Generalized Discrete Truncated Wigner Approximation for Nonadiabtic Quantum-Classical Dynamics
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Generalized Discrete Truncated Wigner Approximation for Nonadiabtic
Quantum-Classical Dynamics
Haifeng Lang,1, 2 Oriol Vendrell,1 and Philipp Hauke2
1)
Theoretical Chemistry, Institute of Physical Chemistry, Heidelberg University, Im Neuenheimer Feld 229,
69120 Heidelberg, Germany
2)
INO-CNR BEC Center and Department of Physics, University of Trento, Via Sommarive 14, I-38123 Trento,
Italy
arXiv:2104.07139v2 [physics.chem-ph] 25 Apr 2021
Nonadiabatic molecular dynamics occur in a wide range of chemical reactions and femtochemistry experi-
ments involving electronically excited states. These dynamics are hard to treat numerically as the system’s
complexity increases and it is thus desirable to have accurate yet affordable methods for their simulation.
Here, we introduce a linearized semiclassical method, the generalized discrete truncated Wigner approxima-
tion (GDTWA), which is well-established in the context of quantum spin lattice systems, into the arena
of chemical nonadiabatic systems. In contrast to traditional continuous mapping approaches, e.g. the
Meyer–Miller–Stock–Thoss and the spin mappings, GDTWA samples the electron degrees of freedom in
a discrete phase space, and thus forbids an unphysical unbounded growth of electronic state populations.
The discrete sampling also accounts for an effective reduced but non-vanishing zero-point energy without an
explicit parameter, which makes it possible to treat the identity operator and other operators on an equal
footing. As numerical benchmarks on two Linear Vibronic Coupling models show, GDTWA has a satisfactory
accuracy in a wide parameter regime, independently of whether the dynamics is dominated by relaxation or
by coherent interactions. Our results suggest that the method can be very adequate to treat challenging
nonadiabatic dynamics problems in chemistry and related fields.
PACS numbers: Valid PACS appear here
I. INTRODUCTION clude models from quantum optics41,42 , cold atoms43–45 ,
quantum spin chains5,6,8,10 , spin-boson models10,13,35–38 ,
and non-adiabatic molecular dynamics46 where the Born-
The phase space representation is a powerful Oppenheimer approximation breaks down.12,13,15,33,34 .
tool for computing quantum dynamics, with vari- In essence, TWA approaches treat bosons in the same
ous linearized approximation methods having been way as mapping approaches treat the nuclei degrees of
developed by diverse communities over the years, freedom (DoFs), examples being the phonons in trapped-
from quantum chemists to physicists.1–38 . Physi- ion experiments and bosonic ultracold atoms for TWA,
cists often subsume those methods under the name and the nuclei in chemical reaction and photo-chemical
of Truncated Wigner Approximations (TWA) with experiments for mapping approaches. In contrast, there
many family members1–11 , whereas chemists usu- are several choices for the spin DoF (the electron sub-
ally call them mapping approaches, including the system). Consider an electron subsystem with N elec-
Meyer–Miller–Stock–Thoss (MMST) mapping12–32 and tronic states, |1i , |2i , · · · , |N i. The symmetry group
spin mapping (SM)33–38 . The key idea of these methods of the electron DoF is SU (N ). MMST mapping ap-
is to sample the quantum distribution of the initial states proaches and Schwinger boson cluster TWA (CTWA)8
as the Wigner quasiprobability distribution, and neglect map the electron DoF to a single excitation of N coupled
higher-order quantum corrections of the Moyal bracket, Schwinger bosons, b1 , b2 , · · · , bN , or equivalently N cou-
thus rendering the evolution equations classical. One of pled harmonic oscillators, X1 , P1 , X2 , P2 , · · · , XN , PN . A
the most important reason researchers are interested in severe problem for MMST mapping approaches in the
these approaches is that the simulations using the classi- non-adiabatic dynamics is the physical phase space leak-
cal dynamics are computationally cheap and the Monte age problem, i.e., Schwinger bosons can escape from the
Carlo sampling is trivially parallelizable. Hence, they single excitation phase space under the classical dynam-
can be applied to large systems, which is usually impos- ics. This problem is partially solved by introducing a
sible for the numerically exact full quantum dynamics4,6 . zero-point energy (ZPE) parameter that modifies the in-
Higher-order quantum corrections can also be introduced teraction between electronic and nuclei DoFs12,31,32,34 , or
systematically4,9,27,29 . These approaches are exact in by a projection back to the single excitation Schwinger
the classical limit and the noninteracting limit. They bosons phase space13,15,22,29 . Instead, SM approaches,
can also provide reliable qualitatively correct results for TWA, and Operator CTWA sample the spin DoF in
short time dynamics when the system is not far away the natural phase space of the SU (2)33,34 or SU (N )
from the classical limit, and it is possible to capture the group35–38 . All of the above methods use continuous
long-time detailed-balance behavior39 or hydrodynamic DoFs to describe the electron subsystem. Recently, how-
phenomena6,8,40 for specific models. Typical interest- ever, a novel TWA-related method based on Wooters’
ing systems that are suitable for these approaches in- discrete phase space47,48 for spins, the discrete Trun-2
cated Wigner Approximation (DTWA)5 , has been pro- II. THEORY
posed and successfully generalized to higher spin systems
(GDTWA)6 . DTWA can capture the revivals and entan- We first give the original form of the GDTWA. We then
glement dynamics in quantum spin lattice systems up derive an equivalent form in analogous form to traditional
to an astoundingly long time. Motivated by trapped-ion mapping methods and the Ehrenfest method. This ped-
experiments, it has also been shown that DTWA is ap- agogical rewriting allows us not only to implement the
plicable to spin-boson models under the rotating wave simulations with a lower computational cost; as further
approximation10 . discussed in Sec. III, it also permits us to reveal special
advantages of GDTWA, including the effective non-zero
The goal of this work is to extend the scope of GDTWA reduced ZPE and the absence of physical space leakage.
