of 13
Mixed-Reference
Spin-Flip
Time-Dependent
Density
Functional
Theory:
Multireference
Advantages
with
the Practicality
of Linear
Response
Theory
Woojin
Park,
Konstantin
Komarov,
Seunghoon
Lee,
*
and Cheol
Ho Choi
*
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This:
J. Phys.
Chem.
Lett.
2023,
14, 8896−8908
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ABSTRACT:
The
density
functional
theory
(DFT)
and linear
response
(LR)
time-
dependent
(TD)-DFT
are of the utmost
importance
for routine
computations.
However,
the single
reference
formulation
of DFT
suffers
in the description
of open-shell
singlet
systems
such
as diradicals
and bond-breaking.
LR-TDDFT,
on the other
hand,
finds
difficulties
in the modeling
of conical
intersections,
doubly
excited
states,
and core-level
excitations.
In this Perspective,
we demonstrate
that many
of these
limitations
can be
overcome
by recently
developed
mixed-reference
(MR)
spin-flip
(SF)-TDDFT,
providing
an alternative
yet accurate
route
for such
challenging
situations.
Empowered
by the
practicality
of the LR formalism,
it is anticipated
that MRSF-TDDFT
can become
one of
the major
workhorses
for general
routine
tasks.
O
ne of the most
challenging
goals
in electronic
structure
theory
is to achieve
a balanced
description
of
dynamical
and
nondynamical
(static)
correlation
effects
in a cost-effective
manner.
Although
density
functional
theory
(DFT)
1
has been
the most
successful
and popular
methodology,
it struggles
to
handle
open-shell
singlet
cases
such
as diradicals
and bond
breaking
due to its single-reference
formulation
(see Scheme
1).
2,3
Multireference
theories,
4
on the other
hand,
are
specifically
designed
for
nondynamical
correlation,
which
is
essential
for electronically
degenerate
or near-degenerate
situations
not only of open-shell
species
but also of electronically
excited
states
and conical
intersections.
However,
its missing
dynamical
correlations
necessitate
additional
calculations
in the
forms
of either
perturbations
5
or configuration
interactions,
6
making
the resulting
theories
impractical
for large
systems
or
time-consuming
tasks
like nonadiabatic
molecular
dynamics
(NAMD)
simulations.
7
In fact, our interest
in the development
of new quantum
theories
stems
from
the study
of nonadiabatic
processes.
In addition
to the accurate
descriptions
of ground
electronic
states,
proper
and efficient
quantum
theories
for strongly
correlated
excited
states
have become
important
as the demands
for next-generation
molecules
and materials
increase
rapidly.
8,9
Perhaps
the most
practical
and popular
methodology
for
studying
the excited
states
is the spin-conserving
linear-response
time-dependent
density
functional
theory
(LR-TDDFT).
10
12
It is based
on the time-dependent
Kohn
Sham
(TD-KS)
equation
with the linear-response
formalism
using
a closed-shell
singlet
ground
state
as its reference.
Given
a common
set of
reference
orbitals
(usually
obtained
by Kohn
Sham
DFT),
LR-
TDDFT
can conveniently
produce
a full spectrum
of singly
excited
states
in one calculation
without
any prior
knowledge
of
the nature
of states.
Thus,
it can easily
access
various
properties,
including
excitation
energies
of all excited
states,
as well as
interstate
properties,
like transition
dipole
moment
(TDM),
nonadiabatic
coupling
(NAC),
spin
orbit
coupling
(SOC),
core excitations,
and so on. Despite
all these
advantages,
there
Received:
August
16, 2023
Accepted:
September
21, 2023
Published:
September
28,
2023
Scheme
1. Challenging
Cases
for DFT and
LR-TDDFT
Perspective
pubs.acs.org/JPCL
© 2023
The Authors.
Published
by
American
Chemical
Society
8896
https://doi.org/10.1021/acs.jpclett.3c02296
J. Phys.
Chem.
