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) H00(t) | j exp(ikjt) [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) H00(t) | j exp(ikjt) (ikj)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(ikjt) bj(1)(t) (ħjk)1 bk(1)*(t) (Ek E0) k ≠ 0, j ≠ 0 j≠k + j | H(t) | k exp(ijkt) bj(1)*(t) (ħjk)1 bk(1)(t) (Ek E0) k ≠ 0, j ≠ 0 j≠k = k | H(t) | j exp(ikjt) bj(1)(t) (ħjk)1 bk(1)*(t) (Ek E0) k ≠ 0, j ≠ 0 j≠k + k | H(t) | j exp(ikjt) bk(1)*(t) (ħkj)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(ikjt) bk(1)*(t) bj(1)(t) [ ħk0/(ħjk) + ħj0/(ħkj) ] k ≠ 0, j ≠ 0 j≠k = k | H(t) | j exp(ikjt) bk(1)*(t) bj(1)(t) [ (ħk0 ħj0) / (ħjk) ] k ≠ 0, j ≠ 0 j≠k = (1) j ≠ 0, k ≠ 0 j≠k k | H(t) | j exp(ikjt) 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 = kj bk(1)*(t) bj(1)(t) k | H(t) H00(t) | j exp(ikjt) . (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 (ħ0j)1 | j j≠0 + 2 m | H(t) H00(t) | j j | H(t) | 0 (ħm0)1 (ħj0)1 | m ] j ≠ 0, m≠0 (C.5) for the perturbed ground state and | k(t) = N k [ | k + n | H(t) | k (ħkn)1 | n n≠k + 2 p | H(t) Hkk(t) | n n | H(t) | k (ħnk)1 (ħpk)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 (ħ0j)2 + O (3) ] , (C.7) j≠0 which gives N 0 = [ 1 + 2 0 | H(t) | j j | H(t) | 0 (ħ0j)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 (ħjk)1 k | k + j | H(t) | k (ħkj)1 j | j + 2 j | H(t) | m (ħjm)1 n | H(t) | k (ħkn)1 m | n m≠j n≠k + 2 j | H(t) | m m | H(t) Hjj(t) | k (ħmj)1 (ħkj)1 k | k m≠j + 2 j | H(t) Hkk(t) | n n | H(t) | k (ħnk)1 (ħjk)1 j | j .(C.9) n≠k Thus j(t) | k(t) = j | k + 2 j | H(t) | m m | H(t) | k (ħjm)1 (ħkm)1 m ≠ j, m≠k + 2 j | H(t) | m m | H(t) Hjj(t) | k (ħmj)1 (ħkj)1 m≠j + 2 j | H(t) Hkk(t) | n n | H(t) | k (ħnk)1 (ħjk)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 (ħjm)1 (ħkm)1 m ≠ j, m≠k + 2 j | H(t) | m m | H(t) Hjj(t) | k (ħmj)1 (ħkj)1 m≠j m≠k + 2 j | H(t) | k k | H(t) Hjj(t) | k (ħkj)1 (ħkj)1 + 2 j | H(t) Hkk(t) | m m | H(t) | k (ħmk)1 (ħjk)1 m≠k m≠j + 2 j | H(t) Hkk(t) | j j | H(t) | k (ħjk)1 (ħjk)1 . (C.11) Now 2 j | H(t) | k k | H(t) Hjj(t) | k (ħkj)1 (ħkj)1 + 2 j | H(t) Hkk(t) | j j | H(t) | k (ħjk)1 (ħjk)1 = 2 j | H(t) | k [Hkk(t) Hjj(t)] (ħkj)1 (ħkj)1 + 2 [Hjj(t) Hkk(t)] j | H(t) | k (ħjk)1 (ħjk)1 =0 . (C.12) Also 2 j | H(t) | m m | H(t) | k (ħjm)1 (ħkm)1 m ≠ j, m≠k + 2 j | H(t) | m m | H(t) Hjj(t) | k (ħmj)1 (ħkj)1 m≠j m≠k + 2 j | H(t) Hkk(t) | m m | H(t) | k (ħmk)1 (ħjk)1 m≠k m≠j = 2 j | H(t) | m m | H(t) | k m ≠ j, m≠k [(ħjm)1 (ħkm)1 + (ħmj)1 (ħkj)1 + (ħmk)1 (ħjk)1] . (C.13) We observe that [(ħjm)1 (ħkm)1 + (ħmj)1 (ħkj)1 + (ħmk)1 (ħjk)1] = (ħjm)1 (ħkm)1 (ħkj)1 [ħkj ħkm + ħjm] = 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 (ħk0)1 + 2 k | H(t) | n n | H(t) Hkk(t) | 0 (ħnk)1 (ħ0k)1 (C.17) n≠k k(t) | (1)(t) = ck(1)(t) exp[i (Ek – E0) t/ħ] + k | H(t) | n (ħkn)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(imT) , (D.3) where m denotes m(s = 1). From Eqs. 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