to chemical systems, including a detailed theoretical anal-
ysis and numerical benchmarks. Our theoretical analysis
shows that the discrete phase space used in GDTWA is A. Basics of GDTWA
tailor-made to treat the discrete space of electronic states
in molecules. Additional modifications often required Consider a non-adiabatic Hamiltonian Ĥ describing
to improve the accuracy of the existing mapping ap- N electronic states, |1i , |2i , · · · , |N i, coupled to a nu-
proaches, including a ZPE parameter31,32 , the projection clear DoF (the generalization to several nuclear DoFs is
back to the physical phase space13,15,22,29 , and the dif- straightforward). In the diabatic representation, we can
ferent treatment of identity and traceless operators20,21 , write
are unnecessary in GDTWA. The discrete phase space
itself implicitly solves these mentioned issues. As our nu- p̂2
merical results illustrate, GDTWA achieves an accuracy Ĥ = + V̂ (x̂) (1)
2m
at least as good as existing state-of-the-art mapping ap- N
p̂2 X
proaches, and outperforms them in some of the selected = + |kiVkl (x̂)hl| ,
applications in this article. 2m
kl
This work is organized as follows. In Sec. II, we intro- where m is the mass of the nuclei, x̂ and p̂ are the
duce the GDTWA, first in its original formulation. By nuclear coordinate and momentum operators. In this
rewriting it in a language similar to the formulation of paper, we focusN on initial product states of the form
mapping approaches in chemistry, we show how to imple- ρ(0) = ρnuc (0) ρel (0). These can appear, e.g., in
ment the simulations of GDTWA practically. In Sec. III, molecular systems with only one populated electronic
we compare the GDTWA in the rewritten form with ex- state, such as the ground electronic state, or electroni-
isting fully linearized methods to illustrate how GDTWA cally excited systems prepared by a laser pulse shorter
accounts for an effective ZPE without ZPE parameters, than the time-scale for nuclear displacements.
and we show how GDTWA differs from the partially lin- The density matrix of the electronic DoFs and the
earized methods. In Sec. IV, we benchmark the GDTWA nuclei-electron interaction V̂ (x̂) are matrices with D =
using two Linear Vibronic Coupling (LVC) models fea- N × N elements. We can define D Hermitian operators
turing non-adiabatic dynamics at a conical intersection. Λ̂µ , using the Generalized Gell-Mann Matrices (GGM)
Section V contains our conclusions, and several Appen- for SU (N )49 and the identity matrix Î as a complete ba-
dices complement the main text. sis for the electron DoF,
1
√ (|ki hl| + |li hk|) for 1 ≤ µ ≤ N(N − 1)/2, 1 ≤ l < k ≤ N ,
2
1
√ (|li hk| − |ki hl|) for N(N − 1)/2 < µ ≤ N(N − 1), 1 ≤ l < k ≤ N ,
2i
Λ̂µ = k (2)
1 X
(|li hl| − k |k + 1i hk + 1|) for N(N − 1) < µ ≤ N2 − 1, 1 ≤ k < N ,
p
k(k + 1)
l=1
r
1 Iˆ for µ = D .
N
The explicit form of the Λ̂µ for N = 2 and N = 3 [Λ̂µ , Λ̂ν ] = ifµνξ Λ̂ξ , where fµνξ are the structure con-
are listed in the appendix A. The basis elements are or-
thonormal, tr Λ̂µ Λ̂ν = δµν with the commutation relation3
stants, this stage, the correlators between nuclei and electrons
are taken classical, which amounts to taking the mean-
ifµνξ = tr(Λ̂ξ [Λ̂µ , Λ̂ν ]) , (3) field form of the Heisenberg EOMs in each single tra-
jectory. That approach effectively truncates the order
and the Einstein notation has been used. We are go- of the EOMs. Though the EOMs of GDTWA in each
ing to use these basis elements to derive a semiclassical single trajectory are formally identical to the mean-field
description. method, GDTWA is still a method beyond the mean-
Any operatorP Ôel acting on the electron DoF can be field theory because the quantum fluctuations are par-
expanded as µ cµ Λ̂µ with cµ = tr Ôel Λ̂µ . Then, the tially accounted for in the initial statistical distributions
Hamiltonian in Eq. (1) can be expressed as of the phase space variables, which is similar to tradi-
tional TWA and mapping approaches1–38 .
p̂2 √ X
Ĥ = N Λ̂D + vµ (x̂)Λ̂µ , (4) The sampling of GDTWA for the initial nuclear phase
2m µ variables are identical to the ordinary linearized semiclas-
sical methods,
with vµ (x̂) = tr V̂ (x̂)Λ̂µ . The Heisenberg equation of Z
motions (EOMs) of the operators are η η
Wnuc (x0 , p0 ) = dη hx0 − | ρnuc (0) |x0 + i eip0 η .
