# Geometry of Nonadiabatic Quantum Hydrodynamics

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## Abstract

The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether’s conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called ‘collective’. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite \(\hbar \) by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the \(\hbar \ne 0\) dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called ‘Bohmions’, which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.

## Keywords

Lie group action Momentum map Quantum hydrodynamics Nonadiabatic molecular dynamics## 1 Introduction

### 1.1 Factorized Wave Functions in Quantum Molecular Dynamics

*molecular Schrödinger equation*

*molecular wave function*. The notation is such that \(\{{\boldsymbol{r}}_{k}\} = \{ {\boldsymbol{r}}_{k} : k=1,\ldots ,N_{n}\}\) and \(\{\boldsymbol{x}_{l} \}=\{{\boldsymbol{x}}_{l} : l=1,\ldots ,N_{e}\}\) denote, respectively, \(N_{n}\) nuclear and \(N_{e}\) electronic coordinates. Each \({\boldsymbol{r}} _{k}\) corresponds to a nucleus of mass \(M_{k}\) whilst all electrons have the same mass \(m\). The notation \(\widehat{T}\) and \(\widehat{V}\) in (1) refers to the kinetic energy and potential energy operators, while the subscripts \(n\) and \(e\) denote nuclear and electronic energies, respectively. The subscript \(I\) refers to the interaction potential between nuclei and electrons. More explicitly, one has

*electronic Hamiltonian*. Since \(\widehat{H}_{e}\) also includes the potential energy operators \(\widehat{V}_{n}\) and \(\widehat{V}_{I}\) (both depending on \({\boldsymbol{r}}\)), one may write \(\widehat{H}_{e}=\widehat{H}_{e}( {\boldsymbol{r}})\). The Hamiltonian operator \(\widehat{H}_{e}({\boldsymbol{r}})\) acts on the electronic Hilbert space \(\mathscr{H}_{e}\), identified with the space of \(L^{2}\)-functions of the electronic coordinates, \(\boldsymbol{x}\), which depend parametrically on the nuclear coordinates, \({\boldsymbol{r}}\). Thus, its eigenvalue equation reads

*potential energy surfaces*(PES) [59, 81]. As customary in the chemical physics literature, for simplicity, here we assume a discrete electronic spectrum. Formally, one can solve the molecular Schrödinger equation by writing the so-called

*Born–Huang expansion*[16]

*nuclear wave functions*.

*adiabatic hypothesis*). Within the physical chemistry community, it is widely accepted that stable molecular configurations correspond to the minima of the lowest energy PES, \(E_{0}({\boldsymbol{r}})\), see, e.g., [59].

*mean-field model*is probably the simplest. The standard mean-field factorization ansatz is given by

*exact factorization*(EF), which reads as follows

*collectivizes*in the sense of Guillemin and Sternberg, [32, 33, 51, 56, 57]. This means that equivariant momentum maps transform canonical Hamiltonian dynamics into motion on coadjoint orbits generated by the action of a Lie group on the dual of its Lie algebra. Eventually, Lie–Poisson reduction leads to a new hydrodynamic formulation of nonadiabatic dynamics in which

*hyperbolicity is retained*, rather than setting \(\hbar \to 0\) and passing to the Hamilton–Jacobi equation.

- 1.
The remainder of Sect. 1 introduces background material which links standard elements of quantum mechanics with familiar objects in the setting of geometric mechanics. The fundamental variational principles and symplectic Hamiltonian structure in nonrelativistic quantum mechanics appear in Sect. 1.2.

- 2.
The transformation to quantum hydrodynamics is discussed in Sect. 2.1. In Sect. 2.2, Bohmian trajectories [14] are reinterpreted as Lagrangian paths associated with the quantum hydrodynamic flow. Section 2.4 regularizes the \(\hbar \to 0\) limit of standard quantum hydrodynamics by suitably applying a spatial smoothing operator to the fluid variables of both the collectivized Hamiltonian and the corresponding Lagrangian before taking \(\hbar \to 0\). The resulting smoothed quantum fluid equations are found to admit singular solutions supported on delta functions. We call these singular solutions ‘Bohmions’, because the delta functions on which they are supported move along Lagrangian paths of the regularized quantum fluid Hamiltonians. Section 2.5 shows how the cold-fluid closure of Wigner distributions corresponds to a classical closure of mixed state dynamics arising from the Liouville–von Neumann equation.

- 3.
In the mean-field approximation of coupled nuclear and electronic systems, the wave function is separated into a product of two independent factors, as in equation (4), above. Thus, the mean-field factorization of the wave function neglects the classical-quantum correlations between nuclei and electrons. Section 3 reviews the mean-field model and derives its quantum fluid representation in the geometric mechanics setting.

- 4.
The exact factorization (EF) model [1, 2, 3, 6] captures some of the nuclear and electronic correlation effects which are neglected in the mean field approximation, by letting the electron wave function depend on the nuclear spatial parameters, as in equation (5), above. Section 4 discusses the EF model in both the wave function and density matrix representations, then derives its quantum fluid representation in the geometric mechanics setting.

