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The Requisite Electronic Structure Theory to Describe
Photo-excited Nonadiabatic Dynamics: Nonadiabatic Derivative
Couplings and Diabatic Electronic Couplings
Joseph E. Subotnik,
∗
Ethan C. Alguire, Qi Ou, and Brian R. Landry
Department of Chemistry, University of Pennsylvania,
231 South 34th Street, Philadelphia, Pennsylvania 19104
Shervin Fatehi
Department of Chemistry, University of Utah,
315 South 1400 East, Room 2020, Salt Lake City, Utah 84112
I. CONSPECTUS
Electronically photoexcited dynamics are complicated because there are so many differ-
ent relaxation pathways: fluorescence, phosphorescence, radiationless decay, electon transfer,
etc. In practice, to model photoexcited systems is a very difficult enterprise, requiring accu-
rate and very efficient tools in both electronic structure theory and nonadiabatic chemical
dynamics. Moreover, these theoretical tools are not traditional tools. On the one hand, the
electronic structure tools involve couplings between electonic states (rather than typical sin-
gle state energies and gradients). On the other hand, the dynamics tools involve propagating
nuclei on multiple potential energy surfaces (rather than the usual ground state dynamics).
In this account, we review recent developments in electronic structure theory as directly
applicable for modeling photo-excited systems. In particular, we focus on how one may
evaluate the couplings between two different electronic states. These couplings come in
two flavors. If we order states energetically, the resulting adiabatic states are coupled via
derivative couplings. Derivative couplings capture how electronic wavefunctions change as
a function of nuclear geometry and can usually be calculated with straightforward tools
from analytic gradient theory. One nuance arises, however, in the context of TD-DFT:
how do we evaluate derivative couplings between TD-DFT excited states (which are tricky,
because no wavefunction is available)? This conundrum was recently solved, and we review
the solution below. We also discuss the solution to a second, pesky problem of origin
dependence, whereby the derivative couplings do not (strictly) satisfy translation variance
which can lead to a lack of momentum conservation.
Apart from adiabatic states, if we order states according to their electronic character,
the resulting diabatic states are coupled via electronic or diabatic couplings. The couplings
between diabatic states |Ξ
A
i and |Ξ
B
i are just the simple matrix elements, hΞ
A
|H|Ξ
B
i. A
difficulty arises, however, because constructing exactly diabatic states is formally impossible,
and constructing quasi-diabatic states is not unique. To that end, we review recent advances
in localized diabatization, which is one approach for generating adiabatic-to-diabatic (ATD)
transformations. We also highlight outstanding questions in the arena of diabatization,
especially how to generate multiple globally stable diabatic surfaces.
∗
Electronic address: subotnik@sas.upenn.edu
2
II. INTRODUCTION: THE MATRIX ELEMENTS BEHIND ELECTRONIC
RELAXATION
One of the central goals in modern physical chemistry is to elucidate and quantify path-
ways for electronic relaxation in photo-activated molecules. Innumerable experiments in
time-resolved laser chemistry excite molecules or materials with photons and then probe
the state of the system after a time delay. In a typical UV-Vis experiment, after a photon
has been absorbed and the electronic state has been excited, one would like to know: do
the excited electrons stay still or do they meander in real space, leading to electron trans-
fer (ET)? Will there be electronic excitation transfer (EET) between excitons? Is there a
relevant pathway for intersystem crossing to produce a triplet state, with the potential for
triplet energy transfer (TT)? What is the lifetime of the excited electronic state and where
does that energy go?
The questions above address the fundamental nature of energy conversion between sys-
tems of many nuclei and electrons. And from a practical point of view, often these questions
cannot be answered completely using only spectroscopic data, without any theoretical guid-
ance. Moreover, these questions lie directly at the intersection of two separate and largely
isolated fields in theoretical chemistry– electronic structure theory and chemical dynamics.
In this account, we will highlight recent progress towards understanding electronic relaxation
from the perspective of electronic structure theory, including some results from our research
group. In the applications section below, we will focus on photo-excited, intramolecular ET
and TT, but the electronic structure methodology is quite general.