2 2
x̂˙ t = p̂t /m , (7)
p̂˙t = −∂x̂t vµ (x̂t )Λ̂µ (t) , (5) The novelty of GDTWA is to sample the initial λµ as a
˙ discrete distribution. The details are as follows. First, Λ̂µ
Λ̂µ (t) = fµνξ vν (x̂t )Λ̂ξ (t) . P
can be decomposed as Λ̂µ = aµ aµ |aµ i haµ |, where |aµ i
As in the usual linearized semiclassical methods, are the eigenvectors of Λ̂µ . Then, the initial distribution
GDTWA approximates the observables as statistical av- of λµ (0) is λµ (0) ∈ {aµ } with probabilities
erages over trajectories of the phase space variables
whose equations of motion are classical and formally p(λµ (0) = aµ ) = tr[ρ̂el (0) |aµ i haµ |] . (8)
identical to the quantum Heisenberg EOMs. Define xt ,
pt , and λµ (t) as the time dependent classical phase vari- This distribution can represent arbitrary quantum expec-
ables for x̂, p̂, and Λ̂µ , respectively. Then, their EOMs tation values exactly as a statistical average,
are
X X
ẋt = pt /m , hÔel i = cµ hΛ̂µ i = cµ p(λµ (0) = aµ )aµ . (9)
µ µ,aµ
ṗt = −∂xt vµ (xt )λµ (t) , (6)
λ̇µ (t) = fµνξ vν (xt )λξ (t) , We are now in a position to give theN
formula to eval-
uate arbitrary observables Ô = Ônuc Ôel under the
with initial condition xt=0 = x0 and pt=0 = p0 . At GDTWA framework,
XZ
hÔ(t)i ≈ dx0 dp0 Wnuc (x0 , p0 )Ow,nuc (xt , pt )cµ p(λµ (0) = aµ )λµ (t) , (10)
µ,aµ
where Ow,nuc is the Wigner transformation of the opera- bitrary electronic initial states. However, some specific
tor Ônuc initial electronic states result in a higher accuracy than
Z others. Namely, an increased accuracy is achieved for ini-
η η tial states for which the statistical sampling reproduces
Ow,nuc (x, p) = dη hx − | Ônuc |x + i eipη . (11)
2 2 the initial intra-correlation6 of the electron states, i.e.,
for the observables
In principle, the above sampling can be applied to ar-
Λ̂µ Λ̂ν + Λ̂ν Λ̂µ X X
h i= p(λµ (0) = aµ )p(λν (0) = aν )aµ aν for µ 6= ν, hΛ̂2µ i = p(λµ (0) = aµ )a2µ . (12)
2 a ,a aµ
µ ν
A detailed analysis of the sampling of initial conditions can be found in the Appendix B.4
Generally, it has been proven that the GDTWA sam- accordingly. In fact, Aα (0) is nothing but the quasi-
pling distribution can reproduce the intra-electron corre- phase point operator in the Wootters’ discrete phase
lation for the diagonal states6 |mi hm|, 1 ≤ m ≤ N . For space representation47,48,50 .
convenience, we only consider the initial state |1i h1| in The ansatz of GDTWA in this form is that the Wigner
this article. All the other initial pure states can be con- function is evolved along the classical stationary trajec-
verted to this state by unitary transformations, and all tories
expectation values of observables of mixed states can be XZ
expressed as the summation over the expectation value W (x, p, A, t) ≈ dx0 dp0 wα Wnuc (x0 , p0 )
of pure states. α (15)
O
δ(x − xt )δ(p − pt ) Aα (t),
B. Re-formulation of GDTWA in the language of mapping where the EOMs of the variables are
approaches
ẋt = pt /m ,
In the following, we re-write the GDTWA in a com-
n o
ṗt = −∂xt Tr Aα (t)V̂ (xt ) , (16)
pletely equivalent form that not only reduces the com-
putational cost by reducing the classical DoFs used to Ȧα (t) = i[Aα (t), V̂ (xt )] ,
describe the electronic subsystem from N 2 − 1 to 4N 8 ,
but also reveals important concepts such as ZPE (see with initial conditionNxt=0 = x0 and pt=0 = p0 . Any
Sec. III A), thus enabling a direct comparison to the for- observable Ô = Ônuc Ôel can be evaluated as
malism of linearized semiclassical methods (see Sec. III A Z
and Sec. III B). O
hÔ(t)i ≈ tr dxdpW (x, p, A, t)Ow,nuc (x, p) Ôel
At the core of GDTWA lies a sampling over trajec-
tories. In the original formulation of GDTWA, this is X Z
achieved via sampling over the continuous initial phase = dx0 dp0 wα Wnuc (x0 , p0 )Ow,nuc (xt , pt )
space of the nuclear degree of freedom as well as the α
n o
(α)
discrete electronic initial phase space variables λµ (0), × Tr Aα (t)Ôel .
where we used the index α to label the diverse electronic (17)
initial conditions in the discrete phase space. In the for- The GDTWA in this form, with the EOMs given by
(α)
mulation we are developing here, the role of λµ (0) is Eq. (16) and the expectation values in Eq. (17), has
assumed by the so-called discrete quasi-phase point op- some formal resemblances to the Ehrenfest method. In
erators Aα (0), which are used to describe the electronic both approaches, each trajectory of the nuclei evolves
DoFs using the transformation in the mean potential resulting from the populated elec-
X tronic states. However, there are two main differences
Aα (t) = λ(α)
µ (t)Λ̂µ between these two methods. First, GDTWA trajectories
µ (13) start from a discrete sampling in the space of the quasi-
λ(α) phase point operators rather than from a uniquely de-
µ (t) = tr Aα (t)Λ̂µ .
fined electron state. Second, GDTWA trajectories evolve
For convenience, we will use the notation Aα to express the quasi-phase point operator Aα (t) rather than ρel (t)
Aα (t) in this article when there is no ambiguity. in each individual trajectory.
