Geometric energy transfer in two-component systems

Factoring a wave function into marginal and conditional factors partitions the subsystem kinetic energy into two terms. The first depends solely on the marginal wave function, through its gauge-covariant derivative, while the second depends on the quantum metric of the conditional wave function over the manifold of marginal variables. We derive an identity for the rate of change of the second term. This article is part of the theme issue ‘Chemistry without the Born–Oppenheimer approximation’.


Introduction
Exposing a molecule to a visible or UV electromagnetic pulse excites primarily the electronic degrees of freedom. On a longer time scale, the electrons transfer energy to the nuclei. While energy transfer can be evaluated with knowledge of the electron-nuclear wave function, given a choice of nuclear subsystem, to gain deeper insight into excited state molecular dynamics, optimize control of molecular motion and bond breaking, and engineer molecular scale mechanical motion, it would be valuable to have a formula expressing the energy transfer in terms of simple fundamental quantities.
Using the exact factorization (EF) formalism [1,2], which involves the factorization of a two-component wave function into a product of marginal and conditional amplitudes [3], we have derived Ehrenfest identities [4] for the time rate of change of the kinetic energy, momentum and angular momentum of the nuclear subsystem. By evaluating the commutators in the Heisenberg equations of motion, we found that a novel quantum mechanical force operator appeared in all three identities. In terms of this force operator, all three identities could be put in a form directly analogous to the classical equations of motion dE/dt = F · v, dP/dt = F and dL/dt = R × F.
One wrinkle in those results was the fact that the classical force-times-velocity form could only be achieved for the rate of change of one part of the nuclear kinetic energy, namely the term where |χ (t) is the nuclear wave function that satisfies the effective Schrödinger equation of the exact factorization method; R denotes the set {R i } of N n nuclear coordinates. T n,marg is the kinetic energy of a fictitious closed quantum system acted upon by scalar and vector potentials E(R, t) and A i (R, t) (defined below). In other words, it is simply the kinetic energy one would write down if one were given the Schrödinger equation (1.2) but not told it describes only one component of a two-component system. Since χ is the marginal probability amplitude for the nuclear component of a system of electrons and nuclei, we call T n,marg the marginal nuclear kinetic energy.
The closed system described by equation (1.2) reproduces the true N n -body density and N nbody current density of the nuclear subsystem, but it is fictitious in the sense that off-diagonal elements of the nuclear density matrix and hence non-local observables of the nuclear subsystem cannot be calculated directly from the wave function χ (R, t). A prime example is the true nuclear kinetic energy as obtained from the full wave function |Ψ (t) which obeys the identity [5,6] where the additional contribution is being the vector potential and |Φ(t) the ket corresponding to the electronic wave function Φ(r, t|R) = Ψ (r, R, t)/χ (R, t) conditional on R. The Ehrenfest identity of Li et al. [4] provides a straightforward way to calculate dT n,marg /dt but gives no information about dT n,geo /dt. Our main result is an exact identity for dT n,geo /dt, expressed in terms of primitive quantities. An interesting outcome of the derivation is the appearance of a new rank-3 quantum geometric quantity that cannot be expressed in terms of lower-rank quantities defined on R-space.
The integrand of (1.4) has geometric significance itself, as it can be expressed [7,8] in terms of a tensor contraction between I μν , a symmetric inverse inertia tensor, and a Riemannian metric [9]  which we call the geometric part of the nuclear kinetic energy. As all quantities except I μν are time dependent, we suppress the time argument here and hereafter. In (1.5) and (1.6), we have switched to a tensor calculus notation, i.e. subscripts/superscripts correspond to the covariant/contravariant indices of a tensor on Q, and we have generalized to a Watsonian kinetic energy operatorT n = (1/2)I μν P μ P ν [10] with P μ = −i∂/∂Q μ and an implicit summation convention. The metric in equation (1.5) is the EF counterpart of the quantum metric tensor originally studied in the Born-Oppenheimer (BO) approximation [11][12][13][14]. The quantum metric tensor has recently attracted attention in condensed matter physics, where its applications are too numerous to cite here. The usual laboratory frame kinetic energy operator, cf. equation (1.3), is a special case of the Watsonian kinetic energy, in which the set of Q μ is {R 1x , R 1y , R 1z , R 2x , R 2y , R 2z , . . .} and I μν is diagonal and Q μ -independent, i.e. I μν = I iα,jβ = M −1 i δ ij δ αβ with i labelling the nucleus and α = x, y, z. The Watson form encompasses two additional cases: (1) an isolated molecule after removing the centre-of-mass coordinate [15]; and (2) a system described by a distinguished set of relevant collective coordinates. The phonon modes of a crystalline solid described with Bornvon Karman boundary conditions are an example of case (2) in which Q μ comprise the normal mode amplitudes U qλ with quasi-momentum q and branch λ, I μν = I qλ,q λ = M −1 qλ δ q,−q δ λ,λ is off-diagonal and M qλ is the effective mass of the normal mode [16]. In case (2), I μν will be Q μdependent when Q is non-Euclidean, which can arise e.g. in an approximate reduced description in terms of a restricted set of collective coordinates. In the case of an isolated molecule in which only the centre-of-mass coordinate is removed, it is always possible to choose translationally invariant coordinates such that I μν is Q μ -independent [8,15]; such an I μν is generally nondiagonal but can be further diagonalized. Here, for simplicity, we assume I μν is Q μ -independent but not necessarily diagonal.

