Quantum adiabatic theorem for chemical reactions and systems with time-dependent orthogonalization
QQUANTUM ADIABATIC THEOREM FOR CHEMICALREACTIONS AND SYSTEMS WITH TIME-DEPENDENT ORTHOGONALIZATION
ANDREW DAS ARULSAMY
Abstract.
A general quantum adiabatic theorem with and without the time-dependent orthogonalization is proven, which can be applied to understandthe origin of activation energies in chemical reactions. Further proofs are alsodeveloped for the oscillating Schwinger Hamiltonian to establish the relation-ship between the internal (due to time-dependent eigenfunctions) and external(due to time-dependent Hamiltonian) timescales. We prove that this relation-ship needs to be taken as an independent condition for the quantum adiabaticapproximation. We give four examples, including logical expositions based onthe spin- two-level system to address the gapped and gapless (due to energylevel crossings) systems, as well as to understand how does this theorem allowsone to study dynamical systems such as chemical reactions. MSC: 81P10; 81Q15PACS: 03.65.Ca 1.
Introduction
The quantum adiabatic theorem (QAT) and its approximation (QAA) have beenthe backbone in many areas of quantum physics, namely, in condensed mattertheory (via the Born-Oppenheimer approximation) [1], in atoms, molecules andquantum chemistry [2], in quantum field theory via the Gell-Mann and Low the-orem [3], and presently in the adiabatic quantum computation [4]. The theoremwas first discussed by Ehrenfest [5], and later was formally derived by Born andFock [6] and Kato [7]. Other modern proofs for QAT can be found in Refs. [8, 9,10, 11, 12, 13, 14]. We refer the readers to Refs. [8, 11, 12] for thorough reviewson the development of QAT. Apart from the historical perspectives, Comparat [11]have also proven the existence of two sufficient QAT conditions for the oscillat-ing Schwinger Hamiltonian. Teufel-Spohn [15, 16] and Avron-Elgart [12] on theother hand, have provided the proof for the validity of QAT for systems obeyingthe Born-Oppenheimer approximation, and for gapless systems, respectively. Wewill revisit the two conditions given by Comparat to prove that these conditionsare in fact equivalent to the relationship between T ex and T in . Here, T ex and T in are the respective external and internal timescales. Subsequently, we will discusshow the new theorem proven here addresses the gapless system. Due to numerousapplications of QAT in physics, a proper theorem with explicitly defined conditionsis needed. Here, we prove the existence of such a theorem, which can also be ex-ploited to understand the origin of chemical reactions. To arrive at the standard Date : November 9, 2018.
Key words and phrases.
Quantum adiabatic theorem and approximation; Chemical reactions;Degenerate energy levels; Time-dependent orthogonalization; Internal and external timescales. a r X i v : . [ c ond - m a t . s t r- e l ] S e p ANDREW DAS ARULSAMY
QAT for gapped system, we first let H ( t ), ψ ( t ) and E ( t ) to be the time-dependentHamiltonian, eigenfunction and eigenvalue, respectively, such that [9] H ( t ) ψ n ( t ) = E n ( t ) ψ n ( t ) , (1.1)and { ψ n ( t ) } constitutes a complete orthonormal set that satisfies (cid:104) ψ n ( t ) | ψ m ( t ) (cid:105) = δ nm , (1.2)in which ψ n ( t ) can be expressed as a linear combinationΨ( t ) = (cid:88) n c n ( t ) ψ n ( t ) e iθ n ( t ) , (1.3)with the phase factor θ n ( t ) = − (cid:126) (cid:90) t E n ( t (cid:48) ) dt (cid:48) , (1.4)and { n, m } ∈ N ∗ . We use N ∗ and R to denote the set of positive integers excludingzero and the set of real numbers, respectively. Here, Eq. (1.3) is the general solutionto the time-dependent Schr¨odinger equation i (cid:126) ∂∂t Ψ( t ) = H ( t )Ψ( t ) . (1.5)Therefore, one can show that if the initial Hamiltonian, H init ( t ) evolves very slowlyto a new final Hamiltonian, H fin ( t ), then a particle that starts out in the n th eigenstate (Ψ init n ( t ) = ψ n ( t ) e iθ n ( t ) ) of H init ( t ) is expected to occupy the groundstate (Ψ fin n ( t ) = ψ n ( t ) e iγ n ( t ) e iθ n ( t ) ) of H fin ( t ). Observe that the final eigenstate hasnon-trivially, picked up only a harmless phase factor γ n ( t ) = i (cid:90) t (cid:28) ϕ n ( t (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) ∂∂t (cid:48) ϕ n ( t (cid:48) ) (cid:29) dt (cid:48) . (1.6)In the above formulation, Griffith used Eqs. (1.3) and (1.5) to obtain the requiredcoefficient (in explicit form) [9],˙ c m ( t ) = − c m ( t ) (cid:104) ϕ m ( t ) | ˙ ϕ m ( t ) (cid:105) − (cid:88) n (cid:54) = m c n ( t ) (cid:68) ϕ m ( t ) (cid:12)(cid:12)(cid:12) ˙ H ( t ) (cid:12)(cid:12)(cid:12) ϕ n ( t ) (cid:69) E n ( t ) − E m ( t ) e i ( θ n ( t ) − θ m ( t )) , (1.7)that can be used to find out how slow the evolution should be, so that the initialparticle still occupies the ground state of the new (final) Hamiltonian. Here, c m ( t )and c n ( t ) denote the m th and n th eigenstate coefficients, respectively, such that | c m ( t ) | + | c n ( t ) | = 1. From Eq. (1.7), we can notice that both E n ( t ) and E m ( t )exist at all times, and their eigenfunctions are orthogonal. This means that if E n ( t )and E m ( t ) are degenerate at certain point of time, then Eq. (1.7) cannot be usedto derive the adiabatic criterion. The reason is that the gap that exists when t = t does not exist for t = t . Now, assuming non-degeneracy, the QAT criterion isgiven by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:68) ϕ m ( t ) (cid:12)(cid:12)(cid:12) ˙ H ( t ) (cid:12)(cid:12)(cid:12) ϕ n ( t ) (cid:69) E n ( t ) − E m ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) ˙ c m ( t ) , (1.8) UANTUM ADIABATIC THEOREM FOR CHEMICAL REACTIONS 3 which needs to be satisfied for one to exploit QAA. Here, ϕ m ( t ) is the usual t -dependent eigenfunction for the m th eigenstate and E n ( t ) is the t -dependent eigen-value for the n th eigenstate, and the dot represents time derivative. There are twoimportant