2.5 Time dependent density functional theory

Transcription

1 .5 Time dependent density functional theory The main theorems of Kohn-Sham DFT state that: 1. Every observable is a functional of the density (Hohenger-Kohn theorem).. The density can be obtained within a non-interacting model through the Kohn-Sham potential. Above all, DFT has been very successful in practice (using better approximations than LDA for the functional potential). Time dependent density functional theory (TD-DFT) extends DFT to time dependent Hamiltonians. One reason is that this gives a better venue to obtain excited-state properties using linear response theory. A TDSE Tr v(r, t), Ψ(0) Ψ(r 1,..., r N, t), Ψ(0) ρ(r, t), Ψ(0) Figure 1: Standard time dependent (parametric) functionals We will prove two theorems in TD-DFT corresponding to the two points mentioned above. The theorem which corresponds to Hohenger-Kohn in TD- DFT is the Runge-Gross theorem. One important difference (other the time dependence) is that the initial state is fixed. In essence, the Runge-Gross theorem says that the map A from the initial states and the external potential to the probability density is invertible. There is also a theorem that proves ν- representability: for every interacting time-dependent probability density there exists a non-interacting Kohn-Sham potential with corresponds to the same density. This is van Leeuwen s theorem. Before proving these theorems, we are going to do two other things. The literature covering these theorems normally uses using the formalism of second quantization, so we will explain that formalism first. Nevertheless, we will prove both theorems on first quantization. As a warm up to the actual proofs, we will prove the continuity theorem for the many body probability current in first quantization. 68

2 A v. Leeuwen v(r, t), Ψ(0) ρ(r, t), Ψ(0) v KS (r, t), Φ(0) Figure : van Leeuwen s ν-representability theorem.5.1 Quantum field theory formalism In the section on second quantization we focused on the fermionic creation and annihilation operators a j, a j for molecular orbital ψ (j). We saw that singleelectron operators had the following form on second quantization Ô = O pq a pa q = dr 1 dr ψ (p) (r 1 )O(r 1, r )ψ (q) (r )a pa q pq pq = dr 1 dr ψ (p) (r 1 )a p O(r 1, r ) ψ (q) (r )a q. p q Introducing the quantum field operators ˆΨ(r) = p ψ (p) (r) a p the operator is written Ô = dr 1 dr ˆΨ (r 1 ) O(r 1, r ) ˆΨ(r ). This is standard formalism in quantum field theory. For the formalism of second quantization it is standard to choose a complete basis of wave functions with the property ψ (p) (r 1 ) ψ (p) (r ) = δ(r 1 r ). This implies p { ˆΨ (r 1 ), ˆΨ(r )} = pq = p ψ (p) (r 1 ) ψ (q) (r ){a p, a q } ψ (p) (r 1 ) ψ (p) (r ) = δ(r 1 r ). This can probably get you started reading the relevant literature covered in this lecture. Nevertheless, we will prove both the Runge-Gross and the van Leeuwen s theorem in first quantization. 69

3 .5. The probability current The many body probability current in first quantization is given by the expression j(r) = in dr dr N Ψ(r,..., r N ) Ψ(r,..., r N ) Ψ(r 1,..., r N ) Ψ(r 1,..., r N ). In the following we will not write the coordinates for the wave functions, potentials ans so forth. The probability current obeys the continuity equation. We will prove that first. The continuity equation is an statement about the derivative of the probability density, so let s start with that: t ρ(r) = Ndr..N ( t Ψ ) Ψ + Ψ ( t Ψ ). The Schrödinger equations are t Ψ = ihψ and t Ψ = ihψ. The signs are going to be important. Plugging in the Schródinger equations we get ρ(r) = in dr..n (HΨ ) Ψ Ψ (HΨ) t = in dr..n jψ Ψ Ψ jψ. j j We will use the following operations repeatedly j ( j Ψ )Ψ = ( jψ )Ψ + ( j Ψ ) ( j Ψ) j Ψ ( j Ψ) = Ψ ( jψ) + ( j Ψ ) ( j Ψ). We make this substitution on the prior equation to get t ρ(r) = in dr..n j (( j Ψ )Ψ Ψ ( j Ψ)) j Finally, we use the divergence theorem to turn every divergence j {,.., N} into a surface integral at infinity, which gives zero. The only surviving term is t ρ(r) = in dr..n ( Ψ )Ψ Ψ ( Ψ)) = j. This is the continuity equation for the probability current. A note about notation: it is understood that if the gradient does not have an index it refers to the first coordinate. 70

4 .5.3 Runge-Gross Theorem The picture is given by Fig 1. The initial state Ψ(0) is fixed. From the time dependent potential we obtain the wave function Ψ(t) solving the time dependent Schrödinger equation, TDSE. Of course one can trace down to the probability density, and the density can be thought of as a parametric functional of the potential and the initial state through the function A. The content of the Runge-Gross theorem is that this function is injective. Theorem.1 (Runge-Gross). