Supplemental material for “Adiabatic and nonadiabatic contributions to the energy of a
system subject to a time-dependent perturbation: Complete Separation and physical
interpretation,” by Anirban Mandal and Katharine L. C. Hunt
A. Adiabatic perturbation theory
Recent interest in adiabatic transitions has been spurred by the prospects for adiabatic
quantum computing:1-21
Adiabatic quantum computing is potentially applicable when the
solution of a computational problem can be mapped to the ground state | 0P  of a Hamiltonian
HP. If the ground state | 0  of a Hamiltonian H0 can be prepared, and if the Hamiltonian H0 can
be converted gradually to HP over time, then ground state will evolve adiabatically from | 0  to
the solution state | 0P . A time dependence such as H(t) = H0 (1 – t/T) + (t/T) HP is used, so that
H0 converts to HP as t goes from 0 to T, and then | 0  → | 0P  in the limit as t → T and T → ∞.1
In simulations run on standard computers, adiabatic quantum computing has been applied
to solve problems where the states can be represented as strings of digits, and energy “costs” can
be assigned to specific features of the digit string.1-5,7,15,18,21 These features permit the construction of the Hamiltonian HP. Typically the state | 0  is the uniformly weighted superposition
of all possible digit strings in the space defined by the problem.1-5 Examples of problems
approached in this way include the “exact cover” problem,1 which involves finding ways to
satisfy long lists of conditions on multiple elements of the data strings, the search for marked
vertices in a graph,18 and Ramsey-type problems,21 exemplified by the determination of the
minimum value of N(m, n) such that in every graph with N points there are at least m points that
are completely connected to each other pairwise, or at least n points that are connected to no
others in the set of n.21
The simulations suggest that the time required to solve such problems can be reduced
from a super-exponential dependence on the problem size in classical computing to a polynomial
dependence in quantum computing (see, e.g., Ref. 1). The speed-up achievable with adiabatic
quantum computing is equivalent (up to polynomial factors) to the speed-up from other forms of
quantum computing.14,22 Although adiabatic quantum algorithms appear to require exponential
time in the worst cases for NP-complete problems,22 quantum computing has the potential to
significantly expand the set of numerically tractable problems. Physical systems that have been
used in quantum computers include nuclear spins,6,23-27 cold trapped ions,28-32 photons,33-35 and
superconducting loops;36,37 applications have been suggested—and in some cases, implemented
experimentally—in studying conformational energetics of proteins,12 chemical dynamics,38
electronic structure,24,27,32,33,39 and spin systems,17,23,25,30 in addition to problems in graph theory
and number theory.1,11,18,21,40 Reviews have been provided by Childs and van Dam,15 by Kassal,
Whitfield, Perdomo-Ortiz and Aspuru-Guzik20 and by Buluta and Nori.41
Applications in quantum computing have also prompted re-investigation of the conditions
needed for the validity of the adiabatic theorem, including new experimental work.42 One
standard criterion for the validity of the adiabatic approximation has been stated in terms of the
instantaneous eigenstates | n  of the full Hamiltonian at time t, the instantaneous transition
energies En(t) – Em(t), and the time-derivative of the perturbation ∂H(t)/∂t, in this form: For all
distinct states m and n, it is required that |  m | ∂H(t)/∂t | n  | / [En(t) – Em(t)]2 << 1 at all times
t in the interval [0, T], assuming that the perturbation is first applied at t = 0.43 However,
counterexamples have been identified for which this criterion is met, yet the adiabatic theorem
does not hold.44 Extensive analysis and commentary have ensued.45-58 In particular, the standard
criterion is not sufficient when resonant transitions occur.47,52,59 To remove this limitation, a
relatively simple alternate condition has been suggested in terms of the scaled time s, defined by
s = t/T: it is required that |  m | ∂H(s)/∂s | n  | / [En(s) – Em(s)] |2 << T for all distinct states m
and n, at all scaled times s in the interval [0, 1].59 Other forms of rigorous conditions for
adiabatic time evolution have been derived.10,47,52,55,60-64
The adiabatic theorem has been generalized to open systems,65-69 adiabatic evolution with
