A unified theoretical framework formapping models for the multi-stateHamiltonianCite as: J. Chem. Phys. 145, 204105 (2016); https://doi.org/10.1063/1.4967815Submitted: 13 September 2016 . Accepted: 02 November 2016 . Published Online: 28 November 2016
Jian Liu
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THE JOURNAL OF CHEMICAL PHYSICS 145, 204105 (2016)
A unified theoretical framework for mapping modelsfor the multi-state Hamiltonian
Jian Liua)
Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry,College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
(Received 13 September 2016; accepted 2 November 2016; published online 23 November 2016)
We propose a new unified theoretical framework to construct equivalent representations of the multi-state Hamiltonian operator and present several approaches for the mapping onto the Cartesian phasespace. After mapping an F-dimensional Hamiltonian onto an F+1 dimensional space, creation andannihilation operators are defined such that the F+1 dimensional space is complete for any com-bined excitation. Commutation and anti-commutation relations are then naturally derived, whichshow that the underlying degrees of freedom are neither bosons nor fermions. This sets the scenefor developing equivalent expressions of the Hamiltonian operator in quantum mechanics and theirclassical/semiclassical counterparts. Six mapping models are presented as examples. The frameworkalso offers a novel way to derive such as the well-known Meyer-Miller model. Published by AIPPublishing. [http://dx.doi.org/10.1063/1.4967815]
I. INTRODUCTION
There is considerable effort focused on developingapproaches for describing quantum mechanical effects withclassical and semiclassical dynamics. The lack of classicalanalogy with discrete quantum degrees of freedom has pre-sented a challenge such as non-adiabatic dynamics wheremulti-electronic states are involved. In 1979 Meyer and Millerfirst proposed a mapping of an F-electronic-state Hamiltonianoperator onto continuous degrees of freedom.1 In 1997 Stockand Thoss further presented a more rigorous way to derivethe Meyer-Miller Hamiltonian from Schwinger’s theory ofangular momentum.2 Since then the Meyer-Miller Hamilto-nian has provided a useful theoretical framework to develop(approximate) multi-state quantum dynamics methods.
Although the Meyer-Miller mapping model has success-fully been implemented into various examples, it is worthinvestigating other possible mappings. Cotton and Miller havealso pointed out that the Meyer-Miller mapping “is not the mostnatural one” and proposed a spin mapping model (SPM).3 Thepurpose of the paper is to present a new theoretical frameworkto consistently construct equivalent expressions of the multi-state Hamiltonian operator and then yield their mappings ontothe Cartesian phase space such that classical dynamics can beimplemented. The outline of the paper is as follows. Section IIbegins by reviewing the application of Schwinger’s oscillatormodel of angular momentum to the multi-state Hamiltonianoperator. It then introduces a consistent way to define a pair ofcreation and annihilation operators in a 2-state space, whichnaturally leads to their commutation and anti-commutationrelations. Section III presents several equivalent representa-tions of the multi-state Hamiltonian operator and producestheir classical counterparts. Section IV discusses the differ-ence between the proposed classical mapping models and the
spin mapping model of Ref. 3. Our conclusions are given inSection V.
II. MULTI-STATE HAMILTONIAN OPERATORAND CREATION AND ANNIHILATION OPERATORSA. Application of Schwinger’s oscillator modelof angular momentum
Consider a Hamiltonian operator for F orthonormal states,
H =F∑
m,n=1
Hnm |n〉 〈m|. (1)
The Hamiltonian matrix is often a real symmetric one, whereHnm = Hmn. For convenience, the reduced Planck constant isset to ~ = 1 throughout the paper. Stock and Thoss extendedSchwinger’s oscillator model of angular momentum,4,5 sug-gesting that state |n〉 can be mapped as
(2)
such that it is viewed as a single excitation from the vacuumstate
,
i.e.,
|n〉 = a+n���0⟩
. (3)
Here an excitation represents the occupation of the corre-sponding state. The vacuum state ���0
⟩is orthogonal to any
occupied state |n〉. Stock and Thoss followed Schwinger’s orig-inal oscillator model and represented the F continuous degreesof freedom by the conventional harmonic-oscillator creationand annihilation operators
{a+n , an
}, commutation relations of
which are [am, a+n
]= δmn (∀m, n) . (4)
0021-9606/2016/145(20)/204105/14/$30.00 145, 204105-1 Published by AIP Publishing.
204105-2 Jian Liu J. Chem. Phys. 145, 204105 (2016)
The multi-state Hamiltonian operator in Eq. (1) is thensuggested as2
H =F∑
m,n=1
Hnma+n am. (5)
Because a harmonic oscillator contains an infinite number ofenergy levels, the mapping of Eq. (5) is restricted onto theoscillator subspace with a single excitation, i.e., only levels 0and 1 are employed. (See more discussion in Appendix A.)
It would be more natural to map each of the F states onto adegree of freedom that has only two states—the vacuum stateand the occupied one. A crucial step for constructing such amapping is to seek a more natural definition of creation andannihilation operators rather than directly apply Schwinger’soscillator model.
B. Creation and annihilation operators
As the F states of Eq. (1) are orthonormal, one obtains⟨0 ���0
⟩= 1,⟨
n ���0⟩=
⟨0 |n〉 = 0,
〈m |n〉 = δmn ∀m, n ∈ {1, 2, · · · , F} .
(6)
Since only a single excitation is involved, the creation andannihilation operators can be defined as
a+n =���n⟩ ⟨
0��� ,
an =���0⟩ ⟨
n���.(7)
It is straightforward to verify
a+n |n〉 = 0, a+n |m〉 = 0 (n , m) , (8)
an���0⟩= 0, an |n〉 =
���0⟩
, an |m〉 = 0 (n , m) . (9)
Eqs. (6)–(9) suggest a consistent complete space of all exci-tations. Any combination of two creation operators leads tozero, i.e.,
a+n a+n = a+n a+m = 0. (10)
So does any combination of two annihilation operators, i.e.,
anan = anam = 0. (11)
The combinations of a pair of creation and annihilationoperators are
a+n an = |n〉 〈n| ,
a+n am = |n〉 〈m| ,
ana+n = ama+m =���0⟩ ⟨
0��� ,
ana+m = ama+n = 0 (n , m) .
(12)
Substituting Eq. (12) into Eq. (1) leads to Eq. (5) without anyambiguity.
It is trivial to show the commutation relations[a+n , a+n
]= 0,[
a+n , a+m]= 0,[
an, an]= 0,[
an, am]= 0,[
a+n , an]= |n〉 〈n| − ���0
⟩ ⟨0��� = σ
(n)z ,[
a+n , am]= |n〉 〈m| (n , m) ,
(13)
and the anti-commutation relations[a+n , a+n
]+ = 0,[
a+n , a+m]+ = 0,[
an, an]+ = 0,[
an, am]+ = 0,[
a+n , an]+ = |n〉 〈n| +
���0⟩ ⟨
0��� = 1(n)
,[a+n , am
]+ = |n〉 〈m| (n , m) .