Exact factorization formalism
To derive an identity for dT n,geo /dt that does not invoke the BO approximation, we make use of the exact factorization formalism [1][2][3], the essential elements of which we briefly review here.
Starting from the full wave function Ψ (q, Q) = q, Q|Ψ , where q and Q denote the sets of electronic and nuclear coordinates, one defines the nuclear wave function which is the marginal probability amplitude corresponding to the marginal probability and the arbitrary gauge λ(Q). The conditional electronic wave function depends parametrically on Q and satisfies the equation

Derivation of the main result
Starting from equation (1.6), we obtain where J μ is the nuclear current density In the second line of equation (3.1), we have used the continuity equation, and in the third line, we have performed integration by parts and assumed that the boundary term vanishes; this is generally true for finite systems. Focusing on E geo = (1/2)I μν g μν in the second term and noting that I μν is time-independent, our first step is to evaluate ∂g μν /∂t: After contraction with I μν , these two terms give equal contributions to The terms on the right-hand side of equation (2.4) give the following contributions to the first term in equation (3.3): The (1a) term can be put in various forms but none is simpler than the others. The second term on the right-hand side of ∂g (1a) μν /∂t vanishes upon contraction with the symmetric tensor I μν because is real. Turning to the (1b) term and moving (P ν − A ν ) into the bra, we obtain The second term drops out upon contraction with the symmetric tensor I μν because is real. Therefore, the contribution of the (1b) term to ∂E geo /∂t is where we defined (3.10) The first term simplifies to where we introduced the definition The second term of equation (3.10) gives where we used equation (5.4), derived in §5, to relate D μντ to derivatives of g μν , and hence to ∂ τ E geo , which is an electric-like force originating from E geo . The antisymmetry of B μν makes the contribution from the second term of equation (3.13) vanish. The second term of equation (3.14) will be seen to cancel with the first term of equation (3.1). Putting equations (3.4), (3.8) and (3.14) together, we obtain Substituting this into equation (3.1) yields the final result We used the identity to combine the second and third terms of equation ( the nuclear subsystem T n,marg and the geometric contribution to the nuclear kinetic energy T n,geo as functions of time. The Schrödinger equation is Our strategy is to reverse engineer the functions h 0 (x, t), h 1 (x, t) and h 3 (x, t) such that the dynamics lead to a state with a Gaussian nuclear density which undergoes damped oscillations determined by the functions Substituting into equation (4.1), leads to the following equations: where subscripts denote partial differentiation, e.g. α x = ∂ x α. Choosing a gauge such that χ is real, the first equation is equivalent to the continuity equation ∂ t |χ | 2 = −∂ x J with the nuclear current density and vector potential Assuming |χ (x, t)| 2 is given by equation (4.3), we can invert the continuity equation to obtain Defining w = cos θ , we choose a state Ψ (x, t) with The Hamiltonian parameters that follow from equation (4.5) are and The identity in equation (3.16), adapted to the present model, is In a one-dimensional problem, the metric g becomes the scalar g = 1 4 w 2 and for completeness we record The marginal nuclear kinetic energy can be expressed entirely in terms of ρ = |χ | 2 and J as (4.14) Using the above formulae, we have numerically verified equation (4.11) for M = 10 a.u., η = 0.1 (a.u.) −1 and γ = 40 (a.u.) −1 . Snapshots of the nuclear probability density ρ(x, t) and the Hamiltonian variables h 0 (x, t), h 1 (x, t) and h 3 (x, t) are shown in figure 1 for a series of times. The rapidly spatially varying features of h i (x, t), always pinned near x = 1, are associated with the abrupt change of the electronic state from predominantly state 1 for x < 1 to predominantly state 2 for x > 1, as can be seen in the formula for w(x, t) in equation (4.9).
The time dependence of T n,marg and T n,geo are shown in figure 2a, and the BO-like energy E BO = dx|χ | 2 Φ|Ĥ BO |Φ is shown in figure 2b. The total energy is the sum T n,marg + T n,geo + E BO , and it is not conserved. The peaks in T n,geo occur for the times π/2, 3π/2, 5π/2, . . ., when the nuclear wave packet passes through the location of rapid variation of the electron state near x = 1, where E geo is largest, therefore yielding the maximal contribution to the |χ | 2 -weighted integral in   6). It can be seen that T n,geo is much smaller than T n,marg . While this is typical of molecular ground states, we have found dynamical problems in which T n,geo is as large as T n,marg [17]. Quantitative results for T n,geo have been reported (Suppl. Mat. of Ref. [4]). Figure 3 shows the quantity n 1 − n 2 = |Φ 1 (x, t)| 2 − |Φ 2 (x, t)| 2 , which is the difference in population of the two electronic states. The rapid spatial variation of the electronic state at x = 1 makes E geo relatively larger at these locations. The figure shows that the characteristic distance over which the electronic state switches from state 1 to state 2 becomes smaller as time increases. Hence, E geo and the peaks in T n,geo become larger as time progresses.   Figure 3. Grey scale density plot of the difference in population n 1 − n 2 of the two electronic states for the same parameters as figure 1; white corresponds to n 1 − n 2 = 1, black to n 1 − n 2 = −1.