- 5.
In Sect. 5 a new model is introduced by invoking a factorization ansatz at the level of the molecular density operator. Then, combining the classical closure of nuclear mixed states with the smoothing process presented in Sect. 2.4 leads to an entirely finite-dimensional Hamiltonian system for the interaction of nuclear Bohmion solutions with an ensemble of quantum electronic states. Two different finite-dimensional schemes are presented depending on whether the smoothing process is applied in the Hamiltonian or in the variational formalism.

### 1.2 Quantum Lagrangians, Hamiltonians, and Momentum Maps

This section introduces the standard setting of the Hamiltonian approach to quantum dynamics by focusing on the evolution of *pure quantum states*. Later sections of this work will introduce von Neumann’s density operators and their evolution for mixed states. However, the present section considers only Schrödinger-type equations.

*phase space Lagrangian*, which reads

The Schrödinger equation \(i\hbar \dot{\psi }=\widehat{H}\psi \) follows as the Euler–Lagrange equation for the DF variational principle in (6), in which \(\widehat{H}\) is the self-adjoint Hamiltonian operator constructed from the canonical operators \(\widehat{Q}\) and \(\widehat{P}\) (the so called *canonical observables*). Thus, \(\widehat{H}=\widehat{H}( \widehat{Q},\widehat{P})\) and Open image in new window . Notice that, since \(\widehat{H}\) is self-adjoint, the DF Lagrangian in (6) is \(U(1)\)-invariant so that the condition \(\|\psi \|_{L^{2}}^{2}=1\) is naturally preserved. This amounts to conservation of the total probability. As presented in [11], the Euler–Poincaré formulation [38] of pure state dynamics is derived from the DF variational principle upon letting \(\psi (t)=U(t)\psi _{0}\) with \(U(t)\in \mathcal{U}(\mathscr{H})\). Here, \(\mathcal{U}(\mathscr{H})\) denotes the group of unitary operators on ℋ. In earlier years, this strategy was also exploited in [49, 69] upon restricting \(U(t)\) to be the unitary representation of a finite-dimensional Lie group. For example, if \(U(t)\) is a representation of the Heisenberg group, substituting the ansatz \(\psi (t)=U(t)\psi _{0}\) into the DF variational principle yields canonical Hamiltonian motion on phase space [68].

*Dirac Hamiltonian*, to distinguish it from the Hamiltonian operator, \(\widehat{H}\). Depending on the context, the operator \(\widehat{H}\) and the functional \(h(\psi )\) may both be called the ‘Hamiltonian’. More general systems (such as the nonlinear Schrödinger equation) can be obtained by replacing \(\langle \psi ,\widehat{H}\psi \rangle \) by a suitable functional \(h(\psi )\). In this case, the normalization condition \(\|\psi \|_{L^{2}}^{2}=1\) must be incorporated as a constraint, [64], as

*projective Schrödinger equation*[48] along with the condition

*collectivizes*, in the sense of Guillemin and Sternberg [32, 33, 51, 56, 57], through the momentum maps leading from Schrödinger’s equation to quantum hydrodynamics. A (left) Hamiltonian action of a Lie group \(G\) on a symplectic manifold \((M,\omega )\) induces the

*momentum map*\(J:M\to \mathfrak{g}^{*}\), where \(\mathfrak{g}^{*}\) is the dual space to the Lie algebra \(\mathfrak{g}\) of \(G\). In the special case when \(M\) is a symplectic vector space (so that \(M=V\)), then the momentum map is defined by

*collective Hamiltonian*associated to the group action \(G\). Symplectic momentum maps are Poisson. That is, for smooth functions \(f\) and \(h\), we have \(\{F, H \} = \{f\circ J, h\circ J \} = \{f , h \} \circ J \). This relation defines the

*Lie–Poisson bracket*on \(\mathfrak{g}^{*}\), given in terms of the adjoint action of the Lie algebra on itself, \(\mathrm{ad}: \mathfrak{g}\times \mathfrak{g} \to \mathfrak{g}\), denoted as \(\mathrm{ad}_{\xi }\zeta = [\xi ,\zeta ]\) for any Lie algebra elements \(\xi ,\zeta \in \mathfrak{g}\). Upon denoting the pairing by \(\langle \,\cdot ,\,\cdot \, \rangle _{\mathfrak{g}}: \mathfrak{g}^{*}\times \mathfrak{g}\to \mathbb{R}\), the Lie–Poisson bracket on \(\mathfrak{g}^{*}\) reads as [58]

## 2 Quantum Hydrodynamics

This section illustrates the geometry of the hydrodynamic setting of quantum mechanics, which has its foundations in the Madelung transform [54, 55]. After reviewing the geometry of half-densities and their momentum maps, the latter are exploited to derive the quantum hydrodynamic (QHD) equations.

