(p.465) Appendix L TDDFT in a Lagrangian frame
(p.465) Appendix L TDDFT in a Lagrangian frame
In Chapter 10, we discussed the formal framework and applications of TDCDFT. One of the key points was that memory and spatial nonlocality are closely linked, as illustrated in Fig. 10.2, which shows the motion of a fluid element of the electron liquid. The language of fluid dynamics was used in Section 10.4 to develop a nonadiabatic approximation to the xc vector potential, known as the VK functional, in which A xc is written as the divergence of a viscoelastic stress tensor.
The idea of using concepts of hydrodynamics in TDDFT, which was expressed in the original Runge-Gross paper of 1984, was pursued further by Dobson et al. (1997), Kurzweil and Baer (2004, 2005, 2006, 2008), and Thiele and Kümmel (2009) in the quest for xc functionals with memory. However, the most rigorous hydrody-namic formulation of the general electronic many-body problem and of TD(C)DFT was developed by Tokatly and Pankratov (2003) and Tokatly (2003, 2005a, 2005b, 2006,2007,2009). This work is of great significance for several reasons:
It exemplifies the notion that TDDFT is fundamentally a theory of collective variables—density and current (or velocity)—whose equations of motion must re-flect basic conservation laws. In particular, the balance between external and internal forces in the electron liquid is naturally described in terms of stress tensors. These stress tensors are well-defined objects within a rigorous quantum hydrodynamic theory of interacting many-body systems.
TDDFT can be formulated as a hydrodynamic theory both in the laboratory frame and in a reference frame that moves with the electron fluid elements (the so-called Lagrangian frame). The fundamental variable in this frame is a deformation tensor, gμν, accounting for the internal motion within the fluid elements. The convective motion—which is essentially classical—is treated separately.
In contrast to the formulation in the laboratory frame, the ultranonlocality prob-lem of TDDFT goes away in the comoving frame, and a local approximation exists for nonadiabatic xc effects. A formally exact, nonadiabatic extension of the LDA into the dynamical regime can be defined in this way. In linear response, one re-covers the VK functional in this manner. In the nonlinear regime, the real-time VK functional (Section 10.6.2) emerges in the limit of small deformations.
In this appendix, we will review some of the basics of Tokatly's approach. We won't attempt to cover all the formal and technical details and proofs (see, in particular, Tokatly, 2005b, 2007). Rather, we shall point out the connection with the VK approximation of TDCDFT and illustrate it with some simple numerical examples of one-dimensional model systems (Ullrich and Tokatly, 2006).
(p.466) L.1 Fluid motion in the Lagrangian and laboratory frames
The dynamics of an electronic system can be described by viewing it as a collection of infinitesimal fluid elements. The central idea is that each fluid element carries a unique label, namely its initial position vector ξ at time t 0, which is known as the Lagrangian coordinate. The convective motion of each fluid element can be tracked and uniquely identified by its trajectory.1 The comoving reference frame in which one is “riding along” with each fluid element is called the Lagrangian frame.
The transition between the laboratory frame and the Lagrangian frame involves a nonlinear transformation of coordinates. The trajectory r(ξ, t) of an infinitesimal fluid element in the laboratory frame which evolves from the starting point ξ is formally defined by the following initial-value problem:
The symmetric second-rank metric tensor in ξ-space,
Equations (L.1) and (L.4) determine Cauchy's deformation tensor as a functional of the velocity field u(r, t). An alternative and practically more convenient way to compute ḡ μν for a given velocity field is to solve the equation of motion that governs the dynamics of ḡ μν(r, t) directly in the laboratory frame:
An important property of the deformation tensor is that it allows us to relate the time-dependent density n(r, t) to the initial density distribution, n 0(r):√
(p.467) Let us now illustrate these concepts for a one-dimensional system. The trajectory (L.1) of a fluid element with Lagrangian coordinate ξ simplifies to
x is the position, at time t, of a fluid element that started out at ξ, and u is its velocity. In general, this is a complicated nonlinear differential equation for the trajectory, which can be formally solved by direct integration:
From this, we can determine the time-dependent density in the laboratory frame: we first invert eqn (L.8) to obtain ξ(x, t), then compute the deformation as
In general, of course, this procedure is not very helpful, since the functional form of u(ξ, t) is likely to be unknown. But, as we shall see in the following, it is useful for constructing simple, quasi-one-dimensional analytic examples.