Lett.
2023,
14, 8896
8908
This article is licensed under CC-BY-NC-ND 4.0
are well-known
failures
of this methodology,
e.g., in describing
the long-range
charge
transfer
excitations,
13
17
doubly
excited
states,
18
20
bond
breaking,
21,22
and real and avoided
conical
intersections
(CI)
(see Scheme
1).
23
26
To cope
with some
of
these
difficulties,
DFT-based
theories
such as the state-specific
orbital
optimized
(OO)-DFT
methods
have
been
actively
developed,
27
featuring
an unprecedented
precision
of DFT
methods.
28
However,
its state-specific
orbital
optimization
requires
prior
knowledge
of each
state,
which
is not always
straightforward.
In this regard,
LR theories
are still preferable
because
of their
practicality
and generality.
Some
of the drawbacks
of TDDFT,
in
particular,
the incorrect
description
of the conical
intersections
and the poor description
of multireference
electronic
states,
can
be corrected
by the spin-flip
(SF)-TDDFT,
29
31
as its open-
shell high-spin
triplet
reference
(
M
S
= +1) generates
the ground
singlet
state
as one of its response
states
as well as various
configurations
including
important
doubly
excited
ones
(see
Figure
1). However,
the missing
red configurations
in Figure
1
inevitably
introduce
considerable
spin
contamination.
32
34
Obviously,
a fundamental
solution
to the problem
is to recover
the missing
ones.
However,
unlike
wave
function
theories,
a
significant
challenge
remains
with
respect
to TDDFT
when
going
beyond
the adiabatic
approximation
to account
for more
than
single
excitations.
35
To alleviate
the problem,
the tensor
equation-of-motion
(TEOM)
formalism
was introduced
yield-
ing the spin-adapted
(SA)-SF-DFT.
36
However,
due to the
complexity
of TEOM,
its analytic
energy
gradient
has yet to be
derived.
Instead
of introducing
high-rank
excitations
from
a single
reference,
a second
red reference
(
M
S
=
1) was introduced
(see
Figure
1) as an alternative
way of expanding
the response
space
by some
of us.
37
In the realization
of this idea,
a new mixed
reference
state that has an equiensemble
density
of the
M
S
= +1
and
M
S
=
1 components
of a triplet
state was introduced:
x
x
x
(
)
1
2
(
)
(
)
M
M
0
M
R
0
1
0
1
S
S
=
{
+
}
=+
=
(1)
The two references
are transformed
to a single
hypothetical
reference
by the two spinor-like
open-shell
orbitals
of
s
1
and
s
2
as
shown
in Figure
2a, whose
one-particle
reduced
density
matrix
(RDM)
satisfies
the
idempotent
condition.
Finally,
the linear
response
from
the mixed
hypothetical
reference
yields
a mixed
reference
spin-flip
(MRSF)-TDDFT.
37,38
It is noted
that the
idea
of two references
is reminiscent
of the multireference
concept,
although
the exact
formulations
are fundamentally
different.
The immediate
advantage
of the new approach
is the
expanded
response
space
without
the expensive
multireference
orbital
optimization,
which
profoundly
affects
its characteristics.
To illustrate
it, the major
features
of MRSF-TDDFT
are first
summarized
in the following.
(1) Within
the Tamm
Dancoff
approximation,
39
the use of
mixed-reference
(MR)-RDM
in the linear-response
formalism
yields
two completely
decoupled
linear-response
equations
for
the singlet
and the triplet
excited
states,
respectively
37
A
A
X
X
k
S
T
(
)
,
,
rs
pq
rs
k
pq
rs
k
rs
k
k
pq
k
,
(
)(0)
,
(
)
(
)
(
)
(
)
+
=
=
(2)
where
k
=
S
,
T
represent
singlet
and triplet
states,
A
pq
,
rs
(
k
)(0)
is the
orbital
Hessian
matrix,
and
A
pq
,
rs
(
k
)
is a coupling
matrix
between
the configurations
originating
from
different
components
(
M
S
=
+1 and
M
S
=
1).