assumptions associated to the original QAT given in Eq. (1.8). First,there is no quantum phase transitions or chemical reactions when the initial quan-tum system evolves from t to t . Secondly, the QAT is only applicable for a two-or a multi-level system with well-defined energy gaps (strictly no energy level cross-ings). For example, when one considers a two-level system for simplicity, then thereexist two t -dependent eigenfunctions, observable at all times, and they are alwaysnondegenerate and the two eigenvalues are E n ( t ) (cid:54) = E m ( t ). In fact, we can add asmany eigenfunctions as we possibly could (as denoted by n and m ), but we chosetwo, to keep the discussion straightforward. Of course, later we will go through theexamples with large number of energy levels and energy level crossings.For the sake of argument however, if we assume that E n ( t = t ) (cid:54) = E n ( t = t )(this is definitely different from E n ( t = t ) (cid:54) = E m ( t = t ), as described earlier) thenwe cannot apply the quantum adiabatic approximation [given in Eq. (1.8)] simplybecause both of these eigenvalues ( E n ( t = t ) and E n ( t = t )) are not observablesimultaneously. Recall that t < t . Furthermore, E n ( t = t ) (cid:54) = E n ( t = t ) impliesthe energy gap ( g ) in the form of g = | E n ( t = t ) − E n ( t = t ) | . The energy gap inthis form cannot be defined as an energy gap because ω (cid:48) ( t ) = E n ( t = t ) − E n ( t = t ) (cid:126) (cid:54) = ω ( t ) Bohr = E n ( t ) − E m ( t ) (cid:126) = 1 T in ( t ) . (1.9)In other words, physically valid energy gaps must satisfy ω ( t ) Bohr . Later, we willshow that ω (cid:48) ( t ) can be associated to the possibility of a particle traveling forwardand backward in time. Now, it is also important to note here that if the externaldisturbances with a timescale, (cid:126) ω Bohr T ex ( t ) ∝ (cid:68) ϕ m ( t ) (cid:12)(cid:12)(cid:12) ˙ H ( t ) (cid:12)(cid:12)(cid:12) ϕ n ( t ) (cid:69) , (1.10)approaches ω Bohr due to T ex ≈ T in , then one cannot apply QAA. Now, one shouldnot be carried away into thinking that Eq. (1.8) lacks a small parameter, which isrequired to understand the quantum adiabatic evolution. For example, the small-ness is usually controlled by the dilation operator, τ such that s ex = T ex /τ . In ourformalism however, this smallness is controlled by the relationship between T in and T ex , without the need to introduce any ad-hoc control parameter by hand . We willget to this point in detail in Example 1 when we proof Eq. (1.10). The adiabaticcriteria, T ex (cid:29) T in and Eq. (1.8) have been proven in Ref. [9], however, without theproper definitions for these timescales. The existence of ω (cid:48) ( t ) means that we areequating ϕ n ( t ) with ϕ n ( t = t ) and ϕ m ( t ) with ϕ n ( t = t ) in Eq. (1.8). In this case,we are referring to only one eigenfunction at all times such that ϕ n ( t = t ) evolvesto ϕ n ( t = t ), where ϕ n ( t = t ) and ϕ n ( t = t ) may or may not be orthogonal toeach other. If they are orthogonal, then one obtains ω (cid:48) ( t ). On the other hand, ifthey are not orthogonal, then ω (cid:48) ( t ) does not exist.In addition to our arguments based on Eq. (1.9), Comparat [11] have also shownthat the condition given in Eq. (1.8) is valid provided that the Hamiltonian is real,gapped and non-oscillating. However, the generalized conditions derived there [11], ANDREW DAS ARULSAMY which are valid for any Hamiltonian, do not consider gapless systems nor the Hamil-tonians that allow t -dependent orthogonalization. In addition, the internal andexternal timescales are also not properly taken into account. The t -dependentorthogonalization and large changes to the Hamiltonians (due to large externaldisturbances) are essential because they can lead to continuous or second-orderquantum phase transitions (QPT) during chemical reactions. The readers are re-ferred to Refs. [20, 21, 22, 23] for further details on the concept of QPT and itsapplications in strongly correlated matter.We anticipate that this continuous QPT proposed by Sachdev [20, 21] andSenthil [22, 23] can also occur due to the existence of energy level crossings dueto large external disturbances, with or without the t -dependent orthogonalization,which could be the origin of any chemical reaction between two chemical species.This means that the energy levels of unreacted species must cross by overcoming theactivation energies to form new compounds, and these new compounds (if formed)will have a new set of energy levels or bands. If the formed compounds are ofnon free-electronic systems, then there is a theorem based on the ionization energytheory that states − the energy level spacing of such a system is proportional totheir constituent atomic energy level spacing [24]. Physically, this means that theenergy level spacing of this newly formed compound is proportional to the energylevel spacing of their constituent chemical elements.Before we move on, let us first try to understand the concept of internal andexternal timescales in the QAT. It is well known now that there are different versionsof QATs available for different applications as reviewed by Teufel [8], and Avron-Elgart [12]. However, all the existing QATs are solely based on the scaled time, s ex = t ex /τ ex , s ex ∈ [0 , s ex here is independent of the internal timescale. Thereason for this scenario is that there are no available equations to properly define T ex and T in independently. Anyway, the scaled (dimensionless) time, s ex containsthe dilation time or a time dilation operator, τ ex , such that τ ex allows us to slowdown the time t ex to obtain a slowly varying Hamiltonian with respect to the value1 (when τ ex = t ex ). Hence, t ex is the original (unscaled) physical time for theHamiltonian, and τ ex determines the dilation of this external physical time.In the unscaled formalism, we simply write T ex to denote external timescale, toavoid confusion with t -dependence in