A. An Electronic Structure Theorist’s Best Friend: The Born-Oppenheimer
Approximation
From the perspective of an electronic structure theorist, when photo-excited electrons
relax, they break the Born-Oppenheimer approximation. Mathematically, this Born-
Oppenheimer breakdown is as follows: We begin with the total Hamiltonian as a function
of nuclear (n) and electronic (e) coordinates, where V stands for potential energy and T
stands for kinetic energy. Following standard nomenclature, ~r denotes electronic position
and
~
R denotes nuclear position (indexed by α):
H
T ot
(~r,
~
R) = T
n
(
~
R) + V
nn
(
~
R) + T
e
(~r) + V
ee
(~r) + V
ne
(~r,
~
R)
At this point, the Hamiltonian is partitioned into the nuclear kinetic energy and every-
thing else (the all inclusive electronic Hamiltonian, H
el
).
H
T ot
(~r,
~
R) ≡ T
n
(
~
R) + H
el
(~r,
~
R) (1)
The electronic Hamiltonian is then diagonalized, yielding the many-body adiabatic elec-
tronic states Φ
I
(~r;
~
R) =
D
r
Φ
I
(
~
R)
E
(labeled by I, J):
H
el
(
~
R)
Φ
I
(
~
R)
E
= E
I
(
~
R)
Φ
I
(
~
R)
E
(2)
3
Finally, the true nuclear-electronic wavefunction can be expanded in the basis of adiabatic
electronic eigenstates {Φ
I
}, yielding:
Ψ
T ot
(~r,
~
R) =
X
I
χ
I
(
~
R)Φ
I
(~r;
~
R) (3)
The set {χ
I
} represent nuclear wavefunctions moving along electronic states {Φ
I
} re-
spectively. Plugging Eqn. 3 into the Schrodinger equation, i~
∂
∂t
|Ψ
T ot
i = H
T ot
|Ψ
T ot
i, it is
straightforward to show that
i~
∂
∂t
χ
I
(
~
R) =
−
~
2
2M
∇
2
R
+ E
I
(
~
R)
χ
I
(
~
R) −
X
Jα
~
2
M
α
d
α
IJ
(
~
R)
∂χ
J
∂R
α
−
X
Jα
~
2
2M
α
g
α
IJ
(
~
R)χ
J
(
~
R)
(4)
d
α
IJ
(
~
R) ≡
Z
Φ
I
(~r;
~
R)
∂
∂R
α
Φ
J
(~r;
~
R) d~r (5)
g
α
IJ
(
~
R) ≡
Z
Φ
I
(~r;
~
R)
∂
∂R
α
2
Φ
J
(~r;
~
R) d~r (6)
Intuitively, nuclear wavepackets on different Born-Oppenheimer surfaces are coupled to-
gether by the matrix elements
~
d
IJ
(
~
R) (the derivative coupling) and ~g
IJ
(
~
R) (the second
derivative coupling). To model electronic relaxation, these matrix elements are essential.
B. A Chemical Dynamicist’s Best Friend: A Fixed Diabatic Basis
From the perspective of chemical dynamicists, the adiabatic electronic basis is awkward
because it changes with nuclear position. From this perspective, a better ansatz is the simple
one:
Ψ
T ot
(~r,
~
R) =
X
I
˜χ
I
(
~
R)Ξ
I
(~r) (7)
where the diabatic states {Ξ
I
} are independent of nuclear position and form a static (com-
plete) basis. Plugging Eqn. 7 into the Schrodinger equation, i~
∂
∂t
|Ψ
T ot
i = H
T ot
|Ψ
T ot
i, we
now find:
i~
∂
∂t
˜χ
I
(
~
R) = −
X
α
~
2
2M
α
∂
∂R
α
2
˜χ
I
(
~
R) −
X
J
W
IJ
(
~
R) ˜χ
J
(
~
R) (8)
W
IJ
(
~
R) ≡
Z
Ξ
I
(~r)H
el
(~r;
~
R)Ξ
J
(~r)d~r (9)
The matrix elements W
IJ
(I 6= J) are called diabatic or electronic couplings.
4
C. Outline
Both perspectives above are valid and there will be times when one or the other perspec-
tive is most useful (usually one wants small interstate couplings). An outline of this account
is as follows. In section III, we will highlight recent work aimed at calculating derivative
couplings, and in section IV, we will give an account of recent work to calculate diabatic
couplings. We discuss open questions and future areas for exploration in section VI.
III. DERIVATIVE COUPLINGS
Derivative couplings have long been the “odd man” out in the global field of quantum
chemistry. On the one hand, there is a long literature on computing derivative couplings
going back to the early work of Lengsfield and Yarkony[1, 2]. Yarkony et al originally derived
and implemented the necessary equations for computing derivative couplings in the context
of multiconfigurational self-consistent field theory (MC-SCF)[2]. The resulting expressions
for derivative coupling are quite tedious because of the nature of MC-SCF theory: both
because the MCSCF wavefunctions are rather complicated and because MC-SCF theory
is not invariant to the choice of which occupied (ijk) and which virtual (abc) orbitals are
included in the active space. The earliest applications were towards understanding excited
state-ground state crossings (which are essential for determining whether or not a photo-
excited molecule fluoresces or not).