### 2.1 Half-densities and Momentum Maps

The equivariant momentum map \(\boldsymbol{J}:\operatorname{Den}^{1/2}( \mathbb{R}^{3})\to \mathfrak{X}^{*}(\mathbb{R}^{3})\) for the left action (16) is found as in [26, 46] from the standard definition (12), that is \(\langle {\boldsymbol{J}}( \psi ),{\boldsymbol{u}}\rangle =\,\hbar \langle \mathit{iu}_{\, \operatorname{Den}^{1/2}}(\psi ),\psi \rangle \). Here we have identified the Hilbert space as \(\mathscr{H}=L^{2}(\mathbb{R}^{3})= \operatorname{Den}^{1/2}(\mathbb{R}^{3})\).

*collectivizes*, in the sense of Guillemin and Sternberg [32, 33], through the momentum maps \({\boldsymbol{J}}(\psi )\) and \(|\psi |^{2}\). That is, the Hamiltonian in (18) may be expressed solely in terms of the collective variables \(\boldsymbol{\mu }\) and \(D\), given by \(\langle \psi |\widehat{H}\psi \rangle =h(\boldsymbol{\mu },D)\) with \(\boldsymbol{\mu }={\boldsymbol{J}}(\psi )\) and \(({\boldsymbol{\mu }},D) \in (\mathfrak{X}(\mathbb{R}^{3})\,\circledS \, C^{\infty }(\mathbb{R}^{3}))^{*}\). Here, \(\mathfrak{X}( \mathbb{R}^{3})\,\circledS \, C^{\infty }(\mathbb{R}^{3})\) denotes the semidirect-product Lie algebra of the semidirect-product Lie group \(\operatorname{Diff}(\mathbb{R}^{3})\,\circledS \,C^{\infty }(\mathbb{R} ^{3},S^{1})\), whose elements \((\eta ,\varphi )\) act from the left on the space \(\operatorname{Den}^{1/2}(\mathbb{R}^{3})\) of half-densities as

*push-forward*by the map \(\eta \in \operatorname{Diff}(\mathbb{R}^{3})\); so, \(\eta _{*} D_{0}\) denotes the push-forward of the reference density, \(D_{0}\) by the map \(\eta \). Push-forward by the smooth flow \(\eta \) is called

*advection*in hydrodynamics. In this context, the Lagrangian particle path of a fluid parcel is given by the smooth, invertible, time-dependent map, \(\eta _{t}:\mathbb{R}^{3}\to \mathbb{R}^{3}\), as follows, \(\eta _{t} \boldsymbol{x}_{0}=\boldsymbol{\eta }(\boldsymbol{x}_{0},t) \in \mathbb{R}^{3} \) for initial reference position \(\eta _{0} \boldsymbol{x}_{0}=\boldsymbol{\eta }(\boldsymbol{x}_{0},0)= \boldsymbol{x}_{0}\). After this definition, there should be no confusion between \(\eta _{t}\in {\mathrm{Diff}}(\mathbb{R}^{3})\) and \(\eta _{t} \boldsymbol{x}_{0} = \boldsymbol{\eta }(\boldsymbol{x}_{0},t) \in \mathbb{R}^{3}\). The subscript \(t\) is omitted in most of this paper, for simplicity of notation.

*quantum potential*

### Remark 1

(Effects of the quantum potential)

The appearance of the amplitude of the wave function in the denominator of the quantum potential in (23) implies that its effects do not necessarily fall off with distance. That is, the effects of the quantum potential need not decrease, as the amplitude of the wave function decreases. Moreover, the middle term in (20) is known as the Fisher–Rao norm, which is well-known in information theory. For further discussion of the information geometry in quantum mechanics, see, e.g., [17].

### 2.2 Bohmian Trajectories, Lagrangian Paths & Newton’s Law

*hidden variable*in the Bohmian interpretation of quantum dynamics [14]. Indeed, in this framework the path \({\eta }\) is the fundamental dynamical variable, while the wave function is simply transported in time along the Lagrangian motion of \(\boldsymbol{\eta }({\boldsymbol{x}} _{0},t)\), which in turn satisfies

*Bohmian trajectory*, which is precisely the Lagrangian fluid path of the hydrodynamic picture!

Nonetheless, the Newtonian limit neglects the order \(O(\hbar ^{2})\) quantum dispersion term in the Lagrangian (20) (or, equivalently, (26)) and varies the remainder. The resulting equation for the Bohmian trajectory becomes \(D_{0} (m \ddot{\boldsymbol{\eta }}+\nabla _{\boldsymbol{\eta }}V( \boldsymbol{\eta }) )=0 \). It is clear that the point-particle initial condition \(D_{0}({\boldsymbol{x}}_{0})=\delta ({\boldsymbol{x}} _{0}-{\boldsymbol{q}}_{0})\) is now allowed and thus denoting \({\boldsymbol{q}}(t)=\boldsymbol{\eta }({\boldsymbol{q}}_{0},t)\) and integrating over space yields Newton’s Law \(m\ddot{{\boldsymbol{q}}}+ \nabla V({\boldsymbol{q}})=0\).