H ( t ) and ϕ ( t ) (eigenfunction). Otherwise, wewill have to write H ( T ex ) and ϕ ( T in ), and this notation will confuse the readers intothinking that both H ( T ex ) and ϕ ( T in ) are not normalized to the real-space physicaltime, t , or ϕ ( T in ) is from a different Hamiltonian, and so on. As a consequence, wecan see that the Hamiltonian with scaled time, H ( s ex ) does not take the internaltimescale into account, which can only be taken into account by defining the timevariation in the wave functions or the eigenfunctions for a given H ( s ex ). Similar to s ex , we can also define s in = t in /τ in , where s in ∈ [0 , t in refers to the original physical time of the wave functions, and τ in determinesthe dilation of the internal physical time. Here, s in is independent of the exter-nal timescale. Again, in the unscaled formalism, we write T in to denote internaltimescale.Having understood the different notations used in the scaled and unscaled for-malisms, we can now properly define the time, T ex as the external timescale (char-acteristic time for changes in the Hamiltonian), while T in is the internal timescale UANTUM ADIABATIC THEOREM FOR CHEMICAL REACTIONS 5 (characteristic time for changes in the wave function or eigenfunction). In order tounderstand the existence of these properly defined timescales, we do the following:we can define t in = t ex = t and write s in = t/τ in = t/T in and s ex = t/τ ex = t/T ex .This normalization transforms H ( s ex ) back to H ( t ), and the only thing left to dohere is to find the inequality relation between T ex and T in . However, this H ( s ex ) or H ( t ) still ignores the internal timescale (because it is fixed as a constant), whichleads to an important question − how small should τ ex (or T ex ) be with respect to T in in order to violate the QAA? This question means that we cannot claim a givenHamiltonian varies slowly if τ ex → ∞ because we have to make sure τ in does notgo to ∞ when τ ex → ∞ . We will prove here that the relation between T in and T ex needs to be treated as a separate condition altogether.Now we will discuss why the concept of quantum adiabaticity is correct, and whyany violation that may exist only indicate the non-applicability of the QAA for cer-tain systems. The existence of non-orthogonality in strongly interacting physicalsystems (or non-linear quantum systems) has been discussed by Yukalov [17] asa result of non-linear terms in a given Hamiltonian that destroys the adiabaticevolution. These non-linear terms can originate from various interactions such aselectron-electron, electron-phonon, spin-orbit, spin-spin and so on, which may giverise to numerous energy-level crossings along the time line, t . As anticipated, thismeans that we need to keep track of the orthogonality of the eigenfunctions withrespect to time (which will be explained in the subsequent sub-section). It is to benoted here that strongly correlated systems, including quantum dots and nanos-tructures (non-free-electron systems) do not form gapless systems. For example,the electrons and the phonons cannot be decoupled, unless T electron (cid:28) T phonon [18].Any violation to a given QAT means the existence of a non-adiabatic process,and this does not imply the QAT is incorrect. Violating the QAT deliberately isimportant as a crosscheck because if the QAT is violated for adiabatic processes,then the QAT is either (a) not applicable for that process (for example, applyinggapless or gapped QAT to a system that has time( t )-dependent orthogonalization),or (b) the said QAT is insufficient (again, this does not mean the concept of quantumadiabaticity is incorrect). Consequently, deliberately violating a given QAT can beused as a mean to crosscheck the correctness and sufficiency of the conditions givenin a particular QAT, as well as to have a deeper understanding of the conditionsneeded to guarantee the QAA, for example, with respect to the existence of self-orthogonality [19].Here, we develop a new version of the QAT that (i) allows t -dependent orthog-onalization for each single eigenfunction as an additional condition, which can beused to explain the possibility of a particle traveling backward in time, (ii) allows tounderstand the origin of chemical reactions in the absence of, and in the presenceof t -dependent orthogonalization, and (iii) gives a proper definition for the T ex and T in and establishes the relationship between them as a valid and separate condi-tion. Points (i) to (iii) are the main results of this paper. In addition, we applyour theorem to degenerate energy levels due to energy level crossings to validatethe QAA with properly defined T ex and T in without invoking any Rabi oscillation. ANDREW DAS ARULSAMY Generalized Quantum Adiabatic Theorem
We use H for the complex Hilbert space, while || ϕ || is the norm of an eigenfunc-tion ( ϕ ). Theorem 1.
The quantum eigenfunctions ( or wave functions ) that satisfy thequantum mechanical postulates [25] are represented by the orthonormalized com-plex vectors , | ϕ ( t ) (cid:105) = ( · · · , | ϕ ( t j ) (cid:105) , · · · , | ϕ ( t k ) (cid:105) , · · · , | ϕ ( t z ) (cid:105) ), where | ϕ ( t ) (cid:105) ∈ H and || ϕ ( t ) || = (cid:104) ϕ ( t ) | ϕ ( t ) (cid:105) = (cid:82) R z | ϕ ( t ) | dt = 1. If a quantum system with a time-dependent Hamiltonian, H init ( t ) is initially ( t = t j ) in its ground state ( | ϕ n ( t ) (cid:105) or | ϕ m ( t j ) (cid:105) ) , evolves into a final ( t = t k ) quantum system with a Hamiltonian, H fin ( t ) then the probability for the final quantum system to be in its ground state ( | ϕ m ( t ) (cid:105) or | ϕ m ( t k ) (cid:105) ) is controlled by these three criteria (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:68) ϕ m ( t k ) (cid:12)(cid:12)(cid:12) ˙ H (cid:12)(cid:12)(cid:12) ϕ m ( t j ) (cid:69) − ˙ E m ( t j ) (cid:104) ϕ m ( t k ) | ϕ m ( t j ) (cid:105) E m ( t j ) − E m ( t k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ ˙ c m ( t k ) , (2.