feedback,70 and systems without energy gaps between the ground and excited states.71-73 In the
context of molecular quantum mechanics, Liu and Hunt74 have used nonlocal susceptibility
densities to prove a “force relay” theorem that holds when the electrons respond adiabatically to
a perturbation: The expectation value of the force on the electrons at order n in the perturbation
is equal to the force on the nuclei from the change in the electronic charge density at the next
higher order, n+1.74 This result stems from a hypervirial theorem applied to the momentum
operator for the electrons,75-83 combined with Ehrenfest’s theorem.84
Equilibrium statistical mechanics establishes a connection between quantum mechanical
and thermodynamic adiabaticity.85 A differential change in the energy can be separated into
differentials (which are in general inexact) representing heat and work. The reversible work is
determined by differential changes in the energy levels, with no change in population; while the
reversible heat is determined by differential changes in the state populations, with no change in
the energy values themselves.85 Processes that are adiabatic quantum mechanically correspond
to processes that involve only reversible work, and not heat. The extension of this result to
systems in non-equilibrium steady states has been considered by Keizer.86,87
B. Nonadiabatic transitions
In the Landau-Zener theory of electron transfer88,89 and in the Tully-Preston surface-hopping
model,90,91 the transition probability is expressed in terms of the nuclear velocity in the vicinity
of an avoided crossing, where transitions are expected to be most probable. More generally, in
the “fewest switches” algorithm for molecular dynamics with nonadiabatic electronic transitions
developed subsequently by Tully,92 transitions may occur anywhere the electronic coupling
between states is significant, and among any number of electronic states.
In reaction dynamics several variants of the surface-hopping model have been developed:
Miller and George93 employed a generalized Stückelberg method94 to determine the transition
probabilities. Blais, Truhlar, and Mead95 introduced algorithmic improvements; also, as in the
work of Stine and Muckerman,96 they ensured energy conservation by adjusting the component
of the momentum after the “hop” along the direction determined by the matrix element of the
gradient operator, evaluated between adiabatic states in the appropriate multi-dimensional
coordinate space (when the matrix element is non-zero). A closely related version of the surfacehopping model has been developed by Parlant and Gislason.97 Cline and Wolynes98 have used a
phenomenological collision operator to treat curve-crossing processes in solution; and the
nonadiabatic relaxation of excess electrons in fluids has been analyzed by Space and Coker.99
The Landau-Zener model88,89 and related approaches have been very widely employed.
To cite just a few examples, such approaches have been used in recent studies of dissociative
recombination of DCO+ and DOC+ by Larson et al.;100 within the ab initio multiple-spawning
approach used by Kim et al.,101 in their work on using laser-induced conical intersections (LICIs)
in order to control the photoisomerization of 1,3-cyclo-hexadience; and in the theoretical work of
Palii et al.102 on manipulating the polarization of mixed-valence inorganic compounds using
electric pulses. The Landau-Zener model has been incorporated into the theory of proton
transfer by Cukier and Zhu103 and the theory of hydrogen-atom transfer by Cukier;104 the latter
theory is related to work on proton-coupled electron transfer reviewed by Cukier and Nocera.105
The Zener formalism has been modified by Nakamura and Zhu106,107 to include tunneling, and
the modified approach has been used by Zimmerman et al.108 in calculations on the production of
two triplet excitons following the absorption of a single photon by pentacene or tetracene.
Perturbative and other corrections of the Stueckelberg-Landau-Zener formula have been discussed by Child,109 with applications to the theory of molecular predissociation, collision cross
sections and rate constants. Recent applications of the Landau-Zener theory in physics include
the work of Gaudreau et al.110 on coherent control of three-spin states in triple quantum dots and
the work of Miladinovic et al.111 on the motion of photons in a double optical cavity.