(14)
Here 1(n)
is the identity operator for state |n〉. Importantly,Eqs. (13) and (14) suggest that the underlying degrees offreedom in Eq. (5) are neither bosons nor fermions.
Eqs. (7)–(14) set the scene for developing mapping mod-els. Below we introduce several approaches for mapping theHamiltonian operator of Eq. (5) onto the Cartesian phase spacesuch that classical dynamics can be employed.
III. EQUIVALENT EXPRESSIONS OF THE MULTI-STATE HAMILTONIAN OPERATOR AND THEIRMAPPING MODELS IN THE PHASE SPACE
Define the following operators:
σ(n)x = an + a+n ,
σ(n)y =
an − a+ni
.(15)
It is easy to show
iσ(n)x σ(n)
y = −iσ(n)y σ(n)
x = σ(n)z ,
σ(n)x σ(n)
x = σ(n)y σ(n)
y = σ(n)z σ(n)
z = 1(n)
,(16)
where σ(n)z is given in the fifth equation of Eq. (13). The
commutation relation isi2
[σ(n)
x , σ(n)y
]= σ(n)
z (17)
or takes a more general form
i2
[σ(n)
a , σ(n)b
]= εabcσ
(n)c . (18)
Here the Levi-Civita symbol εabc is equal to 1 for cyclic per-mutations of xyz, equal to −1 for anti-cyclic permutations, andequal to zero if index a and index b are repeated. Similarly, theanti-commutation relations are
12
[σ(n)
a , σ(n)b
]+= δab1
(n). (19)
Here δab is the Kronecker delta. When n , m, one furtherobtains
a+n am = iσ(n)x σ(m)
y = −iσ(n)y σ(m)
x = σ(n)x σ(m)
x
= σ(n)y σ(m)
y (n , m) . (20)
Note that Eqs. (16)–(19) suggest that{σ(n)
x , σ(n)y , σ(n)
z
}repre-
sent Pauli matrices (for a spin 1/2 particle).
A. A mapping model from the analogywith the classical angular momentum
It is trivial to derive from Eqs. (15)–(20)
a+n an =12
(1
(n)+
i2
[σ(n)
x , σ(n)y
]), (21)
204105-3 Jian Liu J. Chem. Phys. 145, 204105 (2016)
a+n am + a+man = i[σ(n)
x , σ(m)y
]= −i
[σ(n)
y , σ(m)x
](n , m) .
(22)
The multi-state Hamiltonian operator of Eq. (5) then becomes
H =∑
n
12
(1
(n)+
i2
[σ(n)
x , σ(n)y
])Hnn
+∑n<m
12
(i
[σ(n)
x , σ(m)y
]− i
[σ(n)
y , σ(m)x
] )Hnm. (23)
Eq. (23) is an equivalent expression of Eq. (1) in quantummechanics.
Because Schwinger’s formulation of angular momentumis employed in the mapping—Eqs. (2) and (3), it is natural toseek an analogy with classical angular momentum rather thanonly stick to Schwinger’s original oscillator model. Note thatclassical angular momentum is defined as
L = x × p. (24)
An alternative version of Eq. (24) is
xapb − xbpa = εabcLc. (25)
Here {xa, xb, xc} = {x, y, z} and {pa, pb, pc} = {px, py, pz}. Theanalogy between Eqs. (18) and (25) suggests a mapping toclassical angular momentum
i2
[σ(n)
a , σ(m)b
]7→ x(n)
a p(m)b − x(m)
b p(n)a (n , m) (26)
and
12
(1
(n)+
i2
[σ(n)
x , σ(n)y
])7→ x(n)p(n)
y − y(n)p(n)x . (27)
Note that 12
(1
(n)+ i
2
[σ(n)
x , σ(n)y
])instead of i
2
[σ(n)
x , σ(n)y
]is
employed in the left-hand side of Eq. (27). This is because thatspin does not depend on spatial coordinates and has no classical
analog. 12
(1
(n)+ σ(n)
z
)rather than σ(n)
z is more natural to be
treated as the z-axis component of a quantum spatial angularmomentum operator such that its analog leads to a classicalangular momentum. (Here Eq. (17) is used.) The eigenvalues
of the 2 × 2 matrix for 12
(1
(n)+ σ(n)
z
)are 0 and 1, which
indicates that the operator represents the z-axis component ofan angular momentum (Lz) which takes either 0 or 1. 0 whenthe state is unoccupied and 1 when it is occupied.
Inserting Eqs. (26) and (27) into Eq. (23) produces theclassical Hamiltonian
H =F∑
n=1
Hnn
(x(n)p(n)
y − y(n)p(n)x
)+
∑n<m
Hnm
[(x(n)p(m)
y − y(n)p(m)x
)+
(x(m)p(n)
y − y(m)p(n)x
)].
(28)
The simplified form is then
H =F∑
n,m=1
Hnm
(x(n)p(m)
y − y(n)p(m)x
). (29)
Its Hamilton’s equations of motion produce
x(n) =∂H
∂p(n)x
= −Hnny(n)−
∑m,n
Hnmy(m),
y(n) =∂H
∂p(n)y
= Hnnx(n) +∑m,n
Hnmx(m),
p(n)x = −
∂H
∂x(n)= −Hnnp(n)
y −∑m,n
Hnmp(m)y ,
p(n)y = −
∂H
∂y(n)= Hnnp(n)
x +∑m,n
Hnmp(m)x ,
(30)
which conserves the classical Hamiltonian Eq. (29) by defini-tion. Eqs. (29) and (30) are noted as Model I in the paper. It iseasy to show that the sum of occupation numbers is a constantof motion for Eq. (30), i.e.,
ddt
F∑n=1
(x(n)p(n)
y − y(n)p(n)x
)= 0. (31)
Interestingly, Model I is reminiscent of the semiclassicalsecond-quantized many-electron Hamiltonian (with only 1-electron interactions) proposed by Li and Miller.6 This kindof analogy suggests that it is possible to obtain a subtleconnection between the mapping model for the multi-stateHamiltonian and that for the many-electron Hamiltonian, aswill be discussed in our future work. (Note that electrons arefermions.)
B. A mapping model from the analogywith the quantum-classical correspondencefor two non-commutable operators
Eq. (15) leads to
a+n =12
(σ(n)
x − iσ(n)y
),
an =12
(σ(n)
x + iσ(n)y
).
