A Demonstration of Consistency between the Quantum Classical Liouville Equation and Berry's Phase and Curvature

TL;DR

This paper demonstrates that the quantum classical Liouville equation inherently includes Berry's phase effects and forces, validating its use for complex Hamiltonians with spin-orbit coupling without additional corrections.

Contribution

It shows that the QCLE naturally incorporates Berry's phase and forces, and that Ehrenfest dynamics also include Berry's phase effects, clarifying their roles in mixed quantum-classical models.

Findings

QCLE includes Berry's phase effects at order 

Ehrenfest dynamics automatically incorporate Berry's phase

Tully's surface hopping model ignores Berry's phase

Abstract

Although the quantum classical Liouville equation (QCLE) arises by cutting off the exact equation of motion for a coupled nuclear-electronic system at order 1 (1 = $\hbar^0$ ), we show that the QCLE does include Berry's phase effects and Berry's forces (which are proportional to a higher order, $\hbar$ = $\hbar^1$ ). Thus, the fundamental equation underlying mixed quantum-classical dynamics does not need a correction for Berry's phase effects and is valid for the case of complex Hamiltonians. Furthermore, we also show that, even though Tully's surface hopping model ignores Berry's phase, Berry's phase effects are included automatically within Ehrenfest dynamics. These findings should be of great importance if we seek to model coupled nuclear-electronic dynamics for systems with spin-orbit coupling, where the complex nature of the Hamiltonian is paramount.

Equations

$$\frac{\partial }{\partial t}{A}_{11}^W(R,P,t)$$

$$\frac{\partial }{\partial t}{A}_{12}^W(R,P,t)$$

$$\frac{-i}{\hslash }\left(V_{11}-V_{22}\right)A_{12}^W-\frac{P^{\alpha }}{M^{\alpha }}{d}_{12}^{\alpha }\left(A_{22}^W-A_{11}^W\right)-\frac{1}{2}{F}_{12}^{\alpha }\left(\frac{\partial A_{11}^W}{\partial P^{\alpha }}+\frac{\partial A_{22}^W}{\partial P^{\alpha }}\right)=0,$$

$$\zeta$$

$$\frac{\partial }{\partial t}{A}_{11}^W(R,P,t)$$

$$\frac{\partial }{\partial t}{A}_{11}^W(R,P,t)=-\frac{P^{\alpha }}{M^{\alpha }}\frac{\partial A_{11}^W}{\partial R^{\alpha }}-F_{11}^{\alpha }\frac{\partial A_{11}^W}{\partial P^{\alpha }}+2\hslash Im\left(d_{21}^{\beta }\frac{P^{\alpha }}{M^{\alpha }}{d}_{12}^{\alpha }\right)\frac{\partial A_{11}^W}{\partial P^{\beta }}$$

$${\vec{F}}_{11}^B=-2\hslash Im\left({\vec{d}}_{21}\frac{P^{\alpha }}{M^{\alpha }}{d}_{12}^{\alpha }\right)=2\hslash Im\left({\vec{d}}_{12}\frac{P^{\alpha }}{M^{\alpha }}{d}_{21}^{\alpha }\right)$$

$$H$$

$$p^y$$

$${\dot{c}}_1$$

$${\dot{c}}_2$$

$$\dot{{\rho }_{21}}=i{\omega }_{12}{\rho }_{21}+\left(\frac{\vec{P}}{M}\cdot {\vec{d}}_{21}\right)({\rho }_{22}-{\rho }_{11})\approx i{\omega }_{12}{\rho }_{21}+\left(\frac{\vec{P}}{M}\cdot {\vec{d}}_{21}\right)$$

$${\rho }_{21}(t)=e^{i{\omega }_{12}t}{\int }_0^t\left(\frac{\vec{P}}{M}\cdot {\vec{d}}_{21}\right)e^{-i{\omega }_{12}\tau }d\tau$$

$${\rho }_{21}(t)$$

$$=-\frac{1}{2}{e}^{i{\omega }_{12}t}\left(-\frac{1}{i{\omega }_{12}}{e^{-i{\omega }_{12}\tau }\frac{\partial \theta (\tau )}{\partial \tau }|}_0^t+\frac{1}{i{\omega }_{12}}{\int }_0^t\frac{{\partial }^2\theta (\tau )}{\partial {\tau }^2}{e}^{-i{\omega }_{12}\tau }d\tau \right)$$