Recently Domcke and Yarkony112 have reviewed the role of conical intersections in
photoinduced molecular dynamics and in spectroscopy, with reference to work by Teller,113
Nikitin and Umanskii,114 and Desouter-Lecomte et al.,115 as well the one-dimensional Zener
model89 and Tully’s least-switching algorithm92 (see also the review of nonadiabatic chemical
dynamics and prospects for the future given in Ref. 116). Tully’s fewest switches algorithm92
has been used by Tavernelli et al.117 in a study of the ultrafast relaxation of a photoexcited singlet
metal-to-ligand charge-transfer state, with the inclusion of solvent effects. The algorithm has
been combined with “on the fly” calculations of the electronic energies, gradients, and
nonadiabatic coupling vectors by Nelson et al.,118 and used in work by Clark et al.119 on speeding
the torsional relaxation of oligofluorenes in solution into the sub-picosecond time regime. Other
recent applications of surface-hopping methods include the work of Lan et al.120 on the photoinduced nonadiabatic decay of benzylidene malonitrile, which involves twisting around a double
bond and the conversion of one of the bond carbon atoms to a pyramidal structure; modeling of
photo-induced dynamics in distyrylbenzene, by Nelson et al.;121 and simulations of excited state
decay following photoisomerization of a bacterial photoreceptor, by Boggio-Pasqua et al.122
Related work has covered photofragmentation and recombination in cluster ions of I2 ∙ Arn,123
nonadiabatic excited state dynamics in pyrazine,124 photodissociation dynamics of F2 in solid
argon,125 methane photodissociation dynamics,126 exciton dissociation and charge separation in
poly-p-phenylenevinylene oligomers,127 environmental effects near conical intersections,128
electron-phonon relaxation in quantum dots,129 and solvent dynamics in proton-coupled electron
transfer.130 Jasper and Truhlar found that using a stochastic model for decoherence improved
agreement between surface-hopping and quantum results for the photodissociation of Na ∙ FH.131
Dang and Herman have recently discussed hops in classically forbidden regions.132
C. Derivation of intermediate results
1) We show that the terms with j ≠ k in the sum j bk(1)*(t) bj(1)(t)  k | H(t)  H00(t) | j 
exp(ikjt) [appearing in the third group of terms in Eq. (38) for E(3)(t)] drop out of the final
expression for E(3)(t). These terms cancel with the contributions to k bk(2)(t) bk(1)*(t) (Ek  E0)
and its complex conjugate that stem from boundary terms Sk(t) obtained from the integration by
parts, in the derivation of Eq. (42) for bk(2)(t). The boundary term Sk(t) is defined by
  k | H(t)  H00(t) | j  exp(ikjt) (ikj)1 bj(1)(t)
Sk(t) = i/ħ
.
(C.1)
j≠0
j≠k
The net contribution from the boundary terms is given by
 
 k | H(t) | j  exp(ikjt) bj(1)(t) (ħjk)1 bk(1)*(t) (Ek  E0)
k ≠ 0, j ≠ 0
j≠k
 
+
 j | H(t) | k  exp(ijkt) bj(1)*(t) (ħjk)1 bk(1)(t) (Ek  E0)
k ≠ 0, j ≠ 0
j≠k
=
 
 k | H(t) | j  exp(ikjt) bj(1)(t) (ħjk)1 bk(1)*(t) (Ek  E0)
k ≠ 0, j ≠ 0
j≠k
 
+
 k | H(t) | j  exp(ikjt) bk(1)*(t) (ħkj)1 bj(1)(t) (Ej  E0) ,
(C.2)
k ≠ 0, j ≠ 0
j≠k
where we have interchanged the summation indices j and k in the second set of terms. Adding
the terms, we obtain
 
 k | H(t) | j  exp(ikjt) bk(1)*(t) bj(1)(t) [ ħk0/(ħjk) + ħj0/(ħkj) ]
k ≠ 0, j ≠ 0
j≠k
=
 
 k | H(t) | j  exp(ikjt) bk(1)*(t) bj(1)(t) [ (ħk0  ħj0) / (ħjk) ]
k ≠ 0, j ≠ 0
j≠k
= (1)
 
j ≠ 0, k ≠ 0
j≠k
 k | H(t) | j  exp(ikjt) bk(1)*(t) bj(1)(t) .
(C.3)
Then we compare the final result in Eq. (C.3) with the total of the j ≠ k terms contained in the
third summation in Eqs. (38) and (39) for E(3)(t),
S = kj
bk(1)*(t) bj(1)(t)  k | H(t)  H00(t) | j  exp(ikjt) .
(C.4)
j ≠ 0, k ≠ 0
j≠k
The final result in Eq. (C.3) is clearly the additive inverse of S in (C.4). This proves Eq. (43) in
the main text. Equations (44) and (45) then follow immediately.