The sampling of the initial condition Aα (0) is achieved To implement the simulation, we require the spec-
via a sampling of the initial λ(α) (0) as in Eq. (8), which tral decomposition for the quasi-phase point operator
using the transformation Eq. (13) translates into Aα . It is easy to check that the spectral decompo-
sition of Eq. (14) is Aα (0) = λ+ |Ψα α
+ (0)i hΨ+ (0)| +
α α
δ2 −iσ2 δN −iσN
λ− |Ψ− (0)i hΨ− (0)|, where the eigenvalues are
1 2 ··· 2
δ2 +iσ2
0 ··· 0 √
Aα (0) =
2
,
(14) 1± 2N − 1
.. .. .. .. λ± = , (18)
. . . . 2
δN +iσN
2 0 ··· 0
with the amplitudes of the associated eigenvectors
with δi , σi = ±1 being independent and identically dis- s
tributed discrete uniform variables on the integers ±1. λ2±
h1|Ψα
± (0)i = ,
The initial density matrix of the electron
P subsystem is λ2± + (N − 1)/2
expanded as ρel (0) = |1i h1| = α wα Aα (0), where
s
wα = 2−2(N −1) for all α. The GDTWA sampling strat- α λ2± δj + iσj
hj|Ψ± (0)i = 2 ∀j > 1 .
egy for the electron subsystem is converted to generating λ± + (N − 1)/2 2λ±
the initial discrete phase points by sampling δi and σi (19)5
The eigenvalues of the quasi-phase point operator can with γ the ZPE parameter, usually chosen from zero (zero
be interpreted as quasi-probabilities, since λ+ + λ− = 1, ZPE treatment) to one (full ZPE treatment), and |Ψα (t)i
λ+ > 0 and λ− < 0. Such quasi-probabilities constitute the normalized electronic wave function. Further, Rα is
the spectrum of Aα , and are conserved during the prop- the square root of the radius of the mapping variables,
agation. We can propagate |Ψα ± (t)i rather than Aα (t) which in the ordinary harmonic oscillator MMST map-
using the EOMs ping notation, with position Xn and momentum Pn for
state n, is defined by
d
i |Ψα (t)i = V̂ (xt ) |Ψα
± (t)i (20) √
dt ± Xn (t) + iPn (t) = 2Rα hn|Ψα (t)i , (23)
and Aα (t) = λ+ |Ψα α α α
+ (t)i hΨ+ (t)| + λ− |Ψ− (t)i hΨ− (t)|.
This completely equivalent reformulation reduces the
number of electronic subsystem DoFs from N 2 − 1 to X
4N . Xn (t)2 + Pn (t)2 = 2Rα
2
. (24)
n
Rα and γ are conserved during the evolution and the
III. DISCUSSION
EOM of |Ψα (t)i is
In this section, we compare the GDTWA with estab- d
i |Ψα (t)i = V̂ (xt ) |Ψα (t)i . (25)
lished fully and partially linearized semiclassical meth- dt
ods. The form of the EOMs of GDTWA is similar to
fully linearized methods but with a computational cost Different mapping approaches use different sampling
close to partially linearized methods. Readers who are strategies for Rα and |Ψα (0)i and evaluate the expec-
only interested in the numerical performance of GDTWA tation values of the observables in different manners.
may skip this section. For each single trajectory, Bα (t) has one non-degenerate
2
eigenvalue Rα − γ/2 and N − 1 degenerate eigenvalues
−γ/2, as can be seen immediately from the definition of
A. Zero point energy treatment within the GDTWA Bα (t) in Eq. (22). In this sense, the ZPE parameter in
approach & absence of physical space leakage the traditional fully linearized method is a negative di-
agonal energy correction term for the nuclei-electron in-
Because of the discrete sampling, GDTWA accounts teractions. The nuclei always see a modified average po-
for a non-zero effective reduced ZPE without introducing tential energy during the evolution in each single trajec-
an explicit ZPE parameter. It is well known that both tory, whence mapping approaches with a non-zero ZPE
full ZPE (approaches based on MMST mapping with- parameter already account for some quantum effects in
out empirical ZPE parameters) and zero ZPE (Ehrenfest their EOMs.
method) are harmful for numerical accuracy31,32 . One Though Eq. (21) and Eq. (16) are formally iden-
possible solution to this problem is to introduce an ad- tical, it is impossible to express Aα in the form
Rα2 ˆ and thus to construct the ZPE-
|Ψα (t)i hΨα (t)| − γ2 I,
justed ZPE-parameter to make the classical dynamics
and phase space of the mapping variables of the harmonic parameter,√except for the case of N = 2, in which
oscillators of the electronic DoFs mimic the spin as much case, γ = 3 − 1. We can nevertheless identify an ef-
as possible31,32,37,38 . GDTWA solves this problem with a fective ZPE-parameter governing the evolution of Aα .