(32)
It is straightforward to obtain
a+n an =14
(σ(n)
x σ(n)x + σ
(n)y σ(n)
y
)+
i4
[σ(n)
x , σ(n)y
], (33)
a+n am =14
(σ(n)
x σ(m)x + σ(n)
y σ(m)y
)+
i4
[σ(n)
x , σ(m)y
](n , m) ,
(34)
or
a+n am + a+man =14
( [σ(n)
x , σ(m)x
]++
[σ(n)
y , σ(m)y
]+
)+
i4
( [σ(n)
x , σ(m)y
]+
[σ(m)
x , σ(n)y
] )(n , m) .
(35)
The multi-state Hamiltonian operator of Eq. (5) then becomes
H =∑
n
(14
(σ(n)
x σ(n)x + σ
(n)y σ(n)
y
)+
i4
[σ(n)
x , σ(n)y
])Hnn
+∑n<m
14
( [σ(n)
x , σ(m)x
]++
[σ(n)
y , σ(m)y
]+
)Hnm
+∑n<m
i4
( [σ(n)
x , σ(m)y
]+
[σ(m)
x , σ(n)y
] )Hnm. (36)
204105-4 Jian Liu J. Chem. Phys. 145, 204105 (2016)
Eq. (36) is another equivalent expression of Eq. (1) in quantummechanics.
Recall the conventional quantum-classical correspon-dence for two non-commutable operators A and B
12
(AB + BA
)7→ A (x, p) B (x, p) ,
[A, B] 7→ 0.(37)
Applying Eq. (37) to Eq. (36) leads to the classical Hamil-tonian
H =∑
n
14
(σ(n)
x σ(n)x + σ
(n)y σ(n)
y
)Hnn
+∑n<m
12
(σ(n)
x σ(m)x + σ(n)
y σ(m)y
)Hnm. (38)
Making a change of variables
x(n) =σ(n)
x√
2,
p(n) =σ(n)
y√
2,
(39)
in Eq. (38) finally yields
H =F∑
n,m=1
12
(x(n)x(m) + p(n)p(m)
)Hnm. (40)
Its Hamilton’s equations of motion are then
x(n) =∂H
∂p(n)= Hnnp(n) +
∑m,n
Hnmp(m),
p(n) = −∂H
∂x(n)= −Hnnx(n)
−∑m,n
Hnmx(m),(41)
which preserves the classical Hamiltonian of Eq. (40). It is triv-ial to verify that the sum of occupation numbers is a constantof motion for Eq. (41), i.e.,
ddt
F∑n=1
(x(n)
)2+
(p(n)
)2
2= 0. (42)
Eqs. (40) and (41) are noted as Model II in the paper.If semiclassical/quasiclassical dynamics is employed, the
commutation relations in Eq. (36) may not be ignored. Usingthe parameters γ and η to describe the effects of i
4
[σ(n)
x , σ(n)y
]
and i4
( [σ(n)
x , σ(m)y
]+
[σ(m)
x , σ(n)y
] ), respectively, one then
obtains the Hamiltonian
HSC =∑
n
12
((x(n)
)2+
(p(n)
)2− γ
)Hnn
+∑n<m
(x(n)x(m) + p(n)p(m)
− η)Hnm. (43)
For instance, because 1/2 is an eigenvalue of the operator
i4
[σ(n)
x , σ(n)y
]=
σ(n)z2 , the value of the parameter γ can be
chosen as 1/2. More generally, an optimum value for γ can
be selected in the regime between the two eigenvalues, i.e.,[−1/
2, 1/2
]. When the commutation relations are not taken
into account (i.e., γ = 0 and η = 0), Eq. (43) approachesthe classical mapping Hamiltonian Eq. (40).
The classical Hamiltonian in Eq. (40) in Model II or thesemiclassical one in Eq. (43) is closely related to the well-known Meyer-Miller Hamiltonian
HMM =∑
n
12
((x(n)
)2+
(p(n)
)2− γ
)Hnn
+∑n<m
(x(n)x(m) + p(n)p(m)
)Hnm, (44)
where the parameter γ is set to 1/2 in Meyer and Miller’s
original version1,2 or chosen to be(√
3 − 1) /
2 or other
optimal values in its applications.7–11 The derivation pro-cedure of Eq. (40) or Eq. (43) is very different fromMeyer and Miller’s or from Stock and Thoss’ seminal workthough.1,2,12 As long as creation and annihilation operatorsare defined in Eq. (7), Eq. (13) demonstrates that conven-tional harmonic-oscillator commutation relations Eq. (4) donot hold.
C. Three mapping models from the analogywith the classical vector
Note that the Pauli matrix σa in physics represents theobservables corresponding to spin along the ath coordinateaxis. When only σx and σy appear in the Hamiltonian operator,it is natural to adopt the mapping of the two Pauli vectors ontotwo 2-dimensional vectors
σ(n)a σ(m)
b 7→(x(n)
a + ip(n)a
) (x(m)
b − ip(m)b
). (45)
The mapping Eq. (45) leads to
σ(n)a σ(n)
a =
[σ(n)
a , σ(n)a
]+
27→
(x(n)
a
)2+
(p(n)
a
)2,
[σ(n)
a , σ(m)a
]+
27→ x(n)
a x(m)a + p(n)
a p(m)a (n , m) ,
(46)
and
i2
[σ(n)
x , σ(n)y
]7→ x(n)p(n)
y − y(n)p(n)x ,
i2
[σ(n)
x , σ(m)y
]7→ x(n)p(m)
y − y(m)p(n)x (n , m) .
(47)
1. Model III
Substituting Eq. (19) into Eq. (23) produces the 3rd equiv-alent representation of the multi-state Hamiltonian operator inquantum mechanics
H =∑
n
14
(σ(n)
x σ(n)x + σ
(n)y σ(n)
y + i[σ(n)
x , σ(n)y
] )Hnn
+∑n<m
12
(i
[σ(n)
x , σ(m)y
]− i
[σ(n)
y , σ(m)x
] )Hnm. (48)
Applying the mapping Eqs. (46) and (47) to Eq. (48) yieldsthe classical Hamiltonian
204105-5 Jian Liu J. Chem. Phys. 145, 204105 (2016)
H =F∑
n=1
(x(n) + p(n)
y
)2+
(y(n) − p(n)
x
)2
4Hnn
+∑n,m
(x(n)p(m)
y − y(m)p(n)x
)Hnm. (49)
Hamilton’s equations of motion then read
x(n) =∂H
∂p(n)x
= Hnn
(p(n)
x − y(n))
2−
∑m,n
Hnmy(m),
y(n) =∂H
∂p(n)y
= Hnn
(x(n) + p(n)
y
)2
+∑m,n
Hnmx(m),
p(n)x = −
∂H
∂x(n)= −Hnn
(x(n) + p(n)
y
)2
−∑m,n
Hnmp(m)y ,
p(n)y = −
∂H
∂y(n)= Hnn
(p(n)
x − y(n))
2+
∑m,n
Hnmp(m)x ,
(50)
which preserves the classical Hamiltonian of Eq. (49).Eqs. (49) and (50) are noted as Model III in the paper. It isstraightforward to verify conservation of the sum of occupationnumbers for Eq. (50), i.e.,
ddt
F∑n=1
14
((x(n) + p(n)
y
)2+
(y(n)− p(n)
x
)2)= 0. (51)
2. Model IV
Similarly, Model IV is constructed from the 4-th equiva-lent representation of the multi-state Hamiltonian operator
H =∑
n
14
(σ(n)
x σ(n)x + σ
(n)y σ(n)
y + i[σ(n)
x , σ(n)y
] )Hnn
+∑n<m
12
( [σ(n)
x , σ(m)x
]++
[σ(n)
y , σ(m)y
]+
)Hnm. (52)
Note that Eq. (20) leads to
a+n am + a+man =12
(i
[σ(n)
x , σ(m)y
]− i
[σ(n)
y , σ(m)x
] )=
12
( [σ(n)
x , σ(m)x
]++
[σ(n)
y , σ(m)y
]+
)(n , m)
(53)
Eq. (52) is then derived from Eqs. (53) and (48). Substi-tuting Eqs. (46) and (47) into Eq. (52) yields the classicalHamiltonian of Model IV
H =F∑
n=1
(x(n) + p(n)
y
)2+
(y(n) − p(n)
x
)2
4Hnn
+∑n<m
((x(n)x(m) + p(n)
x p(m)x
)+
(y(n)y(m) + p(n)
y p(m)y
))Hnm.