37,38
The remaining
X
pq
(
k
)
and
Ω
(
k
)
are the
amplitudes
and excitation
energies,
respectively.
The decoupled
Figure
1.
Upper
panel
shows
the two references
of MRSF-TDDFT
denoted
by black
and red arrows.
The zeroth-order
MR-RDM
which
combines
M
S
= +1 and
1 RDMs
is used in MRSF-TDDFT,
while
only
the
M
S
= +1 RDM
are used
in SF-TDDFT.
In the lower
panel,
the
electronic
configurations
that can be generated
by spin-flip
linear
responses
from
the MR-RDM
are given
by blue,
black,
and red arrows.
The blue ones
are generated
from
both
references,
which
require
a
symmetrization
procedure
to eliminate
OO-type
spin contamination.
The black
and red ones are generated
from
M
S
= +1 and
1 references,
respectively.
By contrast,
those
of SF-TDDFT
are only the blue and
black
ones.
Configurations
that cannot
be obtained
even
in MRSF-
TDDFT
are denoted
by gray arrows.
Figure
2.
(a) The concept
of spinor
transformation,
which
combines
two references
into a single
hypothetical
reference.
(b) The three
axes of
response
theory,
spin-flip
idea, and multireference
components,
where
the shaded
area represents
the MRSF-TDDFT.
The Journal
of Physical
Chemistry
Letters
pubs.acs.org/JPCL
Perspective
https://doi.org/10.1021/acs.jpclett.3c02296
J. Phys.
Chem.
Lett.
2023,
14, 8896
8908
8897
equations
of eq 2 nearly
eliminate
the spin-contamination
of SF-
TDDFT.
Correspondingly,
the ambiguity
with
cumbersome
spin-state
identification
is no longer
needed.
Therefore,
MRSF-
TDDFT
can be readily
applied
to “black-box”
type applications,
such as geometry
optimization,
conical
intersection
search,
and
molecular
dynamics
simulations.
Currently,
the spin-flip
excitations
also
generate
configurations
for quintet
states.
However,
as they
are not sufficiently
generated,
further
developments
are needed
for the more
accurate
descriptions
of quintet
states.
(2) It should
be noted
that the triplet
response
states
from
eq
2 are the triplet
states
with
M
S
= 0, which
is different
from
the
high
spin
reference
triplet
(
M
S
= +1) of MRSF-TDDFT.
Although
they resemble
each
other,
the former
and latter
are
obtained
by response
and variational
calculations,
respectively.
As the latter
reference
triplet
does not have any couplings
with
other
response
states,
it is logical
and recommended
that it
should
not be utilized
together
with
response
states
and the
lowest
response
triplet
should
be used instead.
This practice
is
different
from
LR-TDDFT,
where
the reference
ground
singlet
state has to be utilized.
Figure
3.
(a) The characteristic
topology
of the conical
intersection
(left)
and linear
intersection
(right)
between
two potential
energy
surfaces
of the
S
1
and S
0
states.
While
MRSF-TDDFT
can calculate
both states,
the combination
of LR-TDDFT
and DFT
is needed
for them.
(b) The S
1
(red)
and S
0
(blue)
energies
of CI
tw
BLA
of trans-penta-2,4-dieniminium
cation
(PSB3)
were
calculated
around
a loop with the radius
of 0.01 Å.
41
The branching
plane
vectors
were calculated
using
the algorithm
by Maeda
et al.,
47
which
yields
orthogonal
GDV
and DCV.
(c) NAC
vectors
(or DCVs)
at the MECI
geometries
of PSB3,
where
the left and right images
are obtained
by MRCISD
and MRSF-TDDFT/BH&HLYP,
respectively.