A common alternative method to derive Newton’s Law exploits the analogy with the Hamilton–Jacobi equation of classical mechanics. Since \({\boldsymbol{u}}=m^{-1}\boldsymbol{\mu }/D=\hbar \nabla {\theta }\) according to the momentum map relation for the collective variable \({\boldsymbol{J}}(\psi )\) in (17), the first of these restricted QHD equations happens to recover the Hamilton–Jacobi equation for \(S=\hbar {\theta }\), as for geometrical optics with the classical Hamiltonian \(H({\boldsymbol{q}},{\boldsymbol{p}})=|{\boldsymbol{p}}|^{2}/2M+V( {\boldsymbol{q}})\). This is not necessarily convenient for a fluids interpretation, though, because solutions of Hamilton–Jacobi equations may become singular (e.g., form caustics) even for smooth initial data.

### Remark 2

(Regularization of an “ultraviolet catastrophe” for \(\hbar ^{2} \to 0\))

The Newtonian limit of QHD (22), obtained by simply neglecting the contribution to the Euler–Poincaré equations from the quantum potential in (23) turns out to be problematic. In particular, because the potential (which plays the role of a pressure term) is assumed to be independent of time, the Newtonian limit system (27) is not strictly hyperbolic. This observation is a well known signal, [50], that the solution behaviour in the classical limit \(\hbar ^{2}\to 0\) can become singular, as \(\hbar ^{2}\) multiplies the highest spatial derivative. This is especially clear when the wave function is written in the usual WKB form, as \(\psi =\sqrt{D}\exp (\mathit{iS}/\hbar )\), where \(S\) is the action integral for the Schrödinger equation. Indeed, in the \(\hbar ^{2} \to 0\) limit, the gradient of the quantum potential produces highly oscillatory spatial behaviour. See, e.g., [27, 44] and references therein for discussions of the weak convergence of the rapidly oscillatory solutions obtained in passing the WKB description of the Schrödinger equation to the classical limit as \(\hbar ^{2} \to 0\). As indicated in [44], one convenient way to carry out the limit \(\hbar ^{2}\to 0\) is to apply the Wigner transform of the wave function [78, 82]. The real-valued Wigner function then acts as the quantum-equivalent of the phase-space distribution function in classical mechanics; although the quantum version introduces mathematically technical features, such as Moyal operator brackets, instead of Poisson brackets. One could also treat the rapid oscillations as being stochastic and use probability theory to obtain the expected solution as a classical limit, [70]. Thus, in retrospect, one can appreciate the role of non-zero \(\hbar ^{2}\) in the quantum potential (23) in equations (22) as a dispersive regularization of what would otherwise have led to a type of *ultraviolet catastrophe* [21] for the restricted (Hamilton–Jacobi) QHD in (27), as the solutions of the restricted QHD equations form caustics.

### 2.3 Lie–Poisson Structure of Quantum Hydrodynamics

*in components*by

### 2.4 Regularized QHD and Bohmion Solutions

In remark 2, we have emphasized that the order \(O(\hbar ^{2})\) term in the QHD Hamiltonian (28) may be regarded as a dispersive regularization of the “ultraviolet catastrophe” which occurs in the quantum fluid Hamiltonians as \(\hbar ^{2}\to 0\). The order \(O(\hbar ^{2})\) term is an energy penalty for high gradients \(|\nabla D|^{2}/D=4|\nabla \sqrt{D}|^{2}\) that yields only a weak classical limit as \(\hbar ^{2}\to 0\) [44].

In this section we proceed by regularizing the Lagrangian or Hamiltonian to allow for single-particle solutions. As we have observed in Sect. 2.2, the \(O(\hbar ^{2})\)-terms in QHD prevent the existence of particle-like solutions so that Bohmian trajectories can only be identified with Lagrangian paths following the characteristic curves of the Eulerian fluid velocity. Thus, the \(O(\hbar ^{2})\)-terms in QHD must be treated with particular attention. Instead of adopting semiclassical methods to take the limit \(\hbar ^{2}\to 0\), this section presents an alternative strategy consisting in regularizing the \(O(\hbar ^{2})\)-terms by a smoothening process. More particularly, we shall discuss Bohmian trajectories which can be computed from regularized QHD Hamiltonians and Lagrangians, whose fluid variables have been spatially smoothed; so that their \(\hbar ^{2}\to 0\) limit is no longer singular.

Depending on which terms are regularized, different particle motions may emerge. We present two regularization strategies. The first simply smoothens all the terms in the Hamiltonian, while the second only smoothens the \(O(\hbar ^{2})\)-terms. Although all the equations of motion derived here are Hamiltonian equations on a canonical phase space, they may or may not be in the usual Newtonian form, depending on which regularization scheme is adopted. In particular, the first regularization scheme is adopted in the Hamiltonian framework to regularize the hydrodynamic momentum and density, while the second scheme is based on the variational approach following the standard Bohmian method.