(54)
Hamilton’s equations of motion become
x(n) =∂H
∂p(n)x
= Hnn
(p(n)
x − y(n))
2+
∑m,n
Hnmp(m)x ,
y(n) =∂H
∂p(n)y
= Hnn
(x(n) + p(n)
y
)2
+∑m,n
Hnmp(m)y ,
p(n)x = −
∂H
∂x(n)= −Hnn
(x(n) + p(n)
y
)2
−∑m,n
Hnmx(m),
p(n)y = −
∂H
∂y(n)= Hnn
(p(n)
x − y(n))
2−
∑m,n
Hnmy(m),
(55)
which conserves the classical Hamiltonian of Eq. (54). Conser-vation of the sum of occupation numbers for Eq. (55) can easilybe verified, which shares the same expression as Eq. (51).
3. Model V
When the equivalent representation of the multi-stateHamiltonian operator in Eq. (36) is used, employing the map-ping Eqs. (46) and (47) for Eq. (36) then leads to the classicalHamiltonian of Model V
H =F∑
n=1
(x(n) + p(n)
y
)2+
(y(n) − p(n)
x
)2
4Hnn +
∑n<m
(x(n) + p(n)
y
) (x(m) + p(m)
y
)+
(y(n) − p(n)
x
) (y(m) − p(m)
x
)2
Hnm. (56)
Its Hamilton’s equations of motion are
x(n) =∂H
∂p(n)x
= Hnn
(p(n)
x − y(n))
2+
∑m,n
Hnm
(p(m)
x − y(m))
2,
y(n) =∂H
∂p(n)y
= Hnn
(x(n) + p(n)
y
)2
+∑m,n
Hnm
(x(m) + p(m)
y
)2
,
p(n)x = −
∂H
∂x(n)= −Hnn
(x(n) + p(n)
y
)2
−∑m,n
Hnm
(x(m) + p(m)
y
)2
,
p(n)y = −
∂H
∂y(n)= Hnn
(p(n)
x − y(n))
2+
∑m,n
Hnm
(p(m)
x − y(m))
2,
(57)
which conserves the classical Hamiltonian of Eq. (56). Con-servation of the sum of occupation numbers for Eq. (57) sharesthe same expression as Eq. (51).
D. Other equivalent representations of the multi-stateHamiltonian operator
The theoretical framework (presented in Section IIand Sections III A–III C) yields four equivalent rep-resentations (Eqs. (23), (36), (48), and (52)) of themulti-state Hamiltonian operator (Eq. (1) or Eq. (5)) inquantum mechanics. All these equivalent representationsare expressed in terms of {σ(n)
x , σ(n)y } in the theoretical
framework.
204105-6 Jian Liu J. Chem. Phys. 145, 204105 (2016)
More equivalent representations can be proposed as well.For instance, substituting Eqs. (21) and (53) into Eq. (5) pro-duces the fifth equivalent representation of the multi-stateHamiltonian operator in quantum mechanics
H =∑
n
12
(1
(n)+
i2
[σ(n)
x , σ(n)y
])Hnn
+∑n<m
12
( [σ(n)
x , σ(m)x
]++
[σ(n)
y , σ(m)y
]+
)Hnm. (58)
Applying the similar strategies (introduced in the previous partof the section) to Eq. (58) leads to the classical Hamiltonianof Model VI in the Cartesian phase space
H =F∑
n=1
Hnn
(x(n)p(n)
y − y(n)p(n)x
)+
∑n<m
((x(n)x(m) + p(n)
x p(m)x
)+
(y(n)y(m) + p(n)
y p(m)y
))Hnm.
(59)
Its equations of motion become
x(n) =∂H
∂p(n)x
= −Hnny(n) +∑m,n
Hnmp(m)x ,
y(n) =∂H
∂p(n)y
= Hnnx(n) +∑m,n
Hnmp(m)y ,
p(n)x = −
∂H
∂x(n)= −Hnnp(n)
y −∑m,n
Hnmx(m),
p(n)y = −
∂H
∂y(n)= Hnnp(n)
x −∑m,n
Hnmy(m).
(60)
Conservation of the sum of occupation numbers for Eq. (60)can easily be verified, which shares the same expression asEq. (31).
Similarly, more quantum-classical analogies or more clas-sical mapping models in the Cartesian phase space can beproposed in the theoretical framework.