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:68) ϕ m ( t ) (cid:12)(cid:12)(cid:12) ˙ H ( t ) (cid:12)(cid:12)(cid:12) ϕ n ( t ) (cid:69) E n ( t ) − E m ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ˙ c m ( t ) , and(2.2) T ex ( t ) > T in ( t ) . (2.3) Here t ∈ [ t j , t k ] ∈ R , and (cid:104) ϕ m ( t k ) | ϕ m ( t j ) (cid:105) = δ t k t j = 0 implies ϕ m ( t j ) is orthogonalto ϕ m ( t k ). On the other hand , (cid:104) ϕ m ( t k ) | ϕ m ( t j ) (cid:105) = δ t k t j = 1 implies ϕ m ( t j ) is notorthogonal to ϕ m ( t k ) and { j, k, m, n } ∈ N ∗ . Remark 1 : Equation (2.1) keeps track of all the individual t -dependent eigen-functions in a given system. QAA is strictly valid if ( a ): ≈ ˙ c m ( t k ) in Eq. (2.1) isreplaced with = ˙ c m ( t k ), ( b ): < ˙ c m ( t ) in Eq. (2.2) is replaced with (cid:28) ˙ c m ( t ), and ( c ): > in Eq. (2.3) is replaced with (cid:29) . In fact, we can reversibly switch to strict QAAconditions if needed, or loosen them, or use them to invalidate the QAA in certainsystems. We can now write a statement in regards to QAT: The probability for aparticle that was initially in the ground state of an initial Hamiltonian, H init ( t ), tooccupy any excited state of some final Hamiltonian, H fin ( t ) is approximately zero, if H init ( t ) changes slowly to its final form, H fin ( t ) such that Eqs. (2.1), (2.2) and (2.3)are satisfied. Remark 2 : Time discretizations have been carried out to obtain instantaneouseigenfunctions. In other words, the t -dependent eigenfunction has been time-discretized in the form of ϕ m ( t ) = · · · , ϕ m ( t j ) , · · · , ϕ m ( t k ) · · · , ϕ m ( t z ). Time dis-cretizations are carried out in pairs ( t j and t k ) to obtain two time-discretized eigen-functions ( ϕ m ( t j ) and ϕ m ( t k )) so as to check their orthogonality. The sole purposeof this time-discretization is to capture the t -dependent orthogonalization and/orthe wave function transformation (that will be exposed in the last section). Remark 3 : In addition, it is to be noted here that c ( t j ) = c ( t = t j ), ϕ ( t j ) = ϕ ( t = t j ) and θ ( t j ) = θ ( t = t j ). As for the time derivative variables, ˙ c ( t j ) = ∂c ( t ) ∂t ( t = t j ),˙ ϕ ( t j ) = ∂ϕ ( t ) ∂t ( t = t j ), and so on. Proof for Eq. (2.1): The validity of Eqs. (2.2) and (2.3) have been proven inRef. [9]. The discussion on how Eq. (2.1) is related to chemical reactions is givenin Example 4, while the proof for Eq. (1.10) is given in Example 1. We will start
UANTUM ADIABATIC THEOREM FOR CHEMICAL REACTIONS 7 with a single eigenfunction ( m = 1) for two discretized times, i.e., when t = t j and t = t k . For simplicity, we let j = 0 and k = 1 from here onwards, and then wecan just repeat this procedure for j = 1 and k = 2, j = 2 and k = 3, and so on.Moreover, ϕ m ( t ) is allowed to evolve and to be orthogonal or not orthogonal to ϕ m ( t ) when t = t . The t -dependent Hamiltonian, say for t = t is given by H ( t ) ϕ ( t ) = E ( t ) ϕ ( t ) . (2.4)The t -dependent Schr¨odinger equation is given in Eq. (1.3) in which, Ψ( t ) can beexpressed as a linear combination of ϕ ( t ) and ϕ ( t ) (from Eq. (1.5))Ψ( t ) = t (cid:88) t c ( t ) ϕ ( t ) e iθ ( t ) ⇔ t ∈ [ t , t ] . (2.5)The summation is used between t and t because we have discretized the t -dependent eigenfunction (See Remark 2). Here, we also have c ( t ), which is acoefficient that is related to the probability of finding a particle at a given point oftime ( t or t ), and with a particular eigenvalue, E m ( t ) or E m ( t ), respectively, for m = 1 or 2 or · · · . For example, Eq. (2.1) implies | c m ( t ) | + | c m ( t ) | = 1, whileEq. (2.2) implies | c m ( t ) | + | c n ( t ) | = 1 (for m (cid:54) = n ). Recall here that we are onlyconsidering a single eigenfunction ( m = 1), and due to Remark 2, it is convenientfor us to drop the subscript m . From Eqs. (1.3) and (2.5)˙Ψ = t (cid:88) t (cid:2) ˙ c ( t ) ϕ ( t ) + c ( t ) ˙ ϕ ( t ) + ic ( t ) ϕ ( t ) ˙ θ ( t ) (cid:3) e iθ ( t ) ,i (cid:126) t (cid:88) t (cid:2) ˙ c ( t ) ϕ ( t ) + c ( t ) ˙ ϕ ( t ) + ic ( t ) ϕ ( t ) ˙ θ ( t ) (cid:3) e iθ ( t ) = t (cid:88) t c ( t ) Hϕ ( t ) e iθ ( t ) . (2.6)Using θ ( t ) = − (cid:126) (cid:20) (cid:90) t E ( t (cid:48) ) dt (cid:48) (cid:21) t = t ⇒ ˙ θ ( t ) = − (cid:126) E ( t ) , (2.7)we obtain i (cid:126) t (cid:88) t (cid:2) ˙ c ( t ) ϕ ( t ) + c ( t ) ˙ ϕ ( t ) (cid:3) e iθ ( t ) + i (cid:126) t (cid:88) t ic ( t ) ϕ ( t ) (cid:18) − (cid:126) E ( t ) (cid:19) e iθ ( t ) = t (cid:88) t c ( t ) E ( t ) ϕ ( t ) e iθ ( t ) ,i (cid:126) t (cid:88) t (cid:2) ˙ c ( t ) ϕ ( t ) + c ( t ) ˙ ϕ ( t ) (cid:3) e iθ ( t ) + t (cid:88) t c ( t ) E ( t ) ϕ ( t ) e iθ ( t ) = t (cid:88) t c ( t ) E ( t ) ϕ ( t ) e iθ ( t ) , (2.8) ANDREW DAS ARULSAMY which leads to t (cid:88) t ˙ c ( t ) ϕ ( t ) e iθ ( t ) = − t (cid:88) t c ( t ) ˙ ϕ ( t ) e iθ ( t ) . (2.9)Now comes the crucial part, we will take the inner product with ϕ ( t ), which is aneigenfunction at a later time, t ( t > t ), evolved from ϕ ( t ). Therefore, (cid:104) ϕ ( t ) | t (cid:88) t ˙ c ( t ) | ϕ ( t ) (cid:105) e iθ ( t ) = −(cid:104) ϕ ( t ) | t (cid:88) t c ( t ) | ˙ ϕ ( t ) (cid:105) e iθ ( t ) , (2.10)where (cid:104) ϕ ( t ) | ϕ ( t ) (cid:105) = δ t t . (2.11)From Eq. (2.11), one obtains t (cid:88) t ˙ c ( t ) δ t t e iθ ( t ) = − t (cid:88) t c ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) e iθ ( t ) . (2.12)Now, let us assume that the evolved eigenfunction, ϕ ( t ) is not orthogonal to ϕ ( t )or δ t t = 1. Hence, Eq. (2.12) can be written as˙ c ( t ) δ t t e iθ ( t ) = − t (cid:88) t c ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) e iθ ( t ) ˙ c ( t ) = − t (cid:88) t c ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) e i [ θ ( t ) − θ ( t )] . (2.13)Differentiating Eq. (2.4), taking the inner product with ϕ ( t ) and using (cid:104) ϕ ( t ) | H | ˙ ϕ ( t ) (cid:105) = E ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) we