On the other hand, despite all of the history above, it is safe to say that derivative cou-
plings have not been investigated as thoroughly in the literature as have energy gradients[3].
For quantum chemists interested in molecular structure (as opposed to interstate dynamics),
the derivative couplings are clearly less important quantities than the gradient or hessian.
Moreover, because running nonadiabatic dynamics on the fly was prohibitively expensive un-
til recently[4–8], historically the main application of derivative couplings has been the search
for conical intersections[9–11]. And while locating conical intersections yields intuition about
nonradiative processes, extracting a rate for electronic relaxation is more complicated. For
all of these reasons, the theory of derivative couplings between excited states is still evolving
in the context of electronic structure theory.
A. Derivative Couplings Between Excited States from Response Theory
For wavefunction based electronic structure, derivative couplings can be calculated with
standard analytic gradient theory. For example, configuration interaction singles (CIS)
excited state wavefunctions[12] are just sums of excitations from occupied orbitals {i} to
virtual orbitals {a}:
Ψ
CIS
I
(R)
=
P
ia
t
Ia
i
|Φ
a
i
i. In this case, the derivative coupling is:
d
CIS,α
IJ
≡
Ψ
CIS
I
∂
∂R
α
Ψ
CIS
J
=
X
ia
t
Ia
i
∂
∂R
α
t
Ja
i
+
X
ijab
t
Ia
i
Φ
a
i
∂
∂R
α
Φ
b
j
t
Jb
j
(10)
Eqn. 10 can be easily evaluated with analytic gradient theory[13].
An interesting question now arises in the theory of derivative couplings as related to
response theory. Nowadays, most excited state calculations are run with time-dependent
density functional theory[14–16]. Until recently, only ground state-excited state couplings
![FIG. 5: A plot of donor-acceptor diabatic coupling interpolated as a function of nuclear geometry. Here, we interpolate between the donor and accepor geometries for the methyl-bridged Closs “M” molecule. The diabatic coupling is computed with BoysOV localized diabatization (Eqn. 17). See Ref. [8] for more details. We also plot the electronic excitation density at initial and final geometries. Note that the excitation has moved from donor to acceptor while the molecular geometry has twisted.](/figures/fig-5-a-plot-of-donor-acceptor-diabatic-coupling-37arre7i.webp)
![FIG. 1: The derivative couplings (DC’s) between the first S1 and fourth S4 excited states of LiH as calculated by TDHF. Note that the pseudowavefunction (PW) and response theory (RT) formalisms mostly agree, but the RT matrix elements blows up around 1.9Å“by accident” when E4 − E1 = E1. See Ref. [31]. For now, the PW approach is the only stable means to calculate derivative couplings between TD-DFT excited states.](/figures/fig-1-the-derivative-couplings-dcs-between-the-first-s1-and-rqw5h8z4.webp)
![FIG. 4: A plot of experimental versus theoretical energy transfer rates kTT (as computed with Marcus theory and BoysOV localized diabatization [Eqn. 17]) for the Closs EET molecules. See Ref. [80] for more details.](/figures/fig-4-a-plot-of-experimental-versus-theoretical-energy-3fkezi2l.webp)
![FIG. 3: Mixing angles around the T1 − T2 conical intersection in the g− h plane for benzaldehyde (from Ref. [74]). Note that Boys (Eqn. 16) and ER (Eqn. 18) localized diabatization recover nearly the exact mixing angle (the latter obtained from fitting the local potential energy surface). This data strongly verifies the validity of localized diabatization approaches.](/figures/fig-3-mixing-angles-around-the-t1-t2-conical-intersection-in-18t2y0z7.webp)
![FIG. 7: The g (green) and h (red) vectors for the conical intersection in the Closs M molecule, as calculated by BoysOV localized diabatization (Eqn. 17). Note that the h vector field has been scaled up by a factor of 50. Most of the nuclear displacement in the dominant g direction is in the plane of the rings, suggesting that torsional motion – although important for reaching the conical intersection – is not the dominant motion at the conical intersection. See Ref. [8] for more details.](/figures/fig-7-the-g-green-and-h-red-vectors-for-the-conical-1y0abncw.webp)