$$=\frac{1}{2}\sum _{k=1}^{\infty }\frac{1}{(i{\omega }_{12}{)}^k}\frac{{\partial }^k\theta }{\partial t^k}$$

$$⟨F^y⟩(t)$$

$$p^y$$

$$\frac{\partial }{\partial t}{B}_{12}^W(R,P,t)$$

Full text

I Introduction

Nonadiabatic dynamics are a continuous source of interest and intrigue in the chemical physics community. On the one hand, the fast exchange of energy between nuclear and electronic degrees of freedom violates the Born Oppenheimer (BO) approximation, the bedrock of modern chemistrySzabo and Ostlund (1996). When one violates the BO approximation even moderately, one can find many unexpected effects, the most famous being Berry’s phase effectsBerry (1984). On the other hand, because quantum mechanics is so expensive to propagate, there is a strong impetus to understand nonadiabatic dynamics in a semiclassical fashionTully (1990); Doltsinis (2002); Subotnik et al. (2016); Nelson et al. (2014); Martinez et al. (1996), focusing on quantum electrons and classical nuclei. Thus, for many researchers, the nature of nonadiabatic effects becomes entangled with semiclassical approximations, which leads to only more questions about the fundamental nature of nonadiabatic dynamics.

In the present communication, we want to directly address one such fundamental question in nonadiabatic dynamics: the connection between Berry’s phaseBerry (1984) and the quantum classical Liouville equationKapral and Ciccotti (1999); Martens and Fang (1997). A few words are now appropriate regarding Berry’s phase, both in the context of real and complex Hamiltonians. In general, Berry’s phase effects are usually derived by considering the phase of an electronic wavefunction in the limit of a very slowly evolving potential that mixes together different adiabatic states, and the presence of Berry’s phase can lead to interference effects around degeneracies(e.g. the Aharanov-Bohm effectAharonov and Bohm (1959); Yarkony (1999) and tunneling suppressionXie et al. (2016)). When the Hamiltonian is real, Berry’s phase is effectively a generalization of the Longuet-Higgins phaseLonguet-Higgins et al. (1958); Longuet-Higgins (1961); Herzberg and Longuet-Higgins (1963), and there is an enormous literature in the chemical physics literature regarding the role of Berry’s phase effects around conical intersections Yarkony (1996); Ryabinkin et al. (2017). Of note, however, is that for a complex Hamiltonian, Berry’s phase can yield real effects even without a relevant intersection point; the Berry curvature (see equation 6) will be nonzeroMead and Truhlar (1979); Shankar (1994). Although this case is not usually addressed in the chemical physics literature (where we usually assume that the molecular Hamiltonian is real), the question of curve crossings with complex Hamiltonians has been investigated previously.Mead (1979); Matsika and Yarkony (2001a, b); Matsika and Yarkony (2002) and Takatsuka and Yonehara have written extensively about Berry’s “Lorentz-like” forces in the context of semiclassical, path branching dynamics Takatsuka and Yonehara (2011, 2010).

Let us now turn to the QCLEKapral and Ciccotti (1999); Martens and Fang (1997). The QCLE represents the simplest means to rigorously take the semiclassical limit of a coupled nuclear-electronic systems, treating nuclei classically and electrons quantum mechanically. The basic premise is to take a partial Wigner transform over a set of nuclear degrees of freedom, and then expand the total equation of motion in units of $\hbar.$. The QCLE includes only terms on the order of $\hbar^{-1}$ and $\hbar^{0}=1$; all terms on the order of $\hbar,\hbar^{2},\ldots$ etc. are ignored. Formally, the resulting dynamics have some failures – there is no Jacobi identity and correlation functions will not be invariant to time translationNielsen et al. (2001). Nevertheless, the dynamics are generally considered to be very accurate. In the context of the spin-boson model, the QCLE is exact. In this spirit, the QCLE is the underlying phase space equation against which one would like to compare all other semiclassical approachesSubotnik et al. (2013); Kapral (2016); Kelly and Kapral (2010); Kelly et al. (2012).