can derive ∂∂t ( Hϕ ( t )) = ∂∂t ( E ( t ) ϕ ( t )) , ˙ Hϕ ( t ) + H ˙ ϕ ( t ) = ˙ E ( t ) ϕ ( t ) + E ( t ) ˙ ϕ ( t ) , (cid:104) ϕ ( t ) | ˙ H | ϕ ( t ) (cid:105) + E ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) = ˙ E ( t ) + E ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) . (2.14)Since the single eigenfunction at t remains as a single eigenfunction at t , thereis no available excited eigenfunctions for any transition to occur. Therefore thetransition probability is simply zero. Invoking δ t t = 1, we can rewrite Eq. (2.14)as ( E ( t ) − E ( t )) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) = (cid:104) ϕ ( t ) | ˙ H | ϕ ( t ) (cid:105) − ˙ E ( t ) . (2.15)Substituting Eq. (2.15) into Eq. (2.13), we obtain˙ c ( t ) = − c ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) − (cid:88) t (cid:54) = t c ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) e i [ θ ( t ) − θ ( t )] = − c ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) − (cid:88) t (cid:54) = t c ( t ) (cid:68) ϕ ( t ) (cid:12)(cid:12)(cid:12) ˙ H (cid:12)(cid:12)(cid:12) ϕ ( t ) (cid:69) − ˙ E ( t ) E ( t ) − E ( t ) e i [ θ ( t ) − θ ( t )] . (2.16)Equation (2.16) is straightforward where c ( t ) = c ( t ) = 1 because there is onlyone eigenfunction, and is not orthogonal along the time line for t ∈ [ t , t ]. Let us UANTUM ADIABATIC THEOREM FOR CHEMICAL REACTIONS 9 now invoke the orthogonality ( δ t t = 0), which implies that (from Eq. (2.13))0 = − t (cid:88) t c ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) e iθ ( t ) . (2.17)Using Eqs. (2.17) and (2.14), c ( t ) = − c ( t ) e iθ ( t ) (cid:104) ϕ ( t ) | ˙ ϕ ( t ) (cid:105) (cid:88) t (cid:54) = t (cid:68) ϕ ( t ) (cid:12)(cid:12)(cid:12) ˙ H (cid:12)(cid:12)(cid:12) ϕ ( t ) (cid:69) E ( t ) − E ( t ) , (2.18)which in turn allows us to conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:68) ϕ ( t ) (cid:12)(cid:12)(cid:12) ˙ H (cid:12)(cid:12)(cid:12) ϕ ( t ) (cid:69) E ( t ) − E ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ ⇒ c ( t ) ≈ , or(2.19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:68) ϕ ( t ) (cid:12)(cid:12)(cid:12) ˙ H (cid:12)(cid:12)(cid:12) ϕ ( t ) (cid:69) E ( t ) − E ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) ⇒ c ( t ) (cid:28) . (2.20)Equations (2.19) and (2.20) are equal to the adiabatic criterion given in Eq. (2.1)if ˙ E ( t ) (cid:104) ϕ ( t ) | ϕ ( t ) (cid:105) (cid:54) = 0 (cid:4) As indicated earlier, systems with a single eigenfunction, and in the presenceof t -dependent orthogonalization, we will face the consequence of a particle trav-eling backward in time due to c ( t ) (cid:28) ω (cid:48) ( t )(see Eq. (1.9)). This is physically not possible because we will not be able to or-thogonalize (assuming this can be done) the eigenfunction without disturbing theoccupied electron. In other words, it is not possible because the eigenfunction hereitself represents the wave function of the electron based on the Hermitian quantummechanics [26]. Therefore, Eq. (2.20) strictly implies that the ground state electronhas been excited or ionized. We cannot be sure how this (excitation or ionization)is possible from the criterion given in Eq. (2.1) alone, it could be due to degeneracyor invalid Eq. (2.3) [for example T ex ( t ) < T in ( t )]. Later in Example 4 we will justifythat this phenomenon ( c ( t ) (cid:28)
1) can be related to chemical reactions (with andwithout the t -dependent orthogonalization).3. Applications of Theorem 1
Now we are ready to discuss the applications of Theorem 1, and how this theoremrelates to these examples both mathematically and physically. The first exampleis for spin-1 / / t -dependent external disturbances. Subsequently, we discuss gapless systemdue to energy level crossings, while the last one invokes the t -dependent orthogonal-ization in the spin-1 / t -dependent orthogonalization. Figure 1.
A constant magnetic field, B (magnitude B ) rotates ata constant velocity, Ω, and with a constant opening angle, α . Here, B ( t ) = B [sin α cos(Ω t )ˆ i + sin α sin(Ω t )ˆ j + cos α ˆ k ]. See Example 1for details. Example 1: Spin- two-level system in the presence of magnetic field. The first example is the one derived by Griffith [9]. An electron can have spinup or down, and therefore, in the presence of rotating (constant velocity = Ω)and constant magnetic field with a constant opening angle, α , one can control thetransition probability of this electron (from spin up to down) to satisfy [9] |(cid:104) Ψ( t ) | ϕ down ( t ) (cid:105)| = (cid:20) ΩΛ sin α sin (cid:18) Λ t (cid:19)(cid:21) , (3.1)where Λ = Ω + ω − ω Bohr cos α and ω Bohr = [ E up − E down ] / (cid:126) = 1 /T in .Obviously, in this example we have Ω = 1 /T ex . In the limit T ex (cid:29) T in due to Ω (cid:28) ω Bohr , one obtains the transition probability to spin down to be zero because Λ → ω Bohr . Therefore, (Ω / Λ) → (Ω /ω Bohr ) → t -dependent orthogonalization.In order to prove Eq. (1.10), we write the Hamiltonian [9] for this system (see Fig. 1) H G ( t ) = (cid:126) ω Bohr (cid:18) cos α e − i Ω t sin αe i Ω t sin α − cos α (cid:19) , and its normalized eigenspinors [9] ϕ up ( t ) = (cid:18) cos( α/ e i Ω t sin( α/ (cid:19) and ϕ down ( t ) = (cid:18) e − i Ω t sin( α/ − cos( α/ (cid:19) . Their corresponding eigenvalues are (cid:126) ω Bohr / − (cid:126) ω Bohr /
2, respectively. FromEq. (1.3), the exact solution [9]Ψ( t ) = (cid:20) cos(Λ t/ − (cid:2) i ( ω Bohr − Ω) / Λ (cid:3) sin(Λ t/ (cid:21) cos( α/ e − i Ω t/ (cid:20) cos(Λ t/ − (cid:2) i ( ω Bohr + Ω) / Λ (cid:3) sin(Λ t/ (cid:21) sin( α/ e i Ω t/ . UANTUM ADIABATIC THEOREM FOR CHEMICAL REACTIONS 11