3
The MECI
geometries
were optimized
by MRCISD
in ref 48. The numbers
in parentheses
display
the inner
products
between
the MRSF
and the MRCISD
NAC
vectors
and
the ratio
|
NAC
MRSF
|
/
|
NAC
MRCISD
|
of the norms
of the NAC
vectors,
respectively.
The Journal
of Physical
Chemistry
Letters
pubs.acs.org/JPCL
Perspective
https://doi.org/10.1021/acs.jpclett.3c02296
J. Phys.
Chem.
Lett.
2023,
14, 8896
8908
8898
(3) Both
SF- and MRSF-TDDFT
remarkably
produce
the
ground
singlet
(S
0
) state as one of their response
states,
different
from
LR-TDDFT.
It allows
a way of generating
the multi-
configurational
ground
singlet
state,
overcoming
the single
reference
limitations
of the DFT.
Thus,
both
theories
can
describe
the open-shell
singlet
such
as diradicals,
40
as well as
bond-breakings.
3
The O
O type configurations
in Figure
1,
which
include
the
G
,
D
,
L
and
R
configurations
with blue arrows,
are mostly
responsible
for them.
In particular,
the
α
(O1)
β
(O1)
and
α
(O2)
β
(O2)
spin-flip
transitions
yield the
L
and
R
configurations,
respectively.
As the two configurations
occur
from
the two distinct
spin-flip
transitions,
their contributions
to
the final response
state
of SF-TDDFT
may become
unequal,
which
yields
substantial
spin-contamination.
This problem
was
resolved
in MRSF-TDDFT
by averaging
contributions
of the
same
configuration
originating
from
the different
spin-flip
orbital
transitions
from
the two components,
M
S
= +1 and
M
S
=
1, of the MR state.
37
(4) The ability
to generate
the ground
singlet
state
(S
0
) by
MRSF-TDDFT
also resolves
the topological
conical
inter-
section
problem
of LR-TDDFT
41
as both
S
0
and S
1
are in the
same
response
states
(see Figure
3a). On the other
hand,
the
combination
of DFT
and LR-TDDFT
has to be utilized
in the
case of LR-TDDFT,
which
does
not guarantee
correct
state
couplings.
Further
discussions
can be found
later in the main
text.
(5) The simple
one-electron
spin-flip
excitation
from
the
mixed
triplet
reference
produces
not only
singly
but also various
doubly
excited
configurations
as shown
in Figure
1 (with
respect
to the closed-shell
configuration).
42
The doubles
are critical
components
for the accurate
descriptions
of excited
states,
bond
breakings,
conical
intersections,
etc., which
significantly
expands
the applicability
of MRSF-TDDFT.
(6) Recently,
it has been
demonstrated
that the high
spin
triplet
reference
of MRSF-TDDFT
provides
a simple
way of
ensuring
core-hole
relaxation
for accurate
XAS
(X-ray
absorption),
43
which
has been
one of the main
difficulties
of
LR-TDDFT.
Detailed
discussions
can be found
later in the main
text.
(7) Conceptually,
MRSF-TDDFT
introduced
one more
axis
(middle
vertical
line)
of the multireference
concept
to the
existing
response
theory
and spin-flip
operator
axes as shown
in
Figure
2b. As a result,
MRSF-TDDFT
can take advantage
of
multireference
and the practicality
of linear
response
theory
at
the same
time with the formal
O(
N
4
) scaling
(practically
it can
be below
O(
N
3
) due to the integral
screenings).
44
Although
it is
possible
to further
develop
theories
along
other
axes,
such
as
nonlinear
response
theory
or double
spin-flip
theory,
the mixed-
reference
concept
of MRSF-TDDFT
appears
to be the most
practical
and efficient
way of introducing
additional
explicit
electron
correlations
without
sacrificing
performance.
The
major
computational
bottlenecks
of the response
parts
are the
Davidson
iterations,
45
which
typically
are computationally
less
demanding
than the SCF parts
in the case of response
theory.
