The Runge-Gross Theorem

In time-dependent systems, there is no variational principle on the basis of the total energy for it is not a conserved quantity. There exists, however, a quantity analogous to the energy, the quantum mechanical action 

Another important point, often overlooked in the literature, is that a time-dependent problem in quantum mechanics is mathematically defined as an initial value problem. This stems from the fact that the time-dependent Schrodinger equation is a first-order differential equation in the time coordinate. The wave-function (or the density) thus depends on the initial state, which implies that the Runge-Gross theorem can only hold for a fixed initial state (and that the xc potential depends on that state). In contrast, the static Schrodinger equation is a second order differential equation in the space coordinates, and is the typical example of a boundary value problem. 

From the above considerations the reader could conjecture that the proof of the Runge-Gross theorem is more involved than the proof of the ordinary Hohenberg-Kohn theorem. This is indeed the case. What we have to demonstrate is that if two potentials, v r, t) and v r, t), differ by more than a purely time dependent function c(t), they cannot produce the same time-dependent density, i.e. 

This statement immediately implies the one-to-one correspondence between the potential and the density. In the following we will utilize primes to distinguish the quantities of the systems with external potentials v and v. Due to technical reasons that will become evident during the course of the proof, we will have to restrict ourselves to external potentials that are Taylor expandable with respect to the time coordinate aroimd the initial time to 

Clearly, if the two potentials are different by more than a purely time-dependent function, at least one of the expansion coefficients in their Taylor expansion aroimd to will differ by more than a constant 

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Based on an extension [35] of the Runge-Gross theorems descried in Sect. 2 to arbitrary multi-component systems one can develop [36] a TDDFT for the coupled system of electrons and nuclei described above. In analogy to Sects. 2.1-2.3, one can establish three basic statements First of all, there exists a rigorous 1-1 mapping between the vector of external potentials and the vector of electronic and nuclear densities,... 

For non-interacting particles moving in external potentials t>s(r, f), the Runge-Gross theorem holds as well. Therefore the functional... 

A totally different point of view is proposed by Time-Dependent Density Functional Theory [211-215] (TD-DFT). This important extension of DFT is based on the Runge-Gross theorem [216]. It extends the Hohenberg-Kohn theorem to time-dependent situations and states that there is a one to one map between the time-dependent external potential t>ea t(r, t) and the time-dependent charge density n(r, t) (provided we know the system wavefunction at t = —oo). Although it is linked to a stationary principle for the system action, its demonstration does not rely on any variational principle but on a step by step construction of the charge current. 

For this purpose, Runge and Gross proposed the fundamental DFT theorem for periodically time-dependent electronic states, which is called the Runge-Gross theorem (Runge and Gross 1984). This theorem is based on the following two assumptions for the external potential,... 

The Runge-Gross theorem has a severe problem in the use of the wavefunction giving electron density in the third theorem (Gross et al. 1995) Since time-dependent wavefunctions contain time-dependent phase terms, the one-to-one correspondence of the wavefunction and electron density is established only for a specific phase. This problem can be avoided by representing the wavefunction as a functional of the external potential, for which the one-to-one corre-... 

For proper treatment of atomic and molecular dynamics such as collisions or multiphoton ionization (MPI) processes, etc., it is necessary that both the ionization potential and the excited-state properties be described more accurately. In addition, the treatment of time-dependent processes will require the use of the time-dependent density functional theory (TDDFT). The rigorous formulation of TDDFT is due to the Runge-Gross theorem [14]. For any interacting many-particle quantum system subject to a given time-dependent potential, all physical observables are uniquely determined by knowledge of the time-dependent density and the state of the system at any instant in time. In particular, if the time-dependent potential is turned on at some time f and the system has been in its ground state until f, all observables are unique functionals of only the density. In this case, the initial state of the system at time f will be a unique functional of the... 

The right-hand side of (4.23) differs from zero, which again implies that j r,t) j r,t) for t > to. This concludes the first step of the proof of the Runge-Gross theorem. 

This equality was obtained with the help of Green s theorem. The first term on the right-hand side is zero by assumption, while the second term vanishes if the density and the function Wfc(r) decay in a reasonable manner when r —> 00. This situation is always true for finite systems. We further notice that the integrand no(r) [Vwfc(r)] is always positive. These diverse conditions can only be satisfied if either the density no or Vnfc(r) vanish identically. The first possibility is obviously ruled out, while the second contradicts our initial assumption that Wfc(r) is not a constant. This concludes the proof of the Runge-Gross theorem. 

Other observables, such as the total ionization yield or the ATI spectrum, are much harder to calculate within TDDFT. Even though these observables (as all others) are functionals of the density by virtue of the Runge-Gross theorem, the explicit functional dependence is miknown and has to be approximated. 

We recall that by virtue of the Runge-Gross theorem is a functional of the time-dependent density. Separating g into an exchange part (which is simply 1/2 for a two electron system) and a correlation part,... 

In this chapter we tried to give a brief, yet pedagogical, overview of TDDFT, from its mathematical foundations - the Runge-Gross theorem and the time-... 

Time-dependent density functional theory (TDDFT), in contrast, applies the same philosophy as ground-state DFT to time-dependent problems. Here, the complicated many-body time-dependent Schrodinger equation is replaced by a set of time-dependent single-particle equations whose orbitals yield the same time-dependent density. We can do this because the Runge-Gross theorem proves that, for a given initial wave function, particle statistics and interaction, a given time-dependent density can arise from, at most, one time-dependent external potential. This means that the time-dependent potential (and all other properties) is a functional of the time-dependent density. 

TD-DFT is rooted in the Runge-Gross theorem [60] (which is not valid for the degenerate ground state), allowing the extension of the Hohemberg-Kohn-Sham formulation of the TD-DFT theory to the treatment of time-dependent phenomena ... 

The Runge-Gross theorem states that for a many-body system evolving from a fixed initial state there is a one-to-one correspondence between the external time-dependent potential and the (time-dependent) electron density p(r) = p(r, t). Therefore, the behavior... 

Since, because of the Runge-Gross theorem, the wave function is a functional of the electron density, the action integral of Eq. (4.47) is also a unique functional of the density ... 

Based on the Runge-Gross theorem, it is shown that the three-dimensional density of a many body quantum system is sulScient to describe the TD response of the system to an external perturbation, such as electromagnetic field and vibrational motion, R t). This is known as time dependent density functional theory (TDDFT). In the linear response approximation, TDDFT is frequently used to evaluate electronic excitation energies. Here, we use full TDDFT where the density is explicitly propagated in time. Application of the TD (Dirac) vibrational principle to KS energy generates density evolution ... 

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