IV. SPIN MAPPING MODEL OF COTTON AND MILLER
Note that Eq. (17) leads to an equivalent expression ofEq. (58)
H =∑
n
12
(1
(n)+ σ(n)
z
)Hnn
+∑n<m
12
( [σ(n)
x , σ(m)x
]++
[σ(n)
y , σ(m)y
]+
)Hnm. (61)
Employing the transformation
S(n)a =
σ(n)a
2(a = x, y, or z) , (62)
one obtains an equivalent expression of Eq. (61)
H =∑
n
(12
1(n)+ S(n)
z
)Hnn
+∑n<m
( [S(n)
x , S(m)x
]++
[S(n)
y , S(m)y
]+
)Hnm. (63)
If the approximation[S(n)
a , S(m)a
]+≈ 2S(n)
a S(m)a is employed,
Eq. (63) then becomes
H =∑
n
(12
1(n)+ S(n)
z
)Hnn + 2
∑n<m
(S(n)
x S(m)x + S(n)
y S(m)y
)Hnm,
(64)
which is the quantum Hamiltonian operator used for the spinmapping model in Ref. 3. Cotton and Miller replaced the spinoperator in Eq. (64) with the classical angular momentumvector
S(i) ≡
S(i)x
S(i)y
S(i)z
=
√S2 −
(m(i))2 cos
(q(i)
)√S2 −
(m(i))2 sin
(q(i)
)m(i)
(65)
and then obtained the spin mapping Hamiltonian3
H =∑
i
(12+ S(i)
z
)Hii + 2
∑i<j
(S(i)
x S(j)x + S(i)
y S(j)y
)Hij. (66)
Here{m(i), q(i)
}are the action-angle variables for the ith degree
of freedom, where the parameter S2 is suggested to be thequantum value S2 = 3/4. The equations of motion read
q(i) =∂H
∂m(i)= Hii −
2m(i)√S2 −
(m(i))2
×∑j,i
Hij
√S2 −
(m(j))2 cos
(q(i) − q(j)
),
m(i) = −∂H
∂q(i)= 2
√S2 −
(m(i))2
×∑j,i
Hij
√S2 −
(m(j))2 sin
(q(i) − q(j)
),
(67)
a more compact form3 of which is
dS(i)
dt=
∂H
∂S(i)× S(i). (68)
The spin mapping model is then a nonlinear system, which isvery different from the quadratic Hamiltonian models (ModelsI-VI) that are obtained in the theoretical framework. Note thatthe spin mapping model does not employ an equivalent repre-sentation expressed in terms of
{σ(n)
x , σ(n)y
}for the multi-state
Hamiltonian operator (Eq. (1) or Eq. (5)). This is also differentfrom Models I-VI constructed in the theoretical framework.
The spin mapping model [Eqs. (66) and (65)] can not beverified to be an exact mapping model of the time-dependentSchrodinger equation (TDSE). It works well when the cou-pling terms {Hnm} (n , m) are weak but fails in the strongcoupling region. (See Appendix C.) It is not a surprise. Wepoint out in Section III A that spin does not depend on spa-tial coordinates and has no classical analog. S(n)
z is not naturalto be treated as the z-axis component of a quantum spatialangular momentum operator. That is, Eq. (65) is not a goodanalogy. Cotton and Miller have already demonstrated that thespin mapping model [Eqs. (65) and (66)] is less accurate thanthe Meyer-Miller mapping model in Ref. 3. This is mostlybecause that the spin mapping model is not exact even whenthe nuclear motions are frozen.
204105-7 Jian Liu J. Chem. Phys. 145, 204105 (2016)
V. CONCLUSION REMARKS
In this paper, we present a new unified theoretical frame-work to construct equivalent representations of the multi-stateHamiltonian operator and propose several approaches for themapping onto the Cartesian phase space. Below we list thethree key elements:
(1) Extend Schwinger’s formulation to map the F-dimensional Hamiltonian operator onto an F + 1 dimen-sional space. (That is, Eqs. (2) and (3), as first introducedby Stock and Thoss.2)
(2) Define creation and annihilation operators as inEq. (7) such that the F + 1 dimensional space is completefor all (combined) excitations. Commutation and anti-commutation relations are then naturally constructed.
(3) Derive equivalent representations of the Hamiltonianoperator (in terms of
{σ(n)
x , σ(n)y
}) and propose the cri-
teria for mapping them onto the Cartesian phase spacesuch that classical dynamics can be employed.
Three quantum-classical analogies (or criteria) are pro-posed. Six classical Hamiltonian models (namely, Eqs. (29),(40), (49), (54), (56), and (59)) are then developed asexamples. (Similarly, semiclassical/quasiclassical models canalso be developed although they are not explicitly shownin the present paper.) Each of Models I-VI involves aHamiltonian that has only quadratic terms in the Carte-sian phase space. It suggests that the six different clas-sical mapping models can lead to exact quantum resultsif initial conditions are carefully constructed (as discussedin Appendices B and C). Interestingly, Section III Bpresents a novel derivation for the conventional Meyer-Millermodel.1,2,12
Although the quantum Hamiltonian operator used inRef. 3 [i.e., Eq. (64)] is closely related to an equivalent expres-sion of the multi-state Hamiltonian operator [i.e., Eq. (58)] inthe theoretical framework, the spin mapping model of Cottonand Miller3 [Eqs. (65) and (66)] for Eq. (64) does not have the3rd key element listed above. It can not be verified to be exact.(See Appendix C.)
Finally, we note that the unified framework offers away to develop more equivalent representations of the multi-state Hamiltonian operator and their classical/semiclassicalmapping models that are able to produce exact results. Itwill be interesting in future work to seek an optimal andeconomy classical/semiclassical mapping model for study-ing real complex systems. When the state in Eq. (1) is theelectronic state, Hnm = Hnm (R) are often general functionsof the nuclear coordinates R, so that adding the nuclearkinetic energy operator 1
2 pT M−1p to Eq. (1) leads to thewhole nuclear-electronic Hamiltonian operator. It will alsobe interesting to include the nuclear degrees of freedomin the classical mapping models for the multi-state sys-tem. Further investigation along these directions is certainlywarranted.
ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-ence Foundation of China (NSFC) Grant Nos. 21373018 and
21573007, by the Recruitment Program of Global Experts,by Specialized Research Fund for the Doctoral Programof Higher Education No. 20130001110009, by the Min-istry of Science and Technology of China (MOST) GrantNo. 2016YFC0202803, and by Special Program for AppliedResearch on Super Computation of the NSFC-GuangdongJoint Fund (the second phase). We acknowledge the Beijingand Tianjin supercomputer centers for providing computa-tional resources.
APPENDIX A: CREATION AND ANNIHILATIONOPERATORS IN TERMS OF EIGENSTATESFOR A HARMONIC OSCILLATOR
Consider the Hamiltonian for a unit mass and unit fre-quency harmonic oscillator
H =12
(x2 + p2
). (A1)
Its eigenstates {|j〉} satisfy
H |j〉 =
(j +
12
)|j〉 , j = 0, 1, 2, · · · . (A2)
The system consists of an infinite number of eigenstates. It isheuristic to think how the creation and annihilation operatorscan be represented in terms of eigenstates.