Therefore,
the MRSF-TDDFT
does
not add too much
extra
computational
overhead
to ground-state
DFT
calculations,
a
great
practical
feature
for correlated
theories.
In a preliminary
benchmark
with
our new quantum
code,
46
the response
part
(Davidson
iteration)
took much
less time than SCF as shown
in
Figure
S1a, and the timing
ratio of response/SCF
in Figure
S1b
is on average
0.232.
(8) It is noteworthy
that
not all of the electronic
configurations
shown
in Figure
1 can be recovered.
In fact,
four out of six type C
V configurations
(i.e., those
shown
by
the gray
arrows)
remain
unaccounted
for. Typically,
these
configurations
represent
high-lying
excited
states
and make
insignificant
contributions
to the low-lying
states
of organic
molecules.
37
Thus,
the effect
of the missing
configurations
on
the spin contamination
is expected
to be small.
Overall,
MRSF-TDDFT
introduces
explicit
correlations
to the
implicit
accounts
of exchange
correlation
functionals
of DFT,
attempting
to balance
the
dynamical
and
nondynamical
electron
correlations.
To showcase
the above-mentioned
formal
advantages,
selective
studies
done
by MRSF-TDDFT
are
presented
below.
Perhaps,
one of the most
challenging
electronic
structure
issues
is the correct
description
of conical
intersections
(CIs)
as
shown
in Figure
3a.
49,50
CIs are special
geometries
at which
two
(or more)
adiabatic
electronic
states
become
degenerate,
51
54
which
provide
efficient
pathways
for nonadiabatic
population
transfer.
49,50,55
59
The degeneracy
of the intersection
is lifted
along
two unique
directions,
54,60,61
which
are defined
by the
gradient
difference
(
g
) and derivative
coupling
(
h
) vectors
(GDV
and DCV,
respectively)
given
by
E
E
g
1
2
(
)
S
S
Q
Q
1
0
=
(3a)
h
S
S
Q
1
0
=
|
|
(3b)
for the case of a crossing
between
the ground
(S
0
) and lowest
excited
S
1
singlet
states.
The most
significant
requirement
for
quantum
mechanical
theories
to describe
CIs is that it should
produce
nonvanishing
nonadiabatic
coupling
between
the
intersecting
states.
Surprisingly,
this requirement
for the S
1
/S
0
CIs is violated
by most
of the single-reference
theories
as well as
some
multireference
theories,
such
as single-state
(SS)-
CASPT2.
25,48,62
Multi-state
multireference
computational
methods
63
67
are capable
of producing
the correct
topology
of
CIs,
25,55,68,69
however
at the expense
of very
high
cost
of
computations.
The
popular
LR-TDDFT
methodology
12,70,71
fails to yield the correct
dimensionality
of the S
1
/S
0
CI seam
and
predicts
a linear
crossing
instead
23,25,72
(see the right
panel
of
Figure
3a) as they have
to be computed
by independent
DFT
and TDDFT
theories,
respectively.
23,25,72
Consequently,
the
absence
of couplings
between
S
0
and its excited
states
would
introduce
significant
uncertainties
in the description
of S
1
S
0
internal
conversion
processes.
In contrast,
it is remarkable
that MRSF-TDDFT
is capable
of
producing
the correct
double-cone
topology,
as the spin-flip
excitation
from
a triplet
produces
both S
1
and S
0
as its response
states.
The correct
CI topology
by MRSF-TDDFT
method
was
demonstrated
in a conical
intersection
of trans-penta-2,4-
dieniminium
cation
(PSB3)
(see Figure
3b), where
the nonzero
energy
differences
between
the red (S
1
) and blue (S
0
) loops
as
calculated
along
the circle
around
the CI with
the radius
R
ensure
the conical
topology.
41
The full NAC
vectors
at the CI of
PSB3
were
also presented
in Figure
3c, where
the left and right
images
are obtained
by MRCISD
and
MRSF-TDDFT/
BH&HLYP,
respectively.
The numbers
in parentheses
display
the inner
products
between
the MRSF-TDDFT
and the
MRCISD
NAC
vectors
and the ratio
|
NAC
MRSF
|
/
|
NAC
MRCISD
|
of the norms
of the NAC
vectors,
respectively.
The two numbers
with
near
unity
emphasize
the accuracy
of NAC
vectors
by
MRSF-TDDFT.
Therefore,
MRSF-TDDFT
can produce
not
only the correct
topology
of CIs but also accurate
NAC
vectors.
In a benchmark
with
12 conical
intersections,
the relative
The Journal
of Physical
Chemistry
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2023,
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8908
8899
energies
of CIs and their
geometries
by MRSF-TDDFT
are in
excellent
agreement
with those
of MRCISD
(RMSDs
of 0.41 eV
and 0.067
Å, respectively).
41,73
It has been
shown
that the nature
of conical
intersections
among
excited
states
(S
x
,
x
1) are also strongly
dependent
on
the quality
of quantum
mechanical
theories
even
in the most
Figure
4.
1
1
B
u
+
(red)
and 2
1
A
g
(black)
energy
profiles
along
the minimum-energy
paths
(MEPs)
of
trans
-butadiene
by (a) SA-CASSCF(4,4)
(solid
lines),
δ
-CR-EOMCC(2,3)
(dashed
lines),
and XMS-CASPT2(4,4)
(dotted
lines).
(b) The MEPs
by MRSF/BH&HLYP
(solid
lines)
and TDDFT/
B3LYP
(dashed
lines).
All calculations
taken
from
ref 42 were
done
with cc-pVTZ.
The BLA
coordinate
is defined
as the difference
between
the
average
length
of single
bonds
and the average
length
of double
bonds.
Figure
5.
Time
evolution
of the adiabatic
S
0
(black),
S
1
(red),
and S
2
(blue)
populations
for (a) the first 100 fs and (b) the entire
2 ps duration
of the
NAMD
simulations
taken
from ref 86. The light blue curve
in panel
a and the green
curve
in panel
b represent
fittings
of the S
2
and S
1
populations
by a
monoexponential
function,
respectively.
Panel
c compares
the S
2
(blue)
and S
1
(red)
PES profiles
along
the MRSF
MEPs
(solid
lines)
with the EOM-
CCSD
curves
(dashed
lines).
The molecular
structure
with atom
numbers
is given
in the inset.
Panel
d shows
the S
2
and S
1
PES profiles
obtained
with
the 3SA-CASSCF(10,8)
(solid
lines)
and the eXtended
Multi-State
Complete
Active
Space
second-order
Perturbation
Theory
(XMS-CASPT2,
dashed
lines).
MEPs
on the S
2
(blue)
and S
1
(red)
PESs
optimized
using
the nudged
elastic
band
(NEB)
106,107
method
in connection
with MRSF-
TDDFT
and connecting
the FC region,
the CI
21,
BLA
, and the S
1, min
geometries;
the respective
BLA
values
are given
parenthetically.
The BLA
coordinate
is defined
as the difference
between
the average
increments
of the lengths
of the double
bonds
and the decrease
of the single
bond,
BLA =
R
R
R
(
)
1
2
C
O
C
C
C
C
4
8
5
6
4
5
+
=
=
, where
Δ
R
’s
are displacements
with respect
to the S
0
equilibrium
geometry.
For all other
electronic
structure
methods,
the MRSF-TDDFT
MEP
geometries
are utilized
by employing
a 6-31G
*
basis
set with Cs symmetry
restriction.
Adopted
with permission
from
ref 84. Copyright
2009
American
Chemical.
The Journal
of Physical
Chemistry
Letters
pubs.acs.org/JPCL
Perspective
https://doi.org/10.1021/acs.jpclett.3c02296
J. Phys.
Chem.
Lett.
2023,
14, 8896
8908
8900