With this background in mind, recent work has identified a subtle question with regards to nonadiabatic dynamics, namely: Does the QCLE correctly incorporate Berry’s phase effects? On the one hand, one might assume that Berry’s phase and Berry’s curvature – both proportional to $\hbar$ – can arise only if goes beyond the QCLE to include all $\hbar^{1}$ terms in the expansion. Beside this $\hbar$ expansion argument, note also that Berry’s forces are usually derived by considering the Berry potential (or Berry connection) $A\equiv i\hbar\left<\Phi\middle|\nabla\Phi\right>$ of a nearly adiabatic state $\left|\Phi\right>$ and, through a gauge transformation acting on the nuclear space, converting the Berry potential to a magnetic forceShankar (1994) (just as one changes from the vector potential $A$ to the magnetic field $B$ in electrodynamicsCohen-Tannoudji et al. (1989)). Because gauge transformations of the classical degrees of freedom are not preserved in a quantum-classical treatment, one might assume that Berry’s forces cannot be derived by the QCLE.

On the other hand, recent work by Dou et al. derived the electronic friction tensor starting from the QCLEDou et al. (2017) and found the same friction tensor as calculated by a Berry’s phase calculation with a complex density matrix Lu et al. (2010); Lü et al. (2012)–suggesting that Berry’s phase should be derivable from the QCLE. Furthermore, Berry’s phase effects have already been isolated and studied within the QCLE for real, spin-boson HamiltoniansRyabinkin et al. (2014) (where the QCLE is exact). Thus, in this communication, we seek to tease out the answer to the question: are all of Berry’s phase effects captured by the QCLE, especially for the case of a complex Hamiltonian? Below, we will show clearly that, yes, Berry’s phase is derivable from the QCLE through a simple change of representation, as appropriate in the limit of nearly adiabatic dynamics. We will also show that, while such Berry’s phase effects are not captured by surface hopping dynamics, they are captured (at least partially) by Ehrenfest dynamics.

Our conclusions are important for three reasons. First, because the QCLE has traditionally been regarded as the benchmark for all semiclassical algorithms, the present findings are very reassuring: we may continue to use the QCLE as the gold standard – with real or complex Hamiltonians. There is no need to improve upon the QCLE in the presence of complex Hamiltonians, and in particular we may rest assured that the electronic friction tensor as developed in Ref. Dou et al., 2017 already includes all appropriate Berry’s phase effects. Second, our results should be extremely helpful for understanding and improving upon mixed quantum classical trajectory techniques. Meek and Levine (2016) Recent work has clearly shown that Tully’s fewest switches surface hopping (FSSH) algorithm does not include Berry’s forcesMiao et al. (2019) for the case of imaginary Hamiltonians (though some Berry’s phase effects can be captured with FSSH for real Hamiltonians with real conical intersections Gherib et al. (2015)). Even though FSSH is already a partial solution to the QCLESubotnik et al. (2013); Kapral (2016), the failure of surface hopping to recover complex Berry phase effects sheds light on the approximations made in Refs. Subotnik et al., 2013 and Kapral, 2016, and thus justifies modifying FSSH to better reproduce the QCLE and treat the case of complex HamiltoniansMiao et al. (2019). At the same time, we can also infer that all approximations to the QCLE based around Ehrenfest trajectoriesMeyer and Miller (1979); Dunkel et al. (2008); Huo and Coker (2012); Kim et al. (2008); Hsieh and Kapral (2012); Cotton et al. (2014) already include Berry phase effects and need no such modification. Third and finally, the present results highlight just how Berry phase arises for nuclear motion in the adiabatic limit, starting from a very general nonadiabatic approach but without needing to discuss closed loops in any parameter or function spaceAharonov and Anandan (1987); Zwanziger et al. (1990). Our findings confirm that, at least semiclassically, Berry’s phase effects can be understood in terms of well-understood equations of motion already present in the chemical physics literature and within all regimes – from the highly nonadiabatic to the highly adiabatic. With that in mind, we should also be able to learn exactly when Berry’s phase is appropriate–what terms must be small in order to take the semiclassical adiabatic limit?