Consider the lowest s + 1 eigenstates which span an(s + 1)-dimensional state space. The expectation value ofany physical property of interest is first expressed in the(s + 1)-dimensional state space, then we take the limit s→ ∞to obtain the correct result. The annihilation operator is givenby
a = |0〉 〈1| +√
2 |1〉 〈2| + · · · +√
s |s − 1〉 〈s| (A3)
and its Hermitian conjugate produces the creation operator
a+ = |1〉 〈0| +√
2 |2〉 〈1| + · · · +√
s |s〉 〈s − 1| . (A4)
The number operator
N = a+a (A5)
becomes
N =s∑
j=0
j |j〉 〈j |. (A6)
Eqs. (A3) and (A4) lead to the commutation relation[a, a+
]= 1 − (s + 1) |s〉 〈s| . (A7)
As s tends to infinity, the second term of the right-hand side ofEq. (A7) has no effect when the commutator
[a, a+
]operates
on any physical state.13 For example, the expectation of energyfor a general physical state |Φ〉 is
〈Φ| H |Φ〉 = lims→∞
s∑j=1
|〈Φ| j〉|2(j +
12
). (A8)
Note that the expectation value of energy for a physical stateis always finite. This ensures that either |〈Φ|s〉|2 s or |〈Φ|s〉|2
204105-8 Jian Liu J. Chem. Phys. 145, 204105 (2016)
must approach zero when s tends to infinity. The expectationvalue of
[a, a+
]is
〈Φ|[a, a+
]|Φ〉 = 1 − (s + 1) |〈s |Φ〉|2 . (A9)
The second term of the right-hand side of Eq. (A9) mustvanish as s tends to infinity. That is, the physical-statecommutator [
a, a+]
p = 1 (A10)
is sufficient when acting on any physical state.13
When s is finite, the conventional commutation rela-tion (Eq. (A10)) does not hold. That is to say, if onlyfinite eigenstates are employed in Eqs. (A3) and (A4), theoperators
x =a + a+√
2,
p =a − a+√
2i
(A11)
do not lead to the well-known commutation relation[x, p
]= i. (A12)
APPENDIX B: MODELS I-VI ARE EXACT MAPPINGMODELS OF THE TIME-DEPENDENTSCHRODINGER EQUATION
The amplitudes {cn (t)} for being in the different statesat time t are determined by the standard time-dependentSchrodinger equation (TDSE),
icn (t) =F∑
m=1
Hnmcm (t). (B1)
Make a change of variables
cn (t) = x(n) (t) + ip(n) (t) , (B2)
where both x(n) (t) and p(n) (t) are real. Substituting Eq. (B2)into Eq. (B1), one obtains the equations of motion
x(n) (t) =F∑
m=1
Hnmp(m) (t),
p(n) (t) = −F∑
m=1
Hnmx(m) (t).
(B3)
Eq. (B3) is identical to Eq. (41), which is derived fromthe classical Hamiltonian Eq. (40) in Model II. Model II(Eqs. (40) and (41)) is then an exact mapping model in quan-tum mechanics, irrespective of that it is derived as a classicalmapping model of Eq. (36). We note that the action-angle ver-sion of Eq. (B3) was already presented in Meyer and Miller’soriginal work.1
We then consider Model I. Note that Model I has fourvariables while Model II has only two. Make the change ofvariables
x = q, y = −pq, px = pr , py = r (B4)
in the equations of motion of Model I (Eq. (30)). It is trivial toshow that Eq. (30) becomes
q(n) = Hnnp(n)q +
∑m,n
Hnmp(m)q ,
p(n)q = −Hnnq(n)
−∑m,n
Hnmq(m),
p(n)r = −Hnnr(n)
−∑m,n
Hnmr(m),
r(n) = Hnnp(n)r +
∑m,n
Hnmp(m)r .
(B5)
Note that the equations of motion for{r(n), p(n)
r
}are identical
to those of{q(n), p(n)
q
}. If the initial condition is chosen as
r(n) (0) = q(n) (0) , p(n)r (0) = p(n)
q (0) , (B6)
then two of the four variables in Eq. (B5) are redundant. Makethe transformation
x(n) =q(n) + r(n)
√2
,
p(n) =p(n)
q + p(n)r
√2
.
(B7)
Eq. (B5) then reduces to Eq. (B3), the equations of motionfor the TDSE in the Cartesian phase space. That is, ModelI is also in principle an exact mapping model of the TDSEonto the Cartesian phase space, although a different analogyis employed for developing it.
Similarly, it is trivial to verify that the equations of motionof any model in Models III-VI are the same as Eq. (B3) of theTDSE, when the initial condition
x(n) (0) = p(n)y (0) , y(n) (0) = −p(n)
x (0) (B8)
is employed.Although these different classical mapping models for the
multi-state Hamiltonian operator [Eq. (5)] are generated fromdifferent analogies as shown in Sections III A–III D, all the sixdistinct models (Models I-VI) are equivalent expressions of theTDSE when their initial conditions are carefully constructed.
Finally, we note that not all mapping models for equivalentexpressions of the quantum multi-state Hamiltonian operator[Eq. (5)] are exact. For instance, the spin mapping model ofCotton and Miller3 and the semiclassical mapping model ofSwenson et al.14 do not lead to exact results for Eq. (5). Theunified framework proposed in the paper presents a system-atic approach to construct classical mapping models in theCartesian phase space for the multi-state Hamiltonian operatorwhich are able to produce exact results.
APPENDIX C: ALGORITHMS AND NUMERICALEXAMPLES
Below we show the algorithms for Models I-VI and thentest them with a 3-state model.
204105-9 Jian Liu J. Chem. Phys. 145, 204105 (2016)
1. Algorithms for Models I-VIa. Model I
A symplectic algorithm for propagating the trajectorythrough a time interval ∆t for Eq. (30) is
x(n)← x(n)
−∆t2
*,Hnny(n) +
∑m,n
Hnmy(m)+-
,
p(n)x ← p(n)
x −∆t2
*,Hnnp(n)
y +∑m,n
Hnmp(m)y
+-
,
y(n)← y(n) + ∆t *
,Hnnx(n) +
∑m,n
Hnmx(m)+-
,
p(n)y ← p(n)
y + ∆t *,Hnnp(n)
x +∑m,n
Hnmp(m)x
+-
,
x(n)← x(n)
−∆t2
*,Hnny(n) +
∑m,n
Hnmy(m)+-
,
p(n)x ← p(n)
x −∆t2
*,Hnnp(n)
y +∑m,n
Hnmp(m)y
+-
.
(C1)
If the initial state is |n〉, i.e., the occupation number of state |n〉is 1 while those of the other states are 0,
x(n)p(n)y − y(n)p(n)
x = 1,
x(m)p(m)y − y(m)p(m)
x = 0 (m , n) ,(C2)
the initial condition at time t = 0 is then constructed as
x(n) (0) = cos θ,
y(n) (0) = − sin θ,
p(n)x (0) = sin θ,
p(n)y (0) = cos θ,
x(m) (0) = 0,
y(m) (0) = 0,
p(m)x (0) = 0,
p(m)y (0) = 0 (m , n) .
(C3)
Here θ can be any real number between 0 and 2π.
b. Model II
A symplectic algorithm for propagating the trajectorythrough a time interval ∆t for Eq. (41) is
p(n)← p(n)
−∆t2
*,Hnnx(n) +
∑m,n
Hnmx(m)+-
,
x(n)← x(n) + ∆t *
,Hnnp(n) +
∑m,n
Hnmp(m)+-
,
p(n)← p(n)
−∆t2
*,Hnnx(n) +
∑m,n
Hnmx(m)+-
.