### Hamiltonian Regularization

*regularized quantum hydrodynamics (RQHD)*. This is simply obtained by replacing the variables \({\boldsymbol{\mu }}\) and \(D\) in (28) by the corresponding spatially smoothed variables,

*singular and finite dimensional*along the Lagrangian paths for the diffeomorphism, \(\eta \) in Sect. 2.2.

### Lagrangian Regularization

### 2.5 Density Operators and Classical Closures

*matrix element*, in the physics literature) of the integral operator \(\rho \). In more generality, for an arbitrary sequence \(\{\psi _{n}({\boldsymbol{x}})\}\) of \(N\) square-integrable functions, the

*density operator*is given by

Finally, we emphasize again that the Wigner function in (52) does not identify a quantum state. This is analogous to what happens for the quantum harmonic oscillator: in this case, the Wigner–Moyal equation coincides with the classical Liouville equation thereby allowing for delta-function solutions returning classical motion. However, delta-function Wigner distributions do not correspond to quantum states, as their associated density operator is sign-indefinite.

## 3 Mean-Field Model

This section presents the mean-field model, which is based on the factorization (4). Although this model fails to retain correlation effects between nuclei and electrons, it is of paramount importance as the basis of most common models in nonadiabatic dynamics. As we shall see, the geometry of quantum hydrodynamics can be directly applied to this model, thereby leading to the most basic example of hybrid classical-quantum dynamics.

### 3.1 The Mean-Field Ansatz

*mean-field ansatz*introduces a factorization of the type

At this point, a further approximation is often introduced; namely, one assumes that the nuclear dynamics can be treated as classical. This assumption produces a mixed classical-quantum system. In what follows, we will introduce a geometric approach which restricts the nuclear evolution to classical particle trajectories.

### 3.2 Quantum Hydrodynamics and Nuclear Motion

Equations (66) represent the standard mean-field model as it is usually implemented in molecular dynamics simulations [59] (although here we have focused on the simplest case of one nucleus and one electron). As we can see in the previous equation, the classical-quantum coupling in this model occurs solely through the interaction potential \(V_{I}\).

Unfortunately, this quantum fluid picture of the mean-field model is not satisfactory in many cases, because the mean-field factorization (53) disregards the classical-quantum correlations between nuclei and electrons. A more advanced model capturing part of these correlation effects will be presented in the next section.

## 4 Exact Factorization

*exact factorization*in recent work [1, 2, 3, 6]:

*partial normalization condition*(PNC)

### 4.1 Wave Functions vs. Density Operators

*decoherence*, i.e. quantum mixing). However, we shall continue to refer to them as such, because they retain certain mnemonic relationships. We remark that expectation values of nuclear observables involve integration over the \({\boldsymbol{r}}\)-parameters of the electron “wave functions”. More specifically, the expectation value \(\langle A_{n}\rangle \) for a nuclear observable \(A_{n}({\boldsymbol{r}}, {\boldsymbol{r}}')\) is given by (again, in matrix element notation) \(\langle A_{n}\rangle := \iint \!{\mathrm{d}^{3}}r\,{\mathrm{d}^{3}}r' \!\int {\mathrm{d}^{3}}x\,\chi ^{*}({\boldsymbol{r}}')\psi ^{*}( \boldsymbol{x};{\boldsymbol{r}}')A_{n}({\boldsymbol{r}}',{\boldsymbol{r}}) \psi (\boldsymbol{x};{\boldsymbol{r}})\chi ({\boldsymbol{r}}) \). As we shall see, this structure of the nuclear density operator leads to important consequences in the development of the exact factorization theory.

At this stage, we shall only emphasize that all the relations above also apply naturally in the context of the Born–Oppenheimer approximation [43], thereby indicating again that the interpretation of nuclear and electronic motion in terms of genuine wave functions needs to be revisited. For example, backreaction effects generated by the presence of \(\psi \) in (71) can lead to *nuclear decoherence effects* since indeed one has \(\rho _{n}^{2}\neq \rho _{n}\). This is a general feature of classical-quantum coupling [10], which in fact erodes purity in both the classical and the quantum subsystems.

### 4.2 General Equations of Motion

### 4.3 Local Phases and Gauge Freedom

To specify the evolution completely, one may either transform to gauge invariant functions, such as the electric and magnetic fields in electromagnetism, or one may fix the gauge by imposing one condition per degree of gauge freedom. Gauge fixing is not always the best option, though, because it may obscure physical effects arising due to local breaking of gauge symmetry. An example is the Berry phase, which arises from locally breaking the gauge symmetry of phase shifts in nonrelativistic quantum mechanics [9]. See also [79] for broadly ranging discussions of geometric phases in physics.