fundamentally different logic, i.e., GDTWA never intro- Namely, the ZPE-parameter in the traditional fully lin-
duces such a parameter but tames the ZPE only through earized methods can also be constructed by the fol-
a judiciously designed initial sampling procedure. lowing strategy. Notice that tr(Bα ) = Rα 2
− γ2 N and
2
To illustrate how GDTWA accounts for an effective tr(Bα2 ) = Rα
4
− γRα2
+ γ4 N only depend on Rα and γ.
non-zero reduced ZPE, it is convenient to first review how Hence, the ZPE-parameter in the traditional fully lin-
existing methods including symmetrical quasi-classical earized methods can be expressed as
windowing13,15 and generalized spin mapping37,38 , ac- p
count for the ZPE. The EOMs of fully linearized map- N tr (Bα2 ) − (tr Bα )2 tr Bα
γ= √ − (26)
ping approaches12–24,27,30–32,37,38 can also be written in N N −1 N
the form of Eq. (16),
By formally replacing Aα with Bα in Eq. (26), we obtain
ẋt = pt /m , an effective ZPE-parameter for the GDTWA,
n o
ṗt = −∂xt Tr Bα (t)V̂ (xt ) , (21) √
2 N +1−2 (27)
γeff = .
Ḃα (t) = i[Bα (t), V̂ (xt )] , N
where Interestingly, this reduced effective ZPE coincides with
γ the ZPE in recent works using the SM approach35–38 .
2
Bα (t) = Rα |Ψα (t)i hΨα (t)| − Iˆ , (22) The reason of such identical ZPE is that both GDTWA
26
and SM start from the phase space of the electronic DoFs, Different partially linearized methods have different
rather than the phase space of Schwinger bosons. The formulas to evaluate expectation values and different
ZPE of SM and GDTWA can, however, be different when sampling strategies for the initial radius and electronic
the Hamiltonian is block diagonal, see the discussions in trajectories. The electronic subsystem in each single tra-
the Appendix C. jectory of different partially linearized methods are also
A further feature of the implicit ZPE treatment is that different. A typical electronic subsystem in partially lin-
GDTWA treats the traceless and identity operators of earized methods takes the form |Ψ1,α (t)i hΨ2,α (t)|, which,
electronic states in a unified way. No other trick20,21 or unlike Aα (t) and Bα (t), is not hermitian. Specifically,
a specific implementation for the identity operator35,36 is the sampling of |Ψ1,α (0)i and |Ψ2,α (0)i must be uncorre-
required. In this sense, GDTWA seems a more natural lated. As a comparison, there is no forward and backward
approach to obtain observables of the electronic DoF. electronic trajectories concept in GDTWA. So, the two
Another advantage related to the spin phase space of electronic wavefunctions for GDTWA are the spectral de-
GDTWA is that the method does not suffer from the composition of the quasi-phase point operator. The ini-
physical space leakage problem31,32 , and thus eliminates tial conditions for two electronic states in GDTWA in a
the additional projection that is necessary in the LSC- single trajectory are necessarily correlated. In this sense,
IVR and PBME approaches13,15,22,29 . The EOMs and GDTWA is a method with hybrid features of fully lin-
initial sampling constructions ensure that the Aα (t) tra- earized methods and partially linearized methods, i.e.,
jectories are always trapped in this tailor-made electronic GDTWA has the same form of EOMs as the fully lin-
phase space, similarly to what is achieved for Bα (t) in the earized methods, but two electronic wavefunctions in
recently proposed SM approach35–38 . each single trajectory. In conjunction with the inclusion
of an effective ZPE as well as two electronic states in each
single trajectory, this makes GDTWA an extremely effi-
B. Comparison with partially linearized methods cient and surprisingly reliable numerical method, as we
will see in the numerical computations of the following
The nuclei in both GDTWA and partially linearized section.
methods move on a mean-field potential, which is the
average potential of two effective electronic states, in
each single trajectory. Nevertheless, GDTWA has a sig- IV. NUMERICAL RESULTS
nificantly different logic from traditional partially lin-
earized methods, such as the Forward-Backward Trajec- In this section, we perform numerical benchmarks on
tory solution (FBTS)28,29 , partially Linear Density Ma- the GDTWA for prototypical non-adiabbatic dynam-
trix (PLDM)25,26 , and Spin-PLDM35,36 , as we illustrate ics problems in chemistry. Since each GDTWA trajec-
now. tory evolves the classical nuclei and two coupled elec-
The EOMs of the family of partially linearized methods tronic time-dependent states, its numerical complexity
can be written as25,26,28,29,35,36 is close to the partially linearized approach and slightly
ẋt = pt /m , larger than the fully linearized mapping approach. We
2
may thus expect that GDTWA should be considered
R1,α as an alternative approach to partially linearized meth-
ṗt = − ∂xt hΨ1,α (t)| V̂ (xt ) |Ψ1,α (t)i
2 ods, which is indeed confirmed by the numerics reported
2 in this section. The selected mapping approaches to
R2,α
− ∂xt hΨ2,α (t)| V̂ (xt ) |Ψ2,α (t)i , (28) which we compare in this section are PLDM25 , Spin-
2
d PLDM35,36 with non-focus sampling, and the Ehrenfest46
i |Ψ1,α (t)i = V̂ (xt ) |Ψ1,α (t)i , method. For all the methods we run 106 trajectories
dt
d to ensure convergence, though GDTWA starts to con-
i |Ψ2,α (t)i = V̂ (xt ) |Ψ2,α (t)i , verge already with 104 trajectories, a number compara-
dt ble with the Ehrenfest method. We will show numeri-
where |Ψ1,α (t)i and |Ψ2,α (t)i are the forward and back- cal benchmarks for two LVC models46,51–53 , comparing
ward normalized electronic wavefunctions (or electronic the selected linearized semiclasscial methods with nu-
trajectories), respectively, and R1,α and R2,α are the merically converged Multi-configuration time-dependent
square root of the radius of the corresponding mapping Hartree (MCTDH) calculations54–56 .