For convenience below, we will use Einstein summation notation. Electronic states are indexed by 1,2 and nuclear degrees of freedom are indexed by Greek letters ($\alpha,\beta,\gamma$).

II Theory

Without loss of generality, consider the case of two electronic states 1 and 2. According to the quantum classical Liouville equation (QCLE)Kapral and Ciccotti (1999), to first order in the electron-nucleus mass ratio $(m/M)^{1/2}$, the equations of motion for the partial Wigner transform density operator in an adiabatic basis $A^{W}_{ij}$ are (for the diagonal and off-diagonal components):

[TABLE]

and

[TABLE]

A similar equation holds for $A^{W}_{22}$. Here, $V_{ii}({\vec{R}})$ are the adiabatic potential energy surfaces and $\left\{F_{ij}({\vec{R}})\right\}$ are the set of forces, $F^{\alpha}_{ij}({\vec{R}})\equiv-\left<\Phi_{i}({\vec{R}})|\frac{\partial V}{\partial R^{\alpha}}|\Phi_{j}({\vec{R}})\right>$. {$\left|\Phi_{i}({\vec{R}})\right>$} are an adiabatic basis set of electronic states, and $d^{\alpha}_{ij}({\vec{R}})$ are the derivative couplings, $d^{\alpha}_{ij}({\vec{R}})\equiv F^{\alpha}_{ij}({\vec{R}})/\left(V_{ii}({\vec{R}})-V_{jj}({\vec{R}})\right)$. We note that $d_{ij}=-d_{ji}^{*}$

At this point, we assume that all dynamics are being propagated near the adiabatic limit, with the population of state 1 close to unity (and only barely changing in time). Thus, the coherences are nearly stationary and (hopefully) not evolving much as well. In such a case, we can identify the steady state equation of motion for the coherences in Eq. II by ignoring any evolution of the coherences:Dou et al. (2017)

[TABLE]

which has the solution $A^{W}_{12}=\zeta$, where

[TABLE]

Thereafter, we change variables from $A^{W}_{12}$ to $B^{W}_{12}\equiv A^{W}_{12}-\zeta$. The equations of motion for the populations are modified as follows:

[TABLE]

The equations of motion for the coherences are more involved and given in Appendix A. If we assume that we are in the adiabatic limit moving along adiabat 1, noting $B^{W}_{12}$ vanishes in the adiabatic limit, Eq. II simplifies:

[TABLE]

The total effective force is the usual adiabatic force $\vec{F}_{11}$ plus the Berry magnetic force

[TABLE]

which arises as the curl of the Berry connection and which vanishes for a real Hamiltonian. Clearly, the QCLE already includes the effects of Berry phase.

III Discussion: Implications for Semiclassical Dynamics

Having successfully isolated Berry’s phase within the QCLE, let us now discuss the implications of our findings for mixed quantum classical methods. After all, one can view semiclassical nonadiabatic dynamics methods as approximations to the QCLE, and so one must wonder: do the standard semiclassical approaches (surface hopping and Ehrenfest dynamics) also account for Berry’s phase?

Consider the Hamiltonian that was introduced in Ref. Miao et al., 2019:

[TABLE]

where $\theta(x)\equiv\frac{\pi}{2}\left(erf(Bx)+1\right)$, and $\phi(y)\equiv Wy$.

For this Hamiltonian, the adiabatic surfaces are completely flat. For an incoming wavepacket on surface 2 beginning at $x=-\infty$ and traveling in the $+x$ direction, the exact solution predicts that the wavepacket should bend upwards. If W is small enough, the asymptotic momentum of the transmitted wavepacket should be W. This behavior follows by considering Berry’s force. For the Hamiltonian in Eq. 7, the Berry force is $\vec{F}^{B}_{2}=2\hbar Im\left[\vec{d}_{21}(\frac{\vec{P}}{M}\cdot\vec{d}_{12})\right]=\frac{\hbar W}{2}\partial_{x}\theta\sin{\theta}(-\frac{P^{y}}{M},\frac{P^{x}}{M})$. When $W$ is small enough, we can assume that $P^{x}$ is roughly constant, and so we may calculate the final $y$-direction momentum (at the end of a scattering event) by integrating the $y$-component of the Berry force:

[TABLE]

Of course, if W is not small, the result above is invalid; instead, the wavepacket can actually split apart and a portion of the wavepacket will reflect – even though the adiabats are completely flat.