The gauge freedom under the compensating local phase shifts in (77) implies that \(\langle \psi ',i\partial _{t} \psi '\rangle =\partial _{t}\theta +\langle \psi ,i\partial _{t}\psi \rangle \). Hence, one may choose \(\theta \) at will (gauge fixing) so as to accommodate any value of \(\langle \psi ',i\partial _{t}\psi '\rangle \). For example, one may fix \(2|\chi |^{2}\langle \psi |i\hbar \partial _{t}\psi \rangle =\operatorname{Re} \langle \psi |({\delta h}/ {\delta \psi }) \rangle \), so that the \(\psi \) equation in (76) reads \(2i\hbar |\chi |^{2}\partial _{t} \psi ={\delta h}/{\delta \psi }- i\operatorname{Im} \langle \psi |( {\delta h}/{\delta \psi }) \rangle \psi \). The same type of gauge was chosen in passing from equation (9) to equation (11), earlier.

Another convenient choice consists in fixing \(\langle \psi |i\hbar \partial _{t}\psi \rangle =0\), so that the \(\psi \) equation in (76) becomes \(2i\hbar |\chi |^{2}\partial _{t} \psi ={\delta h}/{\delta \psi }- \langle \psi |({\delta h}/{\delta \psi }) \rangle \psi \). This gauge is called the *temporal gauge* (or *Weyl gauge*) in electromagnetism and it has been adopted recently in [1, 4, 71]. We remark that gauge theory is also important in other aspects of chemical physics; for example, see [53] for applications of gauge theory in molecular mechanics.

### 4.4 The Hamiltonian Functional

*Berry connection*[9] as where the notation \({\boldsymbol{A}}({\boldsymbol{r}},t)\) suggests that this quantity plays a role as a gauge field, analogous to the magnetic vector potential in electromagnetism. Here, we recall that \(\langle \,\cdot \,|\,\cdot \,\rangle \) denotes the natural \(L^{2}\) inner product on \(\mathscr{H}_{e}\) and \(\|\cdot \|\) denotes the corresponding norm (whose values again depend on the nuclear coordinate \({\boldsymbol{r}}\)).

*effective electronic potential*, \(\epsilon (\psi ,\nabla \psi )\), given by The last term in (79) is the trace of the real part of the complex

*quantum geometric tensor*[67] where we denote Open image in new window . The imaginary part of \(Q_{ij}\) is proportional to the Berry curvature \(B_{ij}:= \partial _{i} A_{j} - \partial _{j} A_{i\,}\); namely, \(2\hbar {\mathrm{Im}}(Q_{ij}) = B_{ij}\) [47]. The emergence of the trace of its real part,

*gauge invariant part of the time-dependent potential energy surface*[4, 5, 71].

### 4.5 Hydrodynamic Approach

*electronic Hamiltonian functional*. Upon making use of the effective electronic potential \(\epsilon (\psi , \nabla \psi )\) defined in (79), one obtains the functional derivative

*Berry curvature*, effectively a magnetic field generated by the electrons, and \(\boldsymbol{ E}:=-\, \partial _{t} \boldsymbol{A} - \nabla \langle {\psi |i\hbar \partial _{t}\psi }\rangle \) plays the role of an electric field generated by the electrons. In this analogy, \(\hbar \) in the Berry connection defined in equation (78) plays the role of the coupling constant (charge) in the electromagnetic force on a charged particle.

*temporal gauge*, Open image in new window , and the

*hydrodynamic gauge*Open image in new window . A more explicit expression of \(\boldsymbol{ E}\) can be found by using (88), thereby leading to the equation

*not*conserve the spatial integral of the nuclear momentum density,

### 4.6 Newtonian Limit and Lorentz Force

An important point here is that the customary operation in chemical physics of neglecting the quantum potential term in the Lagrangian (87) can be problematic. Normally, this step would invoke the limit \(\hbar ^{2}\to 0\). However, here this process would also lead to discarding the terms \({M^{-1}}({\hbar ^{2}}\|\nabla \psi \|^{2} - {\boldsymbol{A}^{2}})/{2}\) in the effective electronic potential (79), thereby taking the exact-factorization model into the standard mean-field theory. This crucial issue will be resolved in Sect. 5 by performing the exact factorization at the level of the molecular density operator.

We should also comment on the Lorentz force appearing in (102). We notice that the combination of this electromagnetic-type force with the potential energy contribution \(\nabla \epsilon \) suggests that the conventional picture of nuclei evolving on potential energy surfaces fixed in space may be oversimplified. As we shall see, despite the claims made in [72], the force \(\boldsymbol{E}+{\boldsymbol{u}}\times \boldsymbol{B}\) cannot vanish without requiring major modifications of the electron energy function \(\epsilon (\psi ,\nabla \psi )\). Such modifications would result in singular solution behaviour for the Berry curvature that is unexpected for the exact factorization model.

### 4.7 Circulation Dynamics for the Berry Connection

*nuclear*fluid around closed loops which move with the velocity \({\boldsymbol{u}}= (\boldsymbol{\mu }/D+{\boldsymbol{A}})/M\). To see this, we write the motion equation (97) as the Lie derivative of a circulation 1-form,

*Berry phase*) satisfies

### 4.8 Electron Dynamics in the Nuclear Frame

So far, we have presented the geometric aspects of the exact factorization model which are currently available in the literature. This section takes a step further to consider the evolution of the electron density matrix. Upon following the arguments in [5], we shall write the electron dynamics in the Lagrangian frame moving with the nuclear hydrodynamic flow.