variables. In the ordinary harmonic oscillator MMST The LVC Hamiltonian57,58 in the diabatic basis is given
mapping notation, by
√
Xj,n (t) + iPj,n (t) = 2Rj,α hn|Ψj,α (t)i , (29) 1X X
H= ωj p2j + |ki Wkl hl| , (31)
2 j
k,l
where Wkl is obtained by the Taylor expansion with re-
X
Xj,n (t)2 + Pj,n (t)2 = 2Rj,α
2
, for j = 1, 2. (30)
n spect to the electronic ground state equilibrium geome-7
Figure 1. Second diabatic state population of a three-modes Figure 2. Populations of the tuning coordinates hx1 i and hx6a i
two-states model based on Pyrazine (see table I), computed of the Pyrazine-based Model I. The color notations are identi-
using different methods. The GDTWA result (blue solid line) cal to Fig. 1. The GDTWA (blue solid line) and Spin-PLDM
compares fairly well to the exact quantum dynamics (black (red dashed) results fail to capture the oscillation amplitudes,
solid). While GDTWA and, even more so, the Spin-PLDM but still give a qualitatively fair description on the frequency.
method (red dashed) underestimate the mean value reached In contrast, the Ehrenfest (green dashed) and PLDM methods
at long times, the Ehrenfest method (green dashed) overes- (pink dashed) mismatch the oscillation pattern of the exact
timates it. The PLDM methods (pink dashed) considerably quantum results (black dashed) after a few periods.
overestimates the damping of the oscillations.
try,
1X X (k)
Wkk = Ek + ωj x2j + κj xj , (32)
2 j j
(kl)
X
Wkl = λj xj , k 6= l , (33)
j
where xj and pj are the dimensionless position and mo-
mentum for the vibronic mode j, and ωj is the corre-
sponding frequency. Further, Ek is the vertical transi- Figure 3. Expectation values of second-order correlations
(kl) (k)
tion energy of the diabatic state |ki, and λj and κj of the tuning coordinates hx21 i and hx26a i, and the coupling
are the gradients of Wkl and Wkk , respectively. coordinate hx210a i of the Pyrazine-based Model I. The color
In this article, we focus on the time dependence of ob- notations are identical to Fig. 1. The GdTWA (blue solid
servables for the initial line) and Spin-PLDM (red dashed) results qualitatively pre-
Q 1product state of the vibrational dict the ideal higher-order correlation, while the Ehrenfest
ground state Ψ = j π1/4 exp −x2j /2 and the excited
(green dashed) and PLDM methods (pink dashed) deviate
electronic state, which is a typical setup of femtochem- significantly from the exact quantum results (black dashed).
istry experiments. We consider two typical benchmark
models46,51–53 , as given in the Tables I and II. Model I
is a three-modes two-states model based on Pyrazine. It
includes two tuning coordinates x1 and x6a , and one cou-
pling coordinate x10a , and the initial electron wave func-
tion is prepared in the second diabatic state |2i46 . Model
II is a five-modes three-states model based on Benzene
radical cation. It includes three tuning coordinates x2 ,
x16 , and x18 , and two coupling coordinates x8 and x19 ,
and the electron wave function is initialized in the third
diabatic state |3i46 .
Due to symmetry, all the off-diagonal elements of the
electron density matrix of the two models vanish. In
Fig. 1, we show the population of the second diabatic Figure 4. Populations of all three diabatic states of a five-
state of Model I. The GDTWA result compares fairly modes three-states model based on Benzene radical cation
well to the exact quantum dynamics. It seems to under- (see table II), computed using different semiclassical tech-
estimate the amplitude of oscillations around the mean, niques. The GdTWA result (blue solid line) compares fairly
and reaches a long-time average that lies slightly below well to the exact quantum dynamics (black solid) for all the
the exact value. The functional form seems to be bet- three diabatic states populations, while all the other methods
considered fail to describe the long time populations.
ter reproduced than with the Ehrenfest method, and the8
(k) (k)
Ek ω1 κ1 ω6a κ6a ω10a λ
|1i 3.94 0.126 0.037 0.074 –0.105 0.118
0.262
|2i 4.84 0.126 –0.254 0.074 0.149 0.118
Table I. Parameters of Model I that is based on Pyrazine. All quantities are given in eV.
(k) (k) (k) (12) (23)
Ek ω2 κ2 ω16 κ16 ω18 κ18 ω8 λ8 ω19 λ19
|1i 9.75 0.123 -0.042 0.198 -0.246 0.075 -0.125 0.088 0.12
0.164
|2i 11.84 0.123 -0.042 0.198 0.242 0.075 0.1 0.088 0.12
0.154
|3i 12.44 0.123 -0.301 0.198 0 0.075 0 0.088 0.12
Table II. Parameters of Model II based on Benzene radical cation. All quantities are given in eV.