2) We obtain the normalized states at second order in the perturbation, from time-independent
perturbation theory, for the instantaneous Hamiltonian H(t) = H0 +  H(t). At second order, the
states have the form
| 0(t)  = N
0
[ | 0  +   j | H(t) | 0  (ħ0j)1 | j 
j≠0
+ 2   m | H(t)  H00(t) | j   j | H(t) | 0  (ħm0)1 (ħj0)1 | m  ]
j ≠ 0,
m≠0
(C.5)
for the perturbed ground state and
| k(t)  = N
k
[ | k  +   n | H(t) | k  (ħkn)1 | n 
n≠k
+ 2   p | H(t)  Hkk(t) | n   n | H(t) | k  (ħnk)1 (ħpk)1 | p  ]
n ≠ k,
p≠k
(C.6)
for the perturbed excited state, where N 0 and N k are the normalization factors. Then
 0(t) | 0(t)  = N
2
0
[ 1 + 2   0 | H(t) | j   j | H(t) | 0  (ħ0j)2 + O (3) ] ,
(C.7)
j≠0
which gives
N
0
= [ 1 + 2   0 | H(t) | j   j | H(t) | 0  (ħ0j)2 ]1/2
(C.8)
j≠0
and similarly for N k.
3) We prove that the perturbed states are orthogonal to order 2. From Eqs. (66) and (67), for
distinct states | j(t)  and | k(t) , which may be either ground or excited states,
 j(t) | k(t)  =  j | k  +   j | H(t) | k  (ħjk)1  k | k  +   j | H(t) | k  (ħkj)1  j | j 
+ 2  j | H(t) | m  (ħjm)1  n | H(t) | k  (ħkn)1  m | n 
m≠j n≠k
+ 2   j | H(t) | m   m | H(t)  Hjj(t) | k  (ħmj)1 (ħkj)1  k | k 
m≠j
+ 2   j | H(t)  Hkk(t) | n   n | H(t) | k  (ħnk)1 (ħjk)1  j | j  .(C.9)
n≠k
Thus
 j(t) | k(t)  =  j | k 
+ 2  j | H(t) | m   m | H(t) | k  (ħjm)1 (ħkm)1
m ≠ j,
m≠k
+ 2   j | H(t) | m   m | H(t)  Hjj(t) | k  (ħmj)1 (ħkj)1
m≠j
+ 2   j | H(t)  Hkk(t) | n   n | H(t) | k  (ħnk)1 (ħjk)1 .
(C.10)
n≠k
We split the second summation (the third term on the right-hand side) in Eq. (C.10) into the set
of terms with m ≠ j and m ≠ k, plus the term with m = k. In the third summation, we first change
the summation index from n to m, and then similarly split the third summation into the set of
terms with m ≠ j and m ≠ k, plus the term with m = j. This gives
 j(t) | k(t)  =  j | k 
+ 2  j | H(t) | m   m | H(t) | k  (ħjm)1 (ħkm)1
m ≠ j,
m≠k
+ 2   j | H(t) | m   m | H(t)  Hjj(t) | k  (ħmj)1 (ħkj)1
m≠j
m≠k
+ 2  j | H(t) | k   k | H(t)  Hjj(t) | k  (ħkj)1 (ħkj)1
+ 2   j | H(t)  Hkk(t) | m   m | H(t) | k  (ħmk)1 (ħjk)1
m≠k
m≠j
+ 2  j | H(t)  Hkk(t) | j   j | H(t) | k  (ħjk)1 (ħjk)1 .
(C.11)
Now
2  j | H(t) | k   k | H(t)  Hjj(t) | k  (ħkj)1 (ħkj)1
+ 2  j | H(t)  Hkk(t) | j   j | H(t) | k  (ħjk)1 (ħjk)1
= 2  j | H(t) | k  [Hkk(t)  Hjj(t)] (ħkj)1 (ħkj)1
+ 2 [Hjj(t)  Hkk(t)]  j | H(t) | k  (ħjk)1 (ħjk)1
=0 .
(C.12)
Also
2  j | H(t) | m   m | H(t) | k  (ħjm)1 (ħkm)1
m ≠ j,
m≠k
+ 2   j | H(t) | m   m | H(t)  Hjj(t) | k  (ħmj)1 (ħkj)1
m≠j
m≠k
+ 2   j | H(t)  Hkk(t) | m   m | H(t) | k  (ħmk)1 (ħjk)1
m≠k
m≠j
= 2  j | H(t) | m   m | H(t) | k 
m ≠ j,
m≠k
 [(ħjm)1 (ħkm)1 + (ħmj)1 (ħkj)1 + (ħmk)1 (ħjk)1] .