### Remark 3

(Analogies with complex fluids)

We take this opportunity to make the connection between the hydrodynamic exact factorization system and previous investigations of the geometry of liquid crystal flows, as found in [28, 29, 30, 31, 36, 74]. In this comparison, the electronic wave function \(\psi ( \boldsymbol{r},t)\in C^{\infty }(\mathbb{R}^{3},\mathscr{H}_{e})\) is replaced by the director, an orientation parameter field \({\boldsymbol{n}}( \boldsymbol{r},t) \in C^{\infty }(\mathbb{R}^{3},S^{2})\); the unitary evolution operator \(U(\boldsymbol{r},t)\in C^{\infty }(\mathbb{R}^{3}, \mathcal{U}(\mathscr{H}_{e}))\) becomes a rotation matrix \(R( \boldsymbol{r},t)\in C^{\infty }(\mathbb{R}^{3},\mathit{SO}(3))\); and one still considers the coupling to the fluid velocity \({\boldsymbol{u}}( \boldsymbol{r},t)\) given by the action of diffeomorphisms \(\eta \in \text{Diff}(\mathbb{R}^{3})\). Indeed, with these replacements, one has a reduced Lagrangian of the same type, \(\ell ({\boldsymbol{u}}, \boldsymbol{\xi },D,{\boldsymbol{n}})\), and the resulting Euler–Poincaré equations are equivalent to those in (116).

*algebraic relation*\([i\hbar D \xi - {\delta F}/{\delta \rho },\rho ]=0\). The equation in (116) for density operator \(\rho =\psi \psi ^{\dagger }\) now implies the following Liouville–von Neumann equation,

### Remark 4

(Electron decoherence)

Equation (118) will determine the evolution of the electron density matrix defined in (70), \(\rho _{e}(t) := \int D\rho \,\mathrm{d}^{3}r = \int \tilde{\rho }\, \mathrm{d}^{3}r \). Namely, \(i\hbar \dot{\rho }_{e}(t) = \int [D \widehat{H}_{e}-{\hbar ^{2}}M^{-1}{\mathrm{div}}(D\nabla \rho )/2, \rho ] \,\mathrm{d}^{3}r \). This result implies that spatially uniform pure initial states (such that \(\rho _{e}^{2}=\rho _{e}\)) become mixed states as time proceeds. Thus, in agreement with, e.g., [60], the exact factorization model captures electronic decoherence effects (that is, quantum state mixing) from pure initial states; since the density matrix evolution is no longer unitary.

### 4.9 Hamiltonian Structure

## 5 Density Operator Factorization and Singular Solutions

Our discussion in Sect. 4.7 shows that the only contribution to the circulation of the hydrodynamic flow arises from the Berry connection associated to the electronic function \(\psi \). This is due to the fact that the hydrodynamic velocity \(M^{-1} \boldsymbol{\mu }/D=\hbar \operatorname{Im}(\chi ^{*}\nabla \chi )/| \chi |^{2}\) is an exact differential and therefore has zero vorticity. However, we showed in Sect. 2.5 that this restriction can be relaxed by considering density operators. In the same section we also showed that the classical closure of mixed state dynamics allows for multi-particle trajectories arising from the initial condition (44), which in turn is not compatible with the standard QHD definition \(D_{0}=|\chi _{0}|^{2}\). In this section we shall include mixed state dynamics by extending the exact factorization model to a density operator formulation.

### 5.1 Factorization of the Molecular Density Operator

Now that the variational principle and the Hamiltonian functional are completely characterized, we may proceed by restricting nuclear dynamics to undergo classical motion. To this purpose, we combine the two approaches described in Sects. 2.4 and 2.5, by applying the regularization technique after performing the classical closure for density operators.

### 5.2 Classical Closure and Singular Solutions

Remarkably, Hamiltonian systems with Lie–Poisson brackets of the type (126) for the exact factorization (EF) model have already been studied and their geodesic motions have been shown to admit singular solutions for certain choices of quadratic Hamiltonians, in [41]. Following this previous work, in this Section we will extend the momentum map (39) which led to the Bohmion singular solutions in Sect. 2.4, to a singular solution momentum map for a regularized version of the Lie–Poisson system associated to the Hamiltonian (137). Again, in analogy with Sect. 2.4, here we shall present two different treatments: while the first is uniquely based on the Hamiltonian structure, the second exploits the associated variational principle.

#### 5.2.1 Hamiltonian Regularization in 1D

*not*sum on the index \(a\).