L.1.1 Sloshing mode
Consider a system which is confined within hard walls, –L/2 ≤ (x, ξ) ≤ L/2, with initial density
The next step is to invert eqn (L.13) to determine the trajectories of the fluid elements. This requires solving a quadratic equation, with the result
L.1.2 Breathing mode
To simulate a breathing mode, we assume a linear velocity distribution of the fluid elements,
This is easily inverted:
(p.469) The resulting deformation is
We choose the same initial density distribution n 0(ξ) [eqn (L.11)] as for the sloshing mode, and the resulting time-dependent density of the breathing mode is
Figure L.1 shows snapshots of n(x, t), u(x, t), and ḡ(x, t) for the sloshing and the breathing mode, with amplitudes a = b = 0.5, taken at times t = 0, T/4, T/2, 3T/4, where T = 2π/ω. The deformation ḡ(x, t) is maximum at the turning points of the oscillations (at t = T/4 and 3T/4). We find that the breathing mode has large defor-mations everywhere, i.e., g deviates strongly from 1. The sloshing mode, on the other hand, is strongly deformed only towards the edges, where the density is small. We will see below how this affects the nonadiabaticity of the xc potential for the two modes.
L.2 TDDFT in the Lagrangian frame
In Chapter 3, we introduced the local force balance equation as follows:
In TDCDFT, the xc vector potential A xc ensures that the physical density and current are reproduced by an auxiliary system of noninteracting particles. This means that A xc should produce an effective xc Lorentz force that exactly compensates for any difference between the local stress forces in the real interacting system and in the (p.470) noninteracting system. Accordingly, the xc vector potential should satisfy the following equation:
Equation (L.24) serves as a basic definition of A xc, which automatically accounts for the zero-force and zero-torque conditions.
The xc stress tensor P xc,μν must be distinguished from the xc stress tensor σxc,μν introduced earlier. The main difference lies in the fact that P xc,μν, formally exactly and to all orders in the inhomogeneity, accounts for all dynamical xc effects, whereas the ALDA has been separated out in the definition of σxc,μν. Furthermore, σxc,μν is valid only for small deformations of the electron liquid (in a sense to be defined below).
All that remains to be done is the actual calculation of the dynamic xc stress tensor P xc,μν, which enters into the definition (L.23) of υ xc. And this is where the Lagrangian frame comes in. To see how this works, consider the hydrostatic limit of eqn (L.21):
The first two terms on the left-hand side are the linear acceleration force and the force related to the kinetic energy of the frame (a sort of generalized centrifugal force). The last two terms are the forces due to the internal stress. Equation (L.26) tells us that the net force on any fluid element in the Lagrangian frame is exactly zero, which means that there is no current and the density remains stationary during the time evolution.
There are additional, hidden inertial effects, namely a “geodesic” force and a Coriolis force. These pseudoforces depend on the velocity of each particle as it travels along its geodesic (or trajectory), and are implicitly accounted for in the stress tensor:
In other words, the stress tensor is a functional of the deformation tensor and of a skew-symmetric vorticity tensor. Both are functionals of the velocity, according to the Runge-Gross theorem. It turns out, however, that in the lowest order of the gradient expansion the dependence on the vorticity tensor disappears.
(p.471) The xc stress tensor is also a functional of the deformations, . Any approximation to the xc stress tensor in the Lagrangian frame in terms of g μν and can then be transformed back into the laboratory frame and used there to solve the TDKS equations.
The advantage of working in the Lagrangian frame is that a time-dependent local approximation can be derived in a consistent manner, owing to the stationarity of the density distribution. In this way, one arrives at the time-dependent local-deformation approximation (Tokatly, 2005b, 2006), where
The xc stress tensor becomes a spatially local functional of Green's deformation tensor g μν; back in the laboratory frame, it is a local functional of the Cauchy tensor ḡμν. Equation (L.5) or, equivalently, eqns (L.1) and (L.4) show that in general the deformation tensor is a strongly nonlocal (both in space and in time) functional of the velocity. Therefore, in spite of the fact that the xc stress tensor and, consequently, the xc vector potential are local functionals of ḡ μν, they are nonlocal in terms of velocity or any other variable. This makes ḡ μν(r, t) a convenient choice for a basic variable.