(C4)
If the initial state is |n〉, the occupation number representationis
12
((x(n)
)2+
(p(n)
)2)= 1,
12
((x(m)
)2+
(p(m)
)2)= 0 (m , n) .
(C5)
The initial condition at time t = 0 is then constructed as
x(n) (0) =√
2 cos θ,
p(n) (0) =√
2 sin θ,
x(m) (0) = 0,
p(m) (0) = 0 (m , n) .
(C6)
Here θ can be any real number between 0 and 2π.
c. Model III
A symplectic algorithm for propagating the trajectorythrough a time interval ∆t for Eq. (50) is then proposed as
Step 1: propagate{x(n), p(n)
x
}for a half time interval
∆t2
,
Step 2: propagate{y(n), p(n)
y
}for a time interval ∆t,
Step 3: propagate{x(n), p(n)
x
}for another half time interval
∆t2
.
(C7)
Step 1 and Step 3 share the same procedure
p(n)x ← p(n)
x −∆t8
Hnnx(n),
x(n)← x(n)
−∆t4
*,
12
Hnny(n) +∑m,n
Hnmy(m)+-
,
p(n)x ← p(n)
x −∆t4
*,
12
Hnnp(n)y +
∑m,n
Hnmp(m)y
+-
,
x(n)← x(n) +
∆t4
Hnnp(n)x ,
x(n)← x(n)
−∆t4
*,
12
Hnny(n) +∑m,n
Hnmy(m)+-
,
p(n)x ← p(n)
x −∆t4
*,
12
Hnnp(n)y +
∑m,n
Hnmp(m)y
+-
,
p(n)x ← p(n)
x −∆t8
Hnnx(n),
(C8)
and Step 2 reads
p(n)y ← p(n)
y −∆t4
Hnny(n),
y(n)← y(n) +
∆t2
*,
12
Hnnx(n) +∑m,n
Hnmx(m)+-
,
p(n)y ← p(n)
y +∆t2
*,
12
Hnnp(n)x +
∑m,n
Hnmp(m)x
+-
,
y(n)← y(n) +
∆t2
Hnnp(n)y ,
y(n)← y(n) +
∆t2
*,
12
Hnnx(n) +∑m,n
Hnmx(m)+-
,
p(n)y ← p(n)
y +∆t2
*,
12
Hnnp(n)x +
∑m,n
Hnmp(m)x
+-
,
p(n)y ← p(n)
y −∆t4
Hnny(n).
(C9)
204105-10 Jian Liu J. Chem. Phys. 145, 204105 (2016)
When the initial state is |n〉, i.e.,
14
((x(n) + p(n)
y
)2+
(y(n)− p(n)
x
)2)= 1,
14
((x(m) + p(m)
y
)2+
(y(m)− p(m)
x
)2)= 0 (m , n) ,
(C10)
the initial condition at time t = 0 can be constructed the sameas Eq. (C3).
d. Model IV
A symplectic algorithm for propagating the trajectorythrough a time interval ∆t for Eq. (55) is
Step 1: propagate{p(n)
x , p(n)y
}for a half time interval
∆t2
,
Step 2: propagate{x(n), y(n)
}for a time interval ∆t,
Step 3: propagate{p(n)
x , p(n)y
}for another half time interval
∆t2
,
(C11)
where either Step 1 or Step 3 reads
p(n)x ← p(n)
x −∆t8
Hnnp(n)y ,
p(n)x ← p(n)
x −∆t4
*,
12
Hnnx(n) +∑m,n
Hnmx(m)+-
,
p(n)y ← p(n)
y −∆t4
*,
12
Hnny(n) +∑m,n
Hnmy(m)+-
,
p(n)y ← p(n)
y +∆t4
Hnnp(n)x ,
p(n)x ← p(n)
x −∆t4
*,
12
Hnnx(n) +∑m,n
Hnmx(m)+-
,
p(n)y ← p(n)
y −∆t4
*,
12
Hnny(n) +∑m,n
Hnmy(m)+-
,
p(n)x ← p(n)
x −∆t8
Hnnp(n)y ,
(C12)
and Step 2 is
x(n)← x(n)
−∆t4
Hnny(n),
x(n)← x(n) +
∆t2
*,
12
Hnnp(n)x +
∑m,n
Hnmp(m)x
+-
,
y(n)← y(n) +
∆t2
*,
12
Hnnp(n)y +
∑m,n
Hnmp(m)y
+-
,
y(n)← y(n) +
∆t2
Hnnx(n),
x(n)← x(n) +
∆t2
*,
12
Hnnp(n)x +
∑m,n
Hnmp(m)x
+-
,
y(n)← y(n) +
∆t2
*,
12
Hnnp(n)y +
∑m,n
Hnmp(m)y
+-
,
x(n)← x(n)
−∆t4
Hnny(n).
(C13)
When the initial state is |n〉, the occupation number represen-tation is the same as Eq. (C10) and the corresponding initialcondition at time t = 0 is set the same as Eq. (C3).
FIG. 1. Population dynamics of the 3-state Hamiltonian system. (Its param-eters are given by Eqs. (C15) and (C16).) The initial state is |1〉. The couplingbetween any two states is λ = 0.02 in Eq. (C16). (a) Population of state |1〉 (asa function of time). (b) Population of state |2〉. (c) Population of state |3〉. Solidline: exact results. Solid squares: results of Model I. Solid triangles: resultsof Model II. Solid rhombuses: results of Model III. Solid circles: results ofModel IV. Crosses: results of Model V. Hollow squares: results of Model VI.(~ = 1).
204105-11 Jian Liu J. Chem. Phys. 145, 204105 (2016)
FIG. 2. As in Fig. 1. The coupling between any two states is λ = 0.2.
Similar algorithms can be proposed for Models V and VI,as done for Models III and IV. We do not repeat the procedure.(We also note that Models III-VI can also employ the same
FIG. 3. As in Fig. 1. The coupling between any two states is λ = 2.
algorithm as Eq. (C1) when Eq. (C3) is the initial condition attime t = 0 although the performance is not as good for thesemodels.) It should be emphasized that a single trajectory is
204105-12 Jian Liu J. Chem. Phys. 145, 204105 (2016)
FIG. 4. As in Fig. 1. The coupling between any two states is λ = 20.
sufficient for obtaining dynamics of underlying degrees offreedom in each classical mapping model of Models I-VI.(Note that {Hnm} in Eq. (1) are time-independent in the presentpaper.)
