Exact factorization of the time-dependent electron-nuclear wavefunction

A. Abedi, F. Agostini, C. Proetto

The Born-Oppenheimer (BO) approximation is among the most basic approximations in the quantum theory of molecules and solids. It is based on the fact that electrons usually move much faster than the nuclei. This allows us to visualize a molecule or solid as a set of nuclei moving on the potential energy surface generated by the electrons in a specific electronic state. The total wavefunction is then a product of this electronic state, Φ_R^BO(r), and a nuclear wavefunction χ^BO(R) satisfying the Schrödinger equation

Here, R =(R₁ ...R_Nn), denotes the nuclear configuration and r =(r₁ ...r_Ne) represents the set of electronic positions. The concept of the potential energy surface, given in the BO approximation by

is enormously important in the interpretation of all experiments involving nuclear motion. Likewise, the vector potential

and the Berry phase associated with it, provide an intuitive understanding of the behavior of a system near conical intersections. Here and in the following, 〈..|..|..〉_r denotes the inner product over all electronic coordinates. Berry-Pancharatnam phases are usually interpreted as arising from an approximate decoupling of a system from "the rest of the world", thereby making the system Hamiltonian dependent on some "environmental" parameters. The best example is the BO approximation, where the electronic Hamiltonian H^BO_R depends parametrically on the nuclear positions; i.e., the stationary electronic Schrödinger equation is solved for each fixed nuclear configuration R,yielding R-dependent eigenvalues ε ^BO(R) and eigenfunctions Φ^BO_R. Hence one has to acknowledge the fact that in the traditional treatment of molecules and solids the concepts of the potential energy surface and the Berry phase arise as consequence of the BO approximation. Yet, this is "just" an approximation, and some of the most fascinating phenomena of condensed-matter physics, like superconductivity, appear in the regime where the BO approximation is not valid. This raises the question: If one were to solve the Schrödinger equation of the full electron-nuclear Hamiltonian exactly (i.e. beyond the BO approximation) does the Berry phase and the potential energy surface survive, and if so, how and where does it show up? Moreover, many interesting phenomena occur when molecules or solids are exposed to time-dependent external fields, such as lasers. One may ask: Can one give a precise meaning to a time-dependent potential energy surface and a time-dependent Berry phase?

In a recent Letter ^[1] we are able to answer all of the above questions. We prove that the exact solution of the TDSE,

can be written as a single product

(1)

where &Phi_R(r,t) satisfies the partial normalization condition,

, for any fixed nuclear configuration, R, at any time t. An immediate consequence of the identity (Eq. 1) is that,

is the probability density of finding the nuclear configuration R at time t. The electronic wavefunction, Φ_R(r,t), in the gauge where satisfies the equation

(2)

with and

The nuclear wavefunction obeys the equation

(3)

Via these exact equations the time-dependent potential energy surface (TDPES) :

(4)

and the time-dependent Berry connection

(4)

are defined as rigorous concepts. These two quantities mediate the coupling between the nuclear and the electronic degrees of freedom in a formally exact way. The vector potential can be expressed as

(6)

This equation is interesting in several respects. First, writing χ(R,t) = e^iS(R,t)|χ(R,t)|, the last term on the RHS of Eq. (6) can be represented as ∇_νS(R,t), so it can be gauged away.

Consequently, any true Berry connection must come from the first term. If the exact Ψ(t) is real-valued (e.g. for a non-current-carrying ground state) then the first term on the RHS of Eq (6) vanishes and hence the exact Berry connection vanishes. Second, since Im_r is the true nuclear (many-body) current density, Eq. (6) implies that the gauge-invariant current density, Im(χ^*∇_νχ) + |χ|²A_ν, that follows from Eq. (3) does indeed reproduce the exact nuclear current density.

Hence, the solution χ(R,t) of Eq. (3) is, in every respect, the proper nuclear many-body wavefunction: Its absolute-value squared gives the exact nuclear (N-body) density while its phase yields the correct nuclear (N-body) current density.

To demonstrate the usefulness of the approach we have calculated the TDPES's for a numerically exactly solvable model: the H₂⁺ molecular ion in 1D, subject to a linearly polarized laser field. TDPES's along with the corresponding nuclear density, |χ(R,t)|², are plotted in Fig. 1 at six snapshots of time. For the stronger field the dissociation of the molecule is dramatically reflected in the exact TDPES. Also in the "exact Ehrenfest" approximation where the nuclei are treated as classical particles moving on the exact TDPES, the molecule dissociates (open circles in Fig. 1). However, for the weaker field only exact quantum calculation leads to dissociation while in the "exact Ehrenfest" calculation the system gets stuck in a local minimum of the TDPES (solid circles), suggesting that tunneling is the leading dissociation mechanism. This reveals that the TDPES is a powerful interpretive tool to analyze and interpret different types of dissociation processes such as direct vs. tunneling.

References

[1]

Ali Abedi, Neepa T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010).