To obtain an explicit construction of the local functional P xc,μν[ḡ μν], the solution of a homogeneous time-dependent many-body problem in the Lagrangian frame is required (Tokatly, 2005b). Needless to say, this is a daunting task, but there are two practically important, exactly solvable special cases, which are described below.
L.3 The small-deformation approximation
The many-body problem in a homogeneously deformed Lagrangian ξ-space can be solved perturbatively if the deformation tensor ḡ μν deviates only slightly from the unit tensor δ μν:
By introducing the displacement vector s(r, t) = r – ξ(r, t) and using eqn (L.4), we find that a small δḡ μν corresponds to small gradients of the displacement:
Clearly, small gradients of s(r, t) imply that the velocity gradients are also small, since to lowest order in ∇μ s ν eqn (L.1) reduces to the relation ∂s(r, t)/∂t = u(r, t). Keep in mind, however, that smallness of the deformations does not mean that the displacement or the velocity itself is small (i.e., the system can be far beyond the linear-response regime). An example is the rigid motion of a many-body system in a harmonic potential (see the harmonic potential theorem in Section 6.3.2), where ḡ μν = δ μν but the displacement can be arbitrarily large.
The stress tensor functional for small displacement vectors was derived in Tokatly (2005b). Extension of this derivation to the general regime of small deformations, i.e., the regime of small displacement gradients, is straightforward. The resulting xc stress tensor takes the following form:
(p.472) where is the xc pressure of a homogeneous electron liquid, and δP xc,μν is a nonadiabatic correction which is linear in δḡ μν:
The minus sign reflects an ambiguity in the definition of stress tensors in the continuum mechanics literature. The formulation of TDCDFT presented in Chapter 10 follows the sign convention used in classical elasticity theory, where the divergence of σxc,μν defines a force exerted on a small volume element by surrounding parts of the body. By contrast, the sign of Pxc,μν follows from the momentum flow tensor—a convention that is more common in fluid mechanics and in microscopic theory. The trace of Pxc,μν is thus equal to the local pressure, while its divergence gives the force exerted by an infinitesimal volume element on the surrounding fluid.
In the limit of small displacement and small velocity gradients, the spatial derivatives of a xc on the left-hand side of eqn (L.24) are negligible. Thus, in the regime of small deformations we recover the complete nonlinear form of the VK functional [eqns (10.79) and (10.80)]. This is an important result.
The imaginary parts of the complex elastic moduli are responsible for the dissipative (viscous) effects. However, for the high-frequency (short-time) dynamics, these effects become irrelevant. As a result, the high-frequency limit of the nonadiabatic stress tensor of eqn (L.32) becomes completely local and purely elastic:
The structure of the small-deformation approximation, eqns (L.31) and (L.32), clearly demonstrates that in this regime the nonadiabatic contribution appears as a small correction to the adiabatic dynamics, linear in δḡ μν. If the process is strongly nonadiabatic, however, the deformations cannot be considered small. In fact, the de-viation of the deformation tensor from δ μν can serve as a general measure of nonadi-abaticity.
(p.473) L.4 The nonlinear elastic approximation
It is very difficult to account both for the full nonlinear dependence on ḡ μν and for dissipation. The VK formalism of Chapter 10 includes all xc dissipation effects on a level linear in δḡ μν. If we neglect dissipation, a closed nonlinear local approximation to the stress tensor can be rigorously derived. The reason is that the homogeneous many-body problem admits a simple complete solution in the regime of fast dynamics when dissipation is irrelevant. In this case the xc stress tensor becomes a function of the time-dependent density n(r, t) and Cauchy's deformation tensor ḡ μν(r, t):
The function L μν(ḡ kλ) in eqn (L.36) is defined in Appendix C of Tokatly (2005b). In the limit of small deformations, the nonlinear elastic approximation of eqn (L.36) can be expanded around ḡ μν = δ μν and reduces to the linearized form defined by eqns (L.31) and (L.35). One thus recovers the high-frequency limit of the VK functional.