Again, the term in the canonical equations (141) arising from summands in \(h_{ \mathit{REG}}\) in (139) proportional to \(\hbar ^{2}\) provides extensive, potentially long-range coupling among the singular particle-like solutions, because of the presence of \(\bar{D}\) in the denominators of these terms in the Hamiltonian \(h_{\mathit{REG}}\) in (36). However, as in the previous section, the limit \(\hbar ^{2}\to 0\) in the canonical Bohmion equations (141) in the density matrix formulation is regular.

#### 5.2.2 Lagrangian Regularization in 1D

## 6 Conclusions

In this paper, we have exploited momentum maps to collectivize a sequence of molecular quantum chemistry models for factorized nuclear and electronic wave functions, thereby obtaining a sequence of quantum fluid models with shared semidirect-product Lie–Poisson structures. After reviewing the Born–Oppenheimer product of nuclear and electronic wave functions, we started with mean-field theory, and then passed to a recent development called ‘exact factorization’ (EF) for nonadiabatic correlated electron-nuclear dynamics, which has been reported to describe decoherence of pure electron quantum states into mixed states. In the last part, we extended the exact factorization approach to apply for density operators.

In Sect. 2.4, we mollified the weakly convergent WKB \(\hbar \to 0\) limit by applying a smoothing operator to the quantum variables in the collectivized Hamiltonian for regularized quantum hydrodynamics. This smoothing operation preserved the Hamiltonian structure of the quantum fluid model and it resulted in the discovery of singular delta function solutions called ‘Bohmions’ for smooth quantum fluid Hamiltonians. Depending on which terms are regularized in the Hamiltonian, different sets of Bohmion equations are available.

In the development of the paper, we showed that the Hamiltonian formulation of the collectivized quantum fluid equations for EF possesses the same Lie–Poisson bracket structure as in earlier work on perfect complex fluids (PCF), such as liquid crystals, [28, 29, 30, 31, 36, 74]. The parallel between EF and PCF is that the nuclear fluid velocity vector field Lie-transports both the nuclear probability density and the electron density matrix, while the latter also has its own unitary dynamics in the moving frame of the nuclear fluid. This picture was also extended to present a new PCF dynamical model based on the factorization of the molecular density operator.

In the PCF formulation of the nonadiabatic electron problem, smoothing all terms in (137) yields singular momentum maps corresponding to the ‘peakon’ solutions of the well known EPDiff equation [37]. In one spatial dimension, this more general class of Bohmions is governed by a countably infinite set of canonical Hamiltonian equations in phase space, in analogy to the solitons for the Camassa–Holm equation [19]. The countably infinite phase space system can be truncated to a multi-particle phase space system at any finite number of Bohmions, because the Hamiltonian dynamics does not create new Bohmions. In fact, the Bohmion collectivized solutions discussed in Sects. 2.4 and 5.2 comprise a semidirect-product version of the class of ‘peakon’ solutions for the CH equation [19] which arise from the well-known singular momentum map for the entire class of EPDiff equations [37].

A second approach to Bohmion dynamics was presented in Sect. 5.2.2. This approach was developed in the variational framework by smoothening only the \(O(\hbar ^{2})\)-terms in the Lagrangian (142). Although the analogy to the peakon solutions of the Camassa–Holm equation no longer holds entirely, the resulting dynamical system still consists of a countable set of finite dimensional Hamiltonian equations.

Future work will take further advantage of the analogy between continuum dynamics and the collectivization of quantum dynamics via momentum maps. For example, products of delta functions in different spaces can be introduced, corresponding to Bohmion dynamics for the different factorized wave functions of many interacting molecules. This approach is reminiscent of the closure models arising in time dependent Hartree (TDH) theory [34] for quantum dynamics in nuclear physics. Approaches such as these have long been applied in several fields of science, including molecular chemistry, nuclear physics, and condensed matter physics, as well as in celestial mechanics, in hopes of lifting the “curse of dimensions” which tends to be ubiquitous in many-body problems [13].

## Notes

### Acknowledgements

The authors are grateful to the referees for their constructive comments that helped improving the presentation of this material. Also, the authors appreciated stimulating discussions and correspondence with M. Berry, D. Bondar, A. Close, F. Gay-Balmaz, D. Glowacki, S. Jin, T. Ohsawa, L. Ó Náraigh, D. Ruiz, D. Shalashilin, and S. Wiggins. MF acknowledges the Engineering and Physical Sciences Research Council (EPSRC) studentship grant EP/M508160/1 as well as support from the 2018 Institute of Mathematics and its Applications Small Grant Scheme and University of Surrey FEPS Faculty Research Support Fund. The work by DH was partially supported by EPSRC Standard grant EP/N023781/1 and EPSRC Programme grant EP/P021123/1. The work of CT was supported by Leverhulme Research Project grant number 2014-112. This material is partially based upon work supported by the National Science Foundation Grant No. DMS-1440140 while MF and CT were in residence at the Mathematical Sciences Research Institute, during the Fall 2018 semester. We all acknowledge the London Mathematical Society Scheme 3 Grant 31633 (Applied Geometric Mechanics Network).

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