Now, the example above makes very clear (as shown in Ref. Miao et al., 2019) that the FSSH algorithm does not capture Berry’s phase effects in the case of a complex HamiltonianRyabinkin et al. (2014). FSSH dictates motion along adiabats and the algorithm will not predict any bending or reflection; for this reason, in Ref. Miao et al., 2019, we have recommended augmenting FSSH dynamics with the Berry force $\vec{F}_{B}$ (in Eq. 6) in order to better agree with the QCLE and capture the correct quantum dynamics. Clearly, further benchmarking of such a corrected FSSH approach will be necessary.

At this point, it is worthwhile to consider the natural alternative to FSSH dynamics, namely Ehrenfest dynamics. Does Ehrenfest dynamics correctly account for Berry’s phase, or does it also require a Berry phase correction? We will now argue (analytically and numerically) that Ehrenfest dynamics do already include Berry’s phase; for the Hamiltonian in Eq. 7, in the limit of nearly adiabatic dynamics, Ehrenfest trajectories will bend the correct amount. Thus, despite the many failures of Ehrenfest dynamics (i.e. a lack of branchingTully (1990), a lack of detailed balanceParandekar and Tully (2005); Schmidt et al. (2008), a lack of decoherence Subotnik (2010); Bedard-Hearn et al. (2005); Larsen et al. (2006); Zhu et al. (2004a); Jasper and Truhlar (2011)), a correction for Berry’s phase effects is not needed.

To prove this point, consider the propagation of the wave function during an Ehrenfest trajectory for the Hamiltonian in Eq. 7:

[TABLE]

The time evolution of density matrix element ($\rho_{jk}\equiv c_{j}c_{k}^{*}$) is

[TABLE]

Here $\omega_{12}\equiv(E_{1}-E_{2})/\hbar$ and the adiabatic limit has been invoked such that trajectories are moving along surface 2 at all times ($\rho_{22}\approx 1$). Solving the above ODE with initial condition $\rho_{21}(0)=0$, we find:

[TABLE]

For the Hamiltonian in Eq. 7, one can compute $\vec{d}_{21}=\frac{1}{2}\left(-\partial_{x}\theta,iW\sin{\theta}\right)$. Thus, for the small $W$ case where $P^{x}$ is constant and $P^{y}\approx 0$, we can integrate $\rho_{21}(t)$ by parts (with the fact that $\left.\frac{\partial^{k}\theta}{\partial t^{k}}\right|_{t=0}=0$ for any order of $k$ assuming that we initially start far away from the crossing):

[TABLE]

To approximate the above series, we use the definition of $\theta$ in Eq. 7. For the term associated with $\frac{\partial^{k}\theta}{\partial t^{k}}$, one can show that the order of magnitude is $(\frac{BP^{x}}{\omega_{12}M})^{k}$. Thus, if $\frac{BP^{x}}{\omega_{12}M}$ is small, the first term will dominate the series, and the average force (as well as the final momentum) can be calculated as

[TABLE]

[TABLE]

From this argument, it is clear that Berry phase effects are already included in Ehrenfest dynamics (unlike FSSH) and there is no need for any additional corrections.

Finally, in order to numerically assess the relative value of Berry-corrected FSSH and Ehrenfest dynamics, in Fig. 1 we plot the transmitted $y$ momentum as a function of incoming momentum in both the adiabatic and diabatic regimes for the Hamiltonian in Eq. 7. For this data set, we set $W=5$, $M=1000$, and $B=3.0$, and as far as FSSH is concerned, we rescale all velocities in the $x-$direction whenever a hop occurs. We imagine a particle coming on adiabat $2$ from the left. For comparison, besides Ehrenfest and FSSH, we also plot results for exact dynamics as well as classical adiabatic dynamics with Berry’s forces. Reflection is rare and not important here. As one can see from the figure, Ehrenfest outperforms Berry-corrected FSSH in both the diabatic and adiabatic regimes as far as average momentum, indicating that Ehrenfest dynamics work better than FSSH even after a Berry-phase correction. Clearly, despite its many failuresTully (1990); Parandekar and Tully (2005); Schmidt et al. (2008); Subotnik (2010); Bedard-Hearn et al. (2005); Larsen et al. (2006); Zhu et al. (2004a); Jasper and Truhlar (2011), Ehrenfest dynamics incorporate Berry’s forces naturally and work very well for this problem of flat adiabatic surfaces; FSSH captures the correct trends but has a relatively larger error.