Rank-3 quantum geometric quantities
Here we briefly investigate some properties of the rank-3 quantities C μντ and D μντ that appeared in the last section. Beginning with D μντ , we can show that Thus, D μντ is symmetric with respect to interchange of its second two indices. To further simplify D μντ , we derive the following identity: Taking the real part, we find With this identity and equation (5.1), we obtain Thus, D μντ has been expressed in terms of g μν . In fact, we have where Γ μντ is the Christoffel symbol of the first kind in classical Riemannian geometry. Turning to C μντ , we find the expression This is also symmetric with respect to the interchange of the second two indices. However, due to the presence of the irreducible third-order quantity Im ∂ μ Φ|∂ ν ∂ τ Φ , C μντ cannot be expressed in terms of lower-order geometric quantities and their derivatives. The symbol Γ μντ + iC μντ = (∂ μ − iA μ )Φ|(∂ ν − iA ν )(∂ τ − iA τ )Φ allows one to define a connection [18].

Conclusion
We derived an identity for the rate that energy is transferred to T n,geo , the geometric part of the nuclear kinetic energy. This is the part that derives from the gradient with respect to a nuclear coordinate acting on the parametric dependence of the electronic wave function, the latter arising from the factorization of the full wave function. Our identity complements the Ehrenfest-like identity previously derived for dT n,marg /dt [4], the marginal part of the nuclear kinetic energy. Ehrenfest identities for the expectation values of position and momentum resemble Newton's Laws. Similarly, the Ehrenfest-like identity for dT n,marg /dt has a suggestive force-times-velocity form, paralleling the classical formula for the rate of work done by a force, and therefore appears to lend itself to a classical interpretation of the nuclear motion. Indeed, the terms that appear in the force operator have a close resemblance to the corresponding terms in the force that acts on nuclei when they are treated classically [19,20].
It is not yet clear if identity (3.16) for dT n,geo /dt has a simple classical interpretation. We have not been able to put it in a force-times-velocity form. In the course of evaluating dT n,geo /dt, we derived the equation of motion for the quantum metric g μν . This equation involves a new object, the rank-3 geometric quantity C μντ , which appears to be a purely quantum object.
Putting together the identities for dT n,marg /dt and dT n,geo /dt allows us to calculate the total rate of change of the true nuclear kinetic energy. It is hoped that these identities and the insights derived from them will help scientists control energy transfer in quantum systems.
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