FIG. 5. Comparison between Model I and Model II for the population ofstate |1〉 as a function of time for the 3-state Hamiltonian system in theweak coupling regime. (Its parameters are given by Eqs. (C15) and (C16).)The initial state is |1〉. The coupling between any two states is λ = 0.02 inEq. (C16). Solid line (black): exact results. Dotted line (blue): Model I (timeinterval ∆t = 10−3). Dashed line (red): Model II (time interval ∆t = 10−3).Dotted-dashed line (green): Model II (time interval ∆t = 10−4). Short-dashedline (purple): Model II (time interval ∆t = 10−5). Panel (b) is a blow-up ofPanel (a) for the time regime [2.9, 3.4].
The symplectic algorithms of Models I-VI can also beviewed as robust numerical integrators for solving the TDSE.
2. Three-state model
A simple but non-trivial case is a 3-state system. TheHamiltonian operator is given by
H =3∑
n=1
Hnn |n〉 〈n| +∑m<n
Hmn (|m〉 〈n| + |n〉 〈m|). (C14)
The diagonal terms in Eq. (C14) are
H11 = 10, H22 = 7, H33 = 2, (C15)
204105-13 Jian Liu J. Chem. Phys. 145, 204105 (2016)
and the off-diagonal ones are
H12 = H13 = H23 = λ, (C16)
where λ is a parameter. Its value is set to be 0.02, 0.2, 2, or 20to cover from the weak coupling regime to the strong couplingdomain.
The initial state is chosen to be |1〉 (or the initial den-sity is |1〉 〈1|). The corresponding initial condition in ModelI or III-VI is then given by Eq. (C3), while that in Model IIis constructed as Eq. (C6). The results of any one of the sixclassical mapping models are independent of the value of θ inEq. (C3) or Eq. (C6). Figs. 1–4 present the population of eachbasis state of {|n〉} as a function of time produced by the sixmodels. Comparison to the quantum results shows that eachof the six models leads to exact population dynamics in all testcases from the weak coupling regime to the strong couplingdomain.
The time interval of the trajectory propagation in ModelI or III-VI is often larger than that in Model II for achievingthe same accuracy, especially in the weak coupling regime.For example, Fig. 5 shows that the time interval in Model Iis about 100 times of that in Model II for achieving the sameaccuracy when the coupling is λ = 0.02 in Eq. (C16). Whilethe six different classical mapping models perform similarlyin the strong coupling domain, Model I and Models III-VIdemonstrate better numerical performance than Model II inthe weak coupling regime.
Finally, we compare the spin mapping model (SPM) toModel VI. While both mapping models are developed fromthe equivalent expressions [Eqs. (58) and (61) or Eq. (63)]of the multi-state Hamiltonian operator in quantum mechan-ics, different strategies are employed. SPM replaces the spincomponents by its classical angular momentum counterparts[i.e., Eq. (65)]. The equations of motion for SPM are givenby Eq. (68). Consider a pure state as the initial condition,e.g., the ith state is occupied. The classical mapping model[Eq. (68)] employs m(i) = 1
/2 for the occupied state and
m(j) =−1/2 for unoccupied states (j , i) in Eq. (65), while
the angle variable q(i) or q(j) can take any real value between0 and 2π. A single trajectory is not sufficient for obtain-ing meaningful population dynamics results in SPM. Insteadan ensemble of trajectories with different initial conditionsfor q(i) or q(j) is employed in SPM. (This is different fromModels I-VI where only one trajectory is sufficient in the clas-sical limit.) The initial values of the angle variables can beeither uniformly or randomly chosen between 0 and 2π, aslong as enough number of trajectories are used for obtain-ing converged results. Fig. 6 shows that SPM only worksreasonably well in the weak coupling regime but becomes pro-gressively worse as the coupling terms get stronger. For com-parison, Model VI reproduces exact results in any couplingregimes.
One can further employ Cotton and Miller’s quasi-classical approach for SPM3 for the 3-state Hamiltonian.
FIG. 6. Comparison between Model VI and the spin mapping model for the population of state |1〉 as a function of time for the 3-state Hamiltonian system. (Itsparameters are given by Eqs. (C15) and (C16).) The initial state is |1〉. (a) The coupling between any two states is λ = 0.02 in Eq. (C16). (b) λ = 0.2. (c) λ = 2.(d) λ = 20. Solid line (black): exact results. Dotted line (blue): Model VI. Dashed line (red): spin mapping model (SPM).
204105-14 Jian Liu J. Chem. Phys. 145, 204105 (2016)
FIG. 7. Comparison between classical and quasiclassical dynamics for SPMfor the population of state |1〉 as a function of time for the 3-state Hamiltoniansystem. (Its parameters given by Eqs. (C15) and (C16).) The initial state is|1〉. The coupling between any two states is λ = 2 in Eq. (C16). Solid line(black): exact results. Dotted line (red): classical dynamics for SPM. Dashedline (blue): quasiclassical dynamics for SPM.3
While the initial values of the angle variables are (eitheruniformly or randomly) chosen between 0 and 2π, those ofthe action variables are similarly chosen within a distance
(√3 − 1
) /2 ≈ 0.366 of the quantum half integer value 1/2
or −1/2 for occupied or unoccupied states, respectively. Con-sider the case of Fig. 6(c) as an example, where the couplingbetween any two states is λ = 2 in Eq. (C16). Fig. 7 demon-strates that the quasiclassical approach for SPM does not workwell either in the strong coupling regime. This then explainswell why the quasiclassical approach of SPM is less accuratethan that of the Meyer-Miller mapping model as demonstratedin Ref. 3.
1H.-D. Meyer and W. H. Miller, J. Chem. Phys. 70, 3214–3223 (1979).2G. Stock and M. Thoss, Phys. Rev. Lett. 78(4), 578–581 (1997).3S. J. Cotton and W. H. Miller, J. Phys. Chem. A 119(50), 12138–12145(2015).
4J. Schwinger, in Quantum Theory of Angular Momentum, edited byL. C. Biedenharn and H. Van Dam (Academic, New York, 1965).
5J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, New York,1994).
6B. Li and W. H. Miller, J. Chem. Phys. 137(15), 154107 (2012).7G. Stock and U. Muller, J. Chem. Phys. 111(1), 65–76 (1999).8U. Muller and G. Stock, J. Chem. Phys. 111(1), 77–88 (1999).9A. A. Golosov and D. R. Reichman, J. Chem. Phys. 114(3), 1065–1074(2001).
10S. J. Cotton and W. H. Miller, J. Chem. Phys. 139(23), 234112 (2013).11W. H. Miller and S. J. Cotton, J. Chem. Phys. 142(13), 131103 (2015).12W. H. Miller, J. Phys. Chem. A 113(8), 1405–1415 (2009).13D. T. Pegg and S. M. Barnett, Phys. Rev. A 39(4), 1665–1675 (1989).14D. W. H. Swenson, T. Levy, G. Cohen, E. Rabani, and W. H. Miller, J. Chem.