(C.13)
We observe that
[(ħjm)1 (ħkm)1 + (ħmj)1 (ħkj)1 + (ħmk)1 (ħjk)1]
= (ħjm)1 (ħkm)1 (ħkj)1 [ħkj  ħkm + ħjm] = 0 ,
(C.14)
and finally we obtain
 j | k  = jk
(C.15)
thus establishing that the perturbed states are orthogonal to order 2.
4) We provide intermediate results that are needed to evaluate  k(t) | (t)  to second order in
, using Eq. (62). Up to terms of second order in ,
 (t) | (t) 1/2 = [1 + 2  cn(1)*(t) cn(1)(t)]1/2
n≠0
= [1 – (1/2) 2  cn(1)*(t) cn(1)(t)]
(C.16)
n≠0
 k(t) | (0)(t)  =   k | H(t) | 0  (ħk0)1
+ 2   k | H(t) | n  n | H(t)  Hkk(t) | 0  (ħnk)1 (ħ0k)1
(C.17)
n≠k
 k(t) |  (1)(t)  =  ck(1)(t) exp[i (Ek – E0) t/ħ]
+    k | H(t) | n  (ħkn)1 cn(1)(t) exp[i (En – E0) t/ħ]
(C.18)
n≠k
and
 k(t) | 2 (2)(t)  = 2 ck(2)(t) exp[i (Ek – E0) t/ħ]
(C.19)
D. Comparison with the analysis of Born and Fock
In their proof of the adiabatic theorem, Born and Fock133 derived equations for quantities that are
essentially equivalent to  k(t) | (t) . In this section, we note the differences between the two
approaches, but also demonstrate that the results are consistent.
Born and Fock133 introduced a matrix Ymn(t), with the elements
Ymn(t) =  m(s) | n(t) 
(D.1)
where | m(s)  denotes the mth instantaneous eigenstate of the Hamiltonian at the scaled time s
given by s = t/T, and | n(t)  denotes the full wave function at time t for a system that starts in
the eigenstate | n  at t = 0. Thus our quantity  k | (t)  corresponds to Yk0(t), in their notation.
Born and Fock used a frequency variable k(s) defined by
s
k(s) = (1/ħ)  0 Wk(s) ds ,
(D.2)
where Wk(s) is the kth instantaneous eigenvalue of the full Hamiltonian H(s) at the scaled time
s, and the spectrum is assumed to be discrete, although degeneracies at intermediate times are
not excluded. The quantities Ymn(t) are related to coefficients cmn(s) by
cmn(s) = Ymn(t) exp(imT) ,
(D.3)
where m denotes m(s = 1). From Eqs. (29) and (38) of Ref. 133, Born and Fock defined a
matrix Pmn(s) that induces transitions:
Pmn(s) =  i   m(s) | j   j | dH(s)/ds | k   k | n(s) 
j, k
∙ [Wm(s) – Wn(s)]1 exp[iT(m – n)] .
(D.4)
In Eq. (D.4), the sum over j and k runs over all eigenstates of the unperturbed Hamiltonian H0
and | n(s)  denotes the nth eigenstate of the full Hamiltonian at the scaled time s. The result for
cmn(s) given by Born and Fock to leading order in the expansion in a transition matrix Pmn is133
s
cmn (s) = mn + i
(1)
0
Pmn(s) ds .
(D.5)
From Eqs. (D.2)-(D.5), it is apparent that the result for Ymn(t) given by Born and Fock depends
on the instantaneous eigenstates and instantaneous eigenvalues at all times t between the initial
application of the perturbation (t = 0) and time t.
Additionally, we note that the result for cmn(1)(s) in Eq. (D.5) is of first order in the
transition matrix Pmn; but by construction, cmn(1)(s) includes terms of all orders in the perturbation
parameter . The term of first order in  is readily extracted from Eq. (D.5), however, based on
the observation that dH(s)/ds is itself first order in . To leading order, we find that
| Yk0(t) | 2 = |  k | (t)  | 2 = 2 | bk(1)(t) | 2 ,
which is consistent with Eq. (55) of the current work.
(D.6)
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