Proof for Eq. (1.10): We now evaluate (cid:68) ϕ down ( t ) (cid:12)(cid:12)(cid:12) ˙ H ( t ) (cid:12)(cid:12)(cid:12) ϕ up ( t ) (cid:69) = Ω (cid:20) i α (cid:21) (cid:126) ω Bohr e − i Ω t . (3.2)Since ω Bohr is a constant and Ω = 1 /T ex , we can readily obtain the proportionalitygiven in Eq. (1.10) (cid:4) If ω Bohr is t -dependent, then one obtains (cid:68) ϕ down ( t ) (cid:12)(cid:12)(cid:12) ˙ H ( t ) (cid:12)(cid:12)(cid:12) ϕ up ( t ) (cid:69) E up ( t ) − E down ( t ) ∝ T ex ( t ) (cid:28) . (3.3)Equation (3.3) implies that T ex (cid:29) T in explicitly as explained in the introduction. It is not explicit becausewe can rewrite Eq. (3.3) to obtain (cid:68) ϕ down ( t ) (cid:12)(cid:12)(cid:12) ˙ H ( t ) (cid:12)(cid:12)(cid:12) ϕ up ( t ) (cid:69) ∝ (cid:126) /T in ( t ) T ex ( t ) ∝ (cid:126) T in ( t ) T ex ( t ) (cid:28) , (3.4)which means that we are still not able to find the relationship between T in ( t ) and T ex ( t ). Therefore, Eq. (2.3) needs to be invoked independent of Eq. (2.2). Example 2: Spin- two-level oscillating system. In the above example,Ω( t ) = Ω t , whereas α and ω Bohr are t -independent constants. On the other hand,the oscillating Schwinger Hamiltonian [11, 27] H S ( t ) = (cid:126) ω Bohr ( t )2 (cid:18) cos α ( t ) e − i Ω( t ) sin α ( t ) e i Ω( t ) sin α ( t ) − cos α ( t ) (cid:19) , and Ω, α and ω Bohr are all time-dependent variables. We again assume the twoenergy levels do not cross and there is no t -dependent orthogonalization. Here,Fig. 2 schematically captures the physical system represented by H S ( t ). In thiscase, we have α ( t ) = α ( t ) , α ( t ) , · · · ; ˙ α = ˙ α , · · · ; Ω( t ) = Ω( t ) , · · · ; ˙Ω = ˙Ω , · · · ; ω Bohr ( t ) = ω Bohr ( t ) , · · · ; ˙ ω Bohr ( t ) = ˙ ω Bohr ( t ) , · · · . This means that one will obtainmany α ( t ), ˙ α , Ω( t ), ˙Ω, ω Bohr ( t ), ˙ ω Bohr ( t ), and so on due to t -dependent fluctuationsin the rotating magnetic field (see Fig. 2). Now, to guarantee QAA for H S ( t ), wejust need to invoke Eq. (2.3) such that ω Bohr ( t ) x (cid:29) | ˙ α x | , | ˙ ω Bohr ( t ) y | (cid:29) | ¨ α y | , · · · and ω Bohr ( t ) u (cid:29) Ω u , | ˙ ω Bohr ( t ) v | (cid:29) | ˙Ω v | , · · · where { x, y, u, v } = N ∗ . Using theseinequalities, we prove that the two conditions (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙Ω sin α − i ˙ α ˙Ω cos α − ω Bohr − dd t arg( ˙Ω sin α − i ˙ α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Ω (cid:48) δ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , and(3.5) (cid:90) t (cid:12)(cid:12)(cid:12)(cid:12) dd t (cid:48) Ω (cid:48) δ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) d t (cid:48) (cid:28) , (3.6)derived by Comparat [11] are actually the extended versions of Eq. (2.3). Proof : For H G ( t ), Eq. (3.5) reduces to (cid:12)(cid:12)(cid:12)(cid:12) Ω sin α Ω cos α − ω Bohr (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , (3.7)because ˙ α = 0 = (d / d t )arg(Ω (cid:48) ), Ω( t ) = Ω t and ˙Ω = Ω. Equation (3.7) requires ω Bohr (cid:29)
Ω for QAA to be valid, in accordance with Eq. (2.3). We now leaveEq. (3.5) intact and for QAA to be valid in H S ( t ), one requires ω Bohr (cid:29) | ˙ α | , ω Bohr (cid:29)
Figure 2.
A constant magnetic field, B (magnitude B ) rotateswith some fluctuations at a velocity, Ω( t ) and with a t -dependentopening angle, α ( t ). See Example 2 for details.Ω and | ˙ ω Bohr | (cid:29) | ˙Ω | in which, the first two inequalities satisfy T in (cid:28) T ex , whilethe last one is equivalent to | ˙ T in | (cid:28) | ˙ T ex | . If (d / d t )arg(Ω (cid:48) ) < | ˙ ω Bohr | (cid:29) | (d / d t )arg(Ω (cid:48) ) | or | ¨ T in | (cid:28) | ¨ T ex | , which guarantees ω Bohr (cid:29) | (d / d t )arg(Ω (cid:48) ) | as required from Eq. (3.5). Having proven that, it is nowstraightforward to generalize Eq. (3.6) to obtain (cid:88) q = x,y,u,v (cid:20)(cid:18) T in T ex (cid:19) q + (cid:18) | ˙ T in || ˙ T ex | (cid:19) q + (cid:18) | ¨ T in || ¨ T ex | (cid:19) q + · · · (cid:21) (cid:28) . (3.8)Therefore, Eqs. (3.5) and (3.6) are indeed the extended versions of Eq. (2.3) (cid:4) Example 3: Two-level system with a single energy level crossing (logicalexposition).
For simplicity, we confine ourselves to a two-level system with oneenergy level crossing occurred somewhere between t and t , say t z . Again, weassume there is no t -dependent orthogonalization. In this case, the energy levelcrossing occurs as a result of some t -dependent Yukalov (non-linear)-type interac-tion, V ( t ) [17]. Hence, one can write the solved Hamiltonian for such a systemas H ( t ) ϕ m ( t ) = ( H ( t ) + V ( t )) ϕ m ( t ) = ( h a ( t ) + v a ( t )) ϕ m ( t ) = E a ( t ) ϕ m ( t ) ,H ( t ) ϕ n ( t ) = ( H ( t ) + V ( t )) ϕ n ( t ) = ( h b ( t ) + v b ( t )) ϕ n ( t ) = E b ( t ) ϕ n ( t ) , (3.9)where H ( t ) is the basic Hamiltonian excluding all interactions, whereas V ( t ) con-tains all the interaction terms, and we do not require (cid:104) ϕ n ( t ) | ϕ m ( t ) (cid:105) = 0 or 1.Here, E a ( t ), h a ( t ), v a ( t ), h b ( t ), v b ( t ) and E b ( t ) are all eigenvalues. Degeneracydue to energy level crossing implies E a ( t z ) = E b ( t z ) at a certain point of time( t z ), and for other times, they are not degenerate. If they are always degener-ate, then h a ( t ) = h b ( t ), v a ( t ) = v b ( t ) and E a ( t ) = E b ( t ), which physically mean ϕ n ( t ) = ϕ m ( t ): this means that there are two electrons occupying the same energylevel for all time t . As a consequence, degeneracy due to energy level crossing re-quires h a ( t z ) (cid:54) = h b ( t z ), v a ( t z ) (cid:54) = v b ( t z ) and E a ( t z ) = E b ( t z ). Therefore, for degener-ate systems, one needs to employ Eq. (2.3) such that 1 /T ex ( t z ) (cid:28) ( h a ( t z ) − h b ( t z )) / (cid:126) and 1 /T ex ( t z ) (cid:28) ( v a ( t z ) − v b ( t z )) / (cid:126) where ( v a ( t z ) − v b ( t z )) / (cid:126) and ( h a ( t z ) − h b ( t z )) / (cid:126) equal 1 /T in ( t z ). This is the reason why QAA can still be applied for gapless (dueto energy level crossings) systems without strictly requiring T ex → ∞ . UANTUM ADIABATIC THEOREM FOR CHEMICAL REACTIONS 13
Example 4: Systems with and without the time-dependent orthogonal-ization: applied to chemical reactions.