We now discuss the nonlinear elastic approximation for one-dimensional motion. If all spatial variations are along the x-axis only, the deformation tensor becomes diagonal, with ḡ zz = ḡ yy = 1 and ḡ xx = ḡ(x, t). The xc effects can then be described by an xc scalar potential that is related to the xc stress tensor as follows:
Equation (L.36) for the xx component of the xc stress tensor reduces to the form
Finally, eqn (L.5), which relates the deformation ḡ(x, t) to the velocity u(x, t), simplifies as follows:
Note that L(ḡ → 1) = 1/3 in the limit of zero deformation. P xc, xx then becomes
L.5 Validity of the VK potential and breakdown of the adiabatic approximation
Let us now look at some results for the quasi-one-dimensional model systems introduced above. We take N = 1 a.u. for the sheet density and L = 10 a.u. for the size of the model quantum wells. We want to focus on two things: how compares with the nonlinear VK potential of eqn (10.82), in various frequency regimes, and how they both compare with the ALDA (i.e., how big the nonadiabatic effects are).
Figure L.2 presents for the sloshing and breathing modes. The motion is periodic, and four snapshots taken during one period are shown. In each panel, we compare three frequencies: . Here, is the average of the local plasma frequency for the stationary density (L.11); it is a characteristic frequency of the system, and allows us to identify low-frequency, intermediate-frequency, and high-frequency regimes.
The physical impact of the post-ALDA, nonadiabatic corrections is similar in both modes: they tend to oppose the ALDA potential at the times of maximum deformation of the density distribution. Remember, in this example the density was prescribed, and the xc functionals were simply evaluated with given densities. In reality, of course, the xc potential becomes part of the self-consistent TDKS equation. This means that the elasticity of the electron liquid counteracts the deformations of the density, making the system more rigid and somewhat harder to deform.
But how reliable is the VK functional? In the linear regime, i.e., for motion with small amplitude, we know that it is the exact local, frequency-dependent xc potential. But, in the nonlinear regime, the answer is less clear. In Section 10.6.2 we said that (p.476) eqn (10.82) is valid as long as the spatial gradients of the velocity field are small, which is a somewhat vague statement. Fortunately, we now have the opportunity for a quantitative assessment, namely, by comparison with the elastic approximation , which is “exact” in the high-frequency regime (in the sense that this is the local time-dependent potential which is the exact nonadiabatic extension of the LDA into the dynamical regime).
Figure L.3 (left part) shows the nonadiabatic part of the power
Finally, we show a drastic example of the breakdown of the ALDA. In the right part of Fig. L.3, we plot for large amplitudes a, b = 0.75. The effect is quite dramatic for the breathing mode: here, the nonadiabatic effects are so large that they completely oppose the ALDA.
Let us now summarize the main points that emerge from this analysis:
The hydrodynamic formalism which we have sketched here has been developed further (Tokatly, 2007) to describe situations in which time-dependent electric and magnetic fields are present. In this case, the formalism becomes more compact in a certain sense, since the dependence on the skew-symmetric vorticity tensor disappears (instead, an effective vector potential appears in the theory), and the stress tensor depends only on the deformation g μν. One thus arrives at a time-dependent deformation-functional theory. An interesting feature of this theory is that it allows one to formulate the basic existence and uniqueness proofs in a mathematical form that is quite distinct from the standard TD(C)DFT proofs à la Runge-Gross or van Leeuwen, namely, a form similar to a constrained-search procedure (Tokatly, 2009). This opens up an important new direction in the ongoing attempts to strengthen the mathematical bedrock on which TD(C)DFT is built.
(1) This excludes extreme situations such as shock waves and turbulence, which involve singularities; fortunately, these do not occur in Schrödinger dynamics.
(2) Here, ũ μ and ñ are the velocity and density, transformed into the Lagrangian frame. Upper and lower indices are connected according to standard rules of tensor algebra via uμ = ∑ν gμνu ν.