzene, naphthalene, anthracene) canhave quantum beat in analogy of two-level systems
2. The coherence time can be analyzed by delay-time ofsecond anti-phase pulse → experimental challenge
3. The coherence time is dependent on size: many decay
paths in larger systems
4. Ionic motion triggered by pulse helps quantum beatwhile randomized (thermal) ion motion does not.