In the previous examples, we haveshown how and why QAA is valid for gapped and gapless systems provided thatthey satisfy Eqs. (2.2) and (2.3). These conditions are also shown to be necessaryand sufficient. In the following example however, we will first provide the logicalexpositions of how Eq. (2.1) due to t -dependent orthogonalization can render theQAA valid. All we need to do here is to invoke Eq. (2.1), which now implies (cid:68) ϕ down ( t ) (cid:12)(cid:12)(cid:12) ˙ H ( t ) (cid:12)(cid:12)(cid:12) ϕ up ( t ) (cid:69) E up ( t ) − E down ( t ) ∝ T (cid:48) ex ( t ) ≈ . (3.10)Equation (3.10) tells us that the system has started out with spin-up, ϕ up ( t ) at t = t , and after t -dependent orthogonalization, it ends up with spin-down, ϕ down ( t ) at t = t where t > t . For QAA to be valid, we require large external disturbancessuch that T (cid:48) ex ( t ) ≈ T (cid:48) ex ( t ) ≈ T (cid:48) in ( t ), which is now necessary for the validityof QAA. Without this large disturbances (or rapidly oscillating Hamiltonian), thespin-up electron (at t = t ) cannot occupy the orthogonalized spin-down state at t = t . In this case, c ( t ) (cid:28) Claim : However, one should note here that Eq. (2.1) is also applicable for systemswithout the t -dependent orthogonalization. Proof : Recall Eq. (2.16) with δ t t = 1˙ c m ( t ) = − c m ( t ) (cid:104) ϕ m ( t ) | ˙ ϕ m ( t ) (cid:105) − (cid:88) t (cid:54) = t c m ( t ) (cid:68) ϕ m ( t ) (cid:12)(cid:12)(cid:12) ˙ H (cid:12)(cid:12)(cid:12) ϕ m ( t ) (cid:69) − ˙ E m ( t ) E m ( t ) − E m ( t ) e i [ θ m ( t ) − θ m ( t )] . (3.11)Here, δ t t = 1 implies that there is no t -dependent orthogonalization. First weneglect the second term by requiring (cid:104) ϕ m ( t ) | ˙ H | ϕ m ( t ) (cid:105) − ˙ E m ( t ) is extremelysmall and/or E m ( t ) − E m ( t ) is extremely large, hence˙ c m ( t ) = − c m ( t ) (cid:104) ϕ m ( t ) | ˙ ϕ m ( t ) (cid:105) . (3.12)Its solution c m ( t ) = c m ( t ) e iγ m ( t ) , where(3.13) γ m ( t ) = i (cid:20) (cid:90) t (cid:28) ϕ m ( t (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) ∂∂t (cid:48) ϕ m ( t (cid:48) ) (cid:29) dt (cid:48) (cid:21) t = t . Since the electron occupied the m th eigenstate at t = t , thus c m ( t ) = 1, and when t = t , c m ( t ) = 1, and so on. Therefore, we can now recall Eq. (2.5) and writeΨ m ( t ) = c m ( t ) ϕ m ( t ) e iθ m ( t ) , (3.14)and using Eq. (3.13)Ψ m ( t ) = c m ( t ) ϕ m ( t ) e iθ m ( t ) = c m ( t ) ϕ m ( t ) e iγ m ( t ) e iθ m ( t ) . (3.15)Since δ t t = 1, Eq. (3.15) readsΨ m ( t ) = c m ( t ) ϕ m ( t ) e iγ m ( t ) e iθ m ( t ) , (3.16) as it should be, picking up only an additional phase factor ( e iγ m ( t ) ). If the secondterm on the right-hand side of Eq. (3.11) is not extremely small, then we strictlyrequire (cid:68) ϕ m ( t ) (cid:12)(cid:12)(cid:12) ˙ H (cid:12)(cid:12)(cid:12) ϕ m ( t ) (cid:69) − ˙ E m ( t ) E m ( t ) − E m ( t ) ∝ T (cid:48) ex ( t ) ≈ , (3.17)in accordance with Eq. (3.10) (cid:4) We anticipate that the evolution of the eigenfunction in the presence of, or in theabsence of t -dependent orthogonalization is relevant to understand chemical reac-tions beyond the standard procedures of calculating activation energies, includingthe H +2 molecule ion dissociation [19] and the water molecule splitting [28]. Forexample, chemical reactions (associations or dissociations) strictly require large ex-ternal disturbances that give rise to significant changes to the eigenfunctions toform new compounds or to dissociate to its constituent chemical components [29].In other words, without the significant changes to the eigenfunctions due to largeexternal disturbances, which define the existence of activation energies, the chem-ical reactions cannot occur. For instance, bringing an atomic hydrogen and H + together chemically to produce H +2 requires a new wave function, ϕ ( t, r ) new forthe ground state of H +2 molecule. We take the wave function for the atomic Has ϕ ( t, r ). Thus, the condition given in Eq. (2.1) considers the evolution of thewave function, from ϕ ( t, r ) to ϕ ( t, r ) new . However, the generator ( ˆ G ) defined by ϕ ( t, r ) new = ˆ G ϕ ( t, r ) is not known, and presently G stands for guess as we often doexactly that to obtain the new wave function by means of the linear combination ofatomic orbitals and/or by requiring any arbitrary wave function to give solutionsthat are convergent. Anyway, we can now invoke Eq. (2.1), which is also suitableto check the applicability of the QAT for the chemical reaction, H + H + → H +2 . Inthis case, Eq. (2.1) can be written as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:68) ϕ ( t, r ) new (cid:12)(cid:12)(cid:12) ˙ H chemicalreaction (cid:12)(cid:12)(cid:12) ϕ ( t, r ) (cid:69) − ˙ E m (H) (cid:104) ϕ ( t , r ) new | ϕ ( t , r ) (cid:105) E m (H) − E m (H +2 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ ˙ c m ( t k ) ≈ , (3.18)where E m (H) and E m (H +2 ) are the ground state energies of atomic H and H +2 mole-cule, respectively. The magnitude of approximately one stated in Eq. (3.18) can beachieved during chemical reactions because large external disturbances (by meansof thermal and/or potential energies) are required to initiate any chemical reaction.Such a disturbance will first give rise to excitations of the ground state electronsand therefore, QAA is also applicable for such a dynamical process, provided thatwe know the Hamiltonian ( H ( t ) chemicalreaction ) describing this chemical process explicitly.On the contrary, Eq. (2.2) is not applicable for this process, and as we have statedearlier in the introduction, this does not imply that the QAT has been violated justbecause ϕ ( t, r ) new differs significantly (beyond the phase factor) from ϕ ( t, r ).4. Further Analysis
In the discussion and proofs given above, we should not be falsely led into think-ing that the condition given in Eq. (2.1) imposes orthogonality. On the contrary,Eq. (2.1) allows the final eigenfunction to remain the same, or acquire some phasefactors or be mathematically different from the initial eigenfunction, or even be
UANTUM ADIABATIC THEOREM FOR CHEMICAL REACTIONS 15 different such that the final eigenfunction is orthogonal to the initial one. More-over, the proof for Eq. (2.1) also correctly exposes the intrinsic reason why weneed to guess the eigenfunction, for instance, we have used the linear combinationof the atomic eigenfunction in the form of Eq. (2.5). In other words, we need tounderstand what is the mechanism involved, which gives us the license to rewrite ϕ ( t, r ) as ϕ ( t, r ) new . Well, of course ϕ ( t, r ) new is one of the acceptable solution tothe Hamiltonian that gives the lowest ground state energy. But we still lack thefundamental knowledge that enabled us to construct ϕ ( t, r ) new by hand .For example, we reconsider a H +2 molecular ion such that the single electronmolecular orbital, ϕ ( t, r ) new can be constructed, either from the molecular-orbital(MO), or valence-bond (VB) theory [29]. In MO theory one uses the linear com-bination of atomic orbitals (LCAO) approach to construct the molecular orbital,namely, ϕ ( t, r ) new is constructed from the atomic orbitals of the type, ϕ ( t, r ).In this case, one immediately starts the construction in the molecular framework,meaning, in the presence of both H and H + . In VB theory, one still uses the atomicorbitals, but they are combined non-linearly [29]. Therefore, in both cases, theconstructions of ϕ ( t, r ) new are carried out in the molecular framework [29], eventhough they are generated from the atomic orbitals in different ways.Now, the chemical reaction between H and H + , physico-chemically means thatwe are bringing two isolated atomic H and H + ion closer from infinity, or they maycome closer on their own due to some attraction between them via the Coulombinteraction at finite distances. Either way, once they are close enough, and withsufficient external heat ( Q ), but not higher (than the H − H + bond energy), thesingle electron in H can be excited (or polarized) to the highest possible atomicenergy level and binds to the H + ion, to form H +2 . When one reads the abovechemical reaction carefully, two intrinsic phenomena pop out that can be writtenas two statements. Statement 1 : There is a change in the mathematical property of the wavefunction (beyond the phase factor) when ϕ ( t, r ) for H transforms to ϕ ( t, r ) new forH +2 . Remark 4 : Let us call this the wave function transformation, and this transfor-mation is one form of electronic phase transition arising due to the transition fromthe atomic (or ionic) to molecular system. The formal proof for the existence ofsuch a quantum phase transition in many-body systems is given elsewhere [30].