Lastly, using a Berry force and running purely adiabatic dynamics can be very accurate in the adiabatic regime, i.e. $A=0.10$. That being said, running adiabatic dynamics with a Berry force is awful in the diabatic regime, i.e. $A=0.02$.

IV Conclusions

In this communication, we have demonstrated that, even though the QCLE arises from a cut-off in $\hbar$ at order 0 from the Wigner distribution equation of motion, QCLE dynamics do include Berry’s phase effects (which are of order $\hbar$). As such, even though Berry’s phase effects are not usuallyRyabinkin et al. (2014) studied explicitly with the QCLE, if classical nuclei are sufficient, one can safely study many physical problems with geometric phase using the well-established QCLE and approximations thereof; of course, the bigger problem remains how to solve the QCLE in practice. Here, we have shown that Tully’s surface hopping approximation to the QCLE does not include Berry’s phase effects (when the Hamiltonian is complex), and we have recently made the sensible suggestion to simply add in the Berry force (Eq. 6)Miao et al. (2019). At the same time, we have also shown that Ehrenfest dynamics do contain Berry’s phase and, as such, no extra force is required.

Looking forward, the keen reader should observe that our model problem here (Eq. 7) is an extremely unphysical example whereby one can easily isolate Berry’s phase effects. For most problems with avoided crossings and conical intersectionsLevine and Martínez (2007), the adiabatic force difference will not be constant and surface hopping is usually expected to be a better approximation than Ehrenfest dynamics at recovering long time dynamics (e.g. populations during electron transfer dynamicsLandry et al. (2013)). Further research will need to assess whether FSSH can still be improved and how to incorporate decoherenceWebster et al. (1991, 1994); Schwartz et al. (1996); Wong and Rossky (2002a, b); Prezhdo and Rossky (1997); Prezhdo (1999); Fang and Hammes-Schiffer (1999); Volobuev et al. (2000); Hack and Truhlar (2001); Jasper et al. (2001); Zhu et al. (2004b); Jasper and Truhlar (2005) within a Berry-force modified algorithm. Another important question is how to choose a momentum rescaling direction for surface hopping; here, for the Hamiltonian in Eq. 7, we simply chose $x$ as the rescaling direction but for a more general Hamiltonian, a better ansatz is needed. Unfortunately, preliminary evidence suggests that the algorithms in Ref. Miao et al., 2019 are not yet optimal; perhaps the different form of the QCLE (as present in Eqs. II and A) will be useful for future derivations. At the very least, the equations should yield insight into exactly when one can make the adiabatic approximation and ignore $B_{12}$.

Finally and most importantly, now that we know that Berry’s phase dynamical effects are already included within the QCLE, this communication raises the distinct possibility of using the QCLE (and approximations thereof) to study coupled nuclear-electronic motion on the surfaces of topological materials, where the electronic Hamiltonian is complex and electronic Berry’s phase effects are already known to be of crucial importanceEssin et al. (2009). One must wonder if one will learn something new about nonadiabatic dynamics in such a context.

V Supplementary Material

See supplementary material for the detailed derivations of $\frac{\partial A^{W}_{11}}{\partial t}$ (Eq. II) and $\frac{\partial B^{W}_{12}}{\partial t}$ (Eq. A).

VI Acknowledgement

This material is based upon work supported by the National Science Foundation under Grant No. CHE-1764365. J.E.S. acknowledges a David and Lucille Packard Fellowship. J.E.S. thanks Abe Nitzan, Hsing-Ta Chen, Tao E. Li, Zeyu Zhou, Alec Coffman and Zuxin Jin for interesting conversations.

Appendix A Expression for $\frac{\partial B^{W}_{12}}{\partial t}$

After a great deal of algebra (see the Supporting Information), the equation of motion for $B^{W}_{12}$ can be shown to be:

[TABLE]