Figure 5. Populations of tuning coordinates hx2 i, hx16 i, Figure 7. Second-order correlations of the coupling coordi-
and hx18 i of the Model II that is based on Benzene radical nates hx28 i and hx219 i of Model II. For the dynamics of hx28 i,
cation. The GdTWA result (blue solid line) matches the exact both GDTWA (blue solid line) and Spin-PLDM (red dashed)
quantum dynamics (black solid) best and slightly outperforms match the exact quantum results (black solid), with GDTWA
the Spin-PLDM result (red dashed). slightly outperforming the Spin-PLDM result. For hx219 i, both
methods reproduce qualitative features of the exact evolution
better than the other considered semiclassical techniques.
curve lies closer to the exact result than the curve com-
puted using the Spin-PLDM method. Finally, the PLDM
methods produces the best estimate of the long-time av-
erage, but considerably overestimates the damping of the
oscillations. GDTWA fits the quantum result rather well
at short times and has a fair performance at longer times,
though it does not outperform the other approaches in
this regime. Figure 2 shows the dynamics of the two
tuning coordinates, hx1 i and hx6a i. Though GDTWA
does not entirely capture the correct amplitude, it does
Figure 6. The second-order correlations of the tuning co-
match very well the frequency of the occurring oscilla-
ordinates hx22 i, hx216 i, and hx218 i of Model II. Both GDTWA
(blue solid line) and Spin-PLDM (red dashed) match the ex-
tion. This behavior is similar to the Spin-PLDM method,
act quantum results (black solid) for the dynamics of hx22 i. while PLDM significantly underestimates the oscillation
GDTWA slightly outperforms the Spin-PLDM result (red amplitude and the Ehrenfest method loses half a period
dashed) for the dynamics of hx216 i, while GDTWA is notice- within about five to ten oscillations. Figure 3 presents
ably more accurate than all the other methods for the dynam- the propagation of hx21 i, hx26a i, and hx210a i. In general, we
ics of hx218 i. should not expect the linearized semi-classical methods
to work reliably for such higher-order correlations. As9
the numerical results suggest, Spin-PLDM and GDTWA ACKNOWLEDGMENTS
nevertheless still give qualitatively satisfactory results,
while PLDM and the Ehrenfest method rather quickly We acknowledge support by Provincia Autonoma di
accumulate uncontrolled errors. Trento, the ERC Starting Grant StrEnQTh (Project-ID
The relaxation dynamics of the more complex Model II 804305), Q@TN — Quantum Science and Technology in
is considerably more challenging for the linearized semi- Trento.
classical methods because several states are involved si-
multaneously in the relaxation dynamics. GDTWA is the
only one among the selected semi-classical methods to DATA AVAILABILITY
qualitatively correctly capture the relaxation dynamics,
as seen in the diabatic populations in Figure 4. In Fig- The data that support the findings of this study are
ures 5, 6, and 7, we show the populations of the tuning available within the article.
coordinates as well as their diagonal second-order cor-
relations, and the second-order diagonal correlations of
the coupling coordinates, respectively. PLDM and the Appendix A: Explicit form of Λ̂µ with N = 2 and N = 3
Ehrenfest method display significant deviations from the
exact dynamics. In contrast, GDTWA yields surprisingly The Λ̂µ used in the main text form the basis of SU (N ),
accurate predictions, for some observables even slightly and can thus be represented as N − 1 matrices of size
but noticeably better than Spin-PLDM. N × N , plus the identity matrix.
When N = 2, the basis elements are simply propor-
tional to the Pauli matrices,
! !
1 0 1 1 0 −i
V. CONCLUSIONS Λ̂1 = √ , Λ̂2 = √ ,
2 1 0 2 i 0
! ! (A1)
1 1 0 1 1 0
In this paper, we have introduced a recently devel- Λ̂3 = √ , Λ̂4 = √ .
oped method from the TWA family, GDTWA, to chem- 2 0 −1 2 0 1
ical non-adiabatic systems. The novelty and strength
of GDTWA is to sample the electron DoF in a discrete When N = 3, they are proportional to the Gell–Mann
phase space. We have also re-written the GDTWA in a matrices,
form similar to the Ehrenfest method, with the aim of
showcasing similarities and differences to more conven- 0 1 0 0 0 1
1 1
tional methods. Formally, the EOMs of GDTWA are Λ̂1 = √ 1 0 0 , Λ̂2 = √ 0 0 0 ,
identical to fully linearized mapping approaches. By the 2 2
0 0 0 1 0 0
spectral decomposition of the electron EOM, we demon-
strate that the fundamental difference between GDTWA 0 0 0 0 0 1
1 1
and traditional approaches is that GDTWA has two cou- Λ̂3 = √ 0 0 1 , Λ̂4 = √ 0 0 0 ,
pled correlated electron states in each single classical tra- 2 2
0 1 0 1 0 0
jectory, and hence can be regarded as a partially lin-
earized approach. GDTWA also accounts for an effec- 0 −i 0 0 0 0
1 1
tive ZPE without an explicit ZPE parameter. Numer- Λ̂5 = √ i 0 0 , Λ̂6 = √ 0 0 −i , (A2)
ical benchmarks show the validity of GDTWA for non- 2 2
0 0 0 0 i 0
adiabatic systems. For the two benchmark LVC models
in this paper, GDTWA displays qualitative and quantita- 1 0 0 1 0 0
1 1
tive accuracy compared to the quantum description. For Λ̂7 = √ 0 −1 0 , Λ̂8 = √ 0 1 0 ,
one of the considered models, it even outperforms the 2 6
0 0 0 0 0 −2
Spin-PLDM, which is the only other of the considered
methods to display an at least qualitative agreement for 1 0 0
1
most of the considered situations. Λ̂9 = √ 0 1 0 .
3
Various extensions of the GDTWA are in progress, 0 0 1
namely, the coupling of the system to time-dependent
electromagnetic fields and the extension of GDTWA to
simulations in the adiabatic representation, which will Appendix B: Sampling of the intra-electronic correlation
enable, e.g., on-the-fly simulations in conjunction with
usual electronic structure packages for the electronic The faithful sampling for the intra-electronic correla-
structure. The performance of the method in such sce- tion is crucial for the accuracy of GDTWA for the non-
narios will be reported in future works. adiabatic dynamics. The reason is that, once there is a10
non-zero nuclei-electron coupling, the intra-electron cor- electronic correlations affect the accuracy of GDTWA.
relation terms appear in the higher-order time derivatives After a lengthy but straightforward calculation, we ob-
of the EOMs. We report the detailed analysis for the tain the second- and the third-order time derivative of
diabatic basis in this appendix to show how the intra- λα and Λ̂α ,
d2 λµ (t) pt pt
2
= fµνξ [∂xt vν (xt ) λξ + vν (xt ) fξδ vδ (xt )λ ] , (B1)
dt m m
d2 Λ̂µ (t) ∂x vν (x̂t )p̂t vδ (x̂t )vν (x̂t )p̂t
2
= fµνξ [ Λ̂ξ + fξδ Λ̂ ] + h.c. , (B2)
dt 2m 2m
d3 λµ (t) 2 p2t 1 pt
= f µνξ [∂ x v ν (x t ) λξ − ∂xt vν (xt ) ∂xt vζ (xt )λζ λξ + ∂xt vν (xt ) fξδ vδ (xt )λ ] , (B3)
dt3 t
m2 m m
d3 Λ̂µ (t) ∂x̂4t vν (x̂t ) + 4∂x̂2t vν (x̂t )p̂2t 1 vδ (x̂t )∂x̂t vν (x̂t )p̂t
3
= fµνξ [ 2
Λ̂ξ − ∂x̂t vν (x̂t ) ∂x̂t vζ (x̂t )Λ̂ζ Λ̂ξ + fξδ Λ̂ ] + h.c. , (B4)
dt 8m 2m 2m
where h.c. is the Hermitian conjugate. We focus on the which means the discrete sampling of this state is faithful
short time t ∼ 0 accuracy. As for the separable initial for the intra-electron correlation only if χ = 0, π, or
state ρ(0) the statistical average of Eq. (B1) is identi- ±π/2.
cal to the quantum expectation value of Eq. (B2), the
GDTWA is at least accurate up to O(t2 ). Meanwhile,
the statistical average of Eq. (B3) equals the quantum Appendix C: Different ZPE between SM and GDTWA for
expectation value of Eq. (B4) if Eq. (12), the condition block diagonal Hamiltonians
of faithful statistical sampling of the initial intra-electron
correlations, is fulfilled. Thus, in this case the accuracy Though SM and GDTWA have an identical dimen-
of GDTWA is improved for the short time dynamics, as it sion dependency of the ZPE, they may behave differ-
is ensured to be exact at least up to and including O(t3 ). ently when the Hamiltonian is block diagonal. Con-
We stress that “intra-electron correlation” here de- sider a simple N × N Hamiltonian with the elements
notes only a feature of statistical sampling, to be distin- Hkl = 0 for M < k ≤ N , 1 ≤ l ≤ M and 1 ≤ k ≤ M ,
guished from the correlation between nuclear and elec- M < l ≤ N . The first M diabatic states are decou-
tronic DoFs, or the static correlation and dynamical cor- pled from the other N − M states. Again, we only
relation in the electronic structure theory. We illustrate consider the initial state |1i h1|. As before, we denote
how the discrete sampling fails to represent the intra- the electron phase space variable of the N × N full
electronic correlation at the example of an explicit state electron system as Aα (t) and Bα (t) while the subma-
without the nuclei-electron √ correlation. Consider the trix Aα (t)[1, 2, · · · , M ; 1, 2, · · · , M ] is indicated as AM
α (t)
state |Ψi = (|1i + eiχ |2i)/ 2 for a two-level system, (and analogously for Bα ).
where the discrete sampling gives the probability distri-
Since the first M diabatic states are decoupled from the
bution
others, it is also possible to sample the M ×M subsystem
1 1 ± cos χ directly. We use ÃM M
p(λ1 = ± √ ) = , α (t) and B̃α (t) to represent the elec-
2 2 tron phase space variables obtained by sampling from the
1 1 ± sin χ M ×M subsystem. It is easy to check that the initial dis-
p(λ2 = ± √ ) = , (B5) tributions of AM M
2 2 α (0) and Ãα (0) are identical. Moreover,
1 1 the classical trajectories satisfy AM M
α (t) = Ãα (t) if their
p(λ3 = ± √ ) = . initial conditions are the same. Thanks to the implicit
2 2
ZPE parameter of GDTWA, all the physical quantities
With an explicit calculation, we obtain Λ̂1 Λ̂2 + Λ̂2 Λ̂1
= 0, are invariant independent of whether we use the N × N
2
while full electron system or the M × M subsystem.
The above arguments become much more subtle for the
X sin 2χ
p(λ1 = a1 )p(λ2 = a2 )a1 a2 = , (B6) SM approach with the dimension dependent ZPE param-
a ,a
4 eter. The initial distribution of BαM (0) and B̃αM (0) be-
1 211
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