Statement 2 : In both MO and VB theories, ϕ ( t, r ) new have been constructedbased on the assumption that the chemical reaction has already taken place, sincethe construction of wave functions were carried out within the molecular framework. Remark 5 : This means that, we are lack of the information needed to find thegenerator ( G ) pointed out earlier, and as anticipated, G still remains as a guess,while to those who are not comfortable with this, we may call it “an educatedguess”.In this work, we have actually proven these two statements by proving the ex-istence of the condition stated in Eq. (2.1). For example, we can observe thatEq. (2.1) correctly captures the effect of the chemical reaction by requiring anelectronic phase transition (due to wave function transformation), in which, theexistence of this transition justifies why Remark 2 is necessary. Simply put, eventhough time is a continuous parameter, we cannot construct a continuous t - or r ( t )-dependent eigenfunction for an electron that starts as an atomic orbital of an isolated atomic H that goes through an electronic phase transition to form H +2 .Here, r ( t ) denotes the t -dependent electron coordinate, which means, we do notrequire ϕ ( t, r ) new to be both r and t dependents explicitly.As a consequence, just because ϕ ( t, r ) new cannot be written as a continuousfunction of r ( t ) (starting from isolated H and H + , until the formation of H +2 ), oneshould not assume that the above-stated wave function transformation (or phasetransition) never took place, or should be ignored. The very moment we combinethe atomic orbitals (linearly or otherwise) within the molecular framework provesthe existence of this transformation, which we discovered from Eq. (2.1). Moreover,Eq. (2.1) also leads us to understand why the activation energy for the above chemi-cal reaction exists − because it is needed to initiate the wave function transformation(non-observable) or the electronic phase transition (observable). This is consistentwith the finite-temperature quantum phase transitions proven in Ref. [30].Finally, before we conclude, let us recall the Born-Oppenheimer (BO) argumentsgiven by Teufel [8]. Here, Eq. (2.1) does not apply for an obvious reason − a givenwave function in a particular quantum system, that does not go through a quantumphase transition or t -dependent orthogonalization, does not transform beyond thephase factor with time. Therefore, we just need to invoke Eq. (2.2) and Eq. (2.3)for such systems, which can be done within these well-known theories, MO, VBand the Density Functional Theory (DFT). Briefly, BO approximation is validfor as long as the electrons average kinetic energy equals, or larger than that ofthe ions, mv ≥ M v , which in turn implies v ion (cid:28) v el and T ex (cid:29) T in if M (cid:29) m . Obviously, the above assumption does not violate Eq. (2.3), hence QAAis applicable.It is also true that repulsive interaction (due to electron-electron interaction) canbe invoked to explain the existence of finite energy gap [8, 31], however, energy levelsdo cross due to “lucky coincidences” in many-body systems, as shown in Example3. Even in the presence of such coincidences, we can still check the applicability ofQAA via Eq. (2.3). Now, in the worst case scenario, we will have (refer to Example3) h a ( t ) = h b ( t ) , v a ( t ) = v b ( t ) and E a ( t ) = E b ( t ) , (4.1)then QAA seems to be not applicable. But if we look closely, Eq. (4.1) physicallyimply ϕ n = ϕ m . Hence, one can indeed invoke Eq. (2.3) through the BO approxi-mation and justify the applicability of QAA because the transition between ϕ n and ϕ m does not violate QAA. Of course, mathematically one can construct the eigen-functions such that ϕ n (cid:54) = ϕ m and still satisfy Eq. (4.1). But this is like saying, weare attempting to invalidate QAA, solely for the purpose of violating QAA withoutany physical justification. 5. Conclusions
In conclusion, we have developed a new version of the quantum adiabatic the-orem that takes the time-dependent orthogonalization into account, and even inthe absence of this orthogonalization, we found that these (with and without the t -dependent orthogonalization) effects are very relevant to study the dynamics ofchemical reactions. We also generalized and incorporated the relationship betweenthe internal and external timescales as an additional, but separate condition into UANTUM ADIABATIC THEOREM FOR CHEMICAL REACTIONS 17 the quantum adiabatic theorem. The time-dependent orthogonalization and chem-ical reactions are proven to be captured by Eq. (2.1) self-consistently. In addition,Eq. (2.1) also led us to discover the activation energy in a given chemical reac-tion is needed for the wave function transformation, which can be observed via theelectronic phase transition. While the validity of other conditions are explicitlyexposed to be sufficient in the spin-1/2 two-level system. Therefore, the new the-orem is shown to be valid for both gapped and gapless systems with appropriateexamples.For gapped two-level system, Eqs. (2.2) and (2.3) are found to be sufficient andnecessary, in accordance with Comparat [11]. Whereas, for gapless systems withdegeneracies (due to energy level crossings) and quantized energy levels, Eqs. (2.2)and (2.3) are again sufficient and necessary, in agreement with Avron and El-gart [12]. However, for gapless systems, we need not keep track of the eigenfunc-tions to show they are sufficient, for as long as Eq. (2.3) is valid. Interestinglyhowever, we do not always require T ex → ∞ in the presence of non-linear Yukalov-type [17] interactions. The importance of Eq. (2.3) cannot be overstated as therecent counter-examples [11, 32] and counter-proofs [11, 14, 32] can be traced backto the violation of Eq. (2.3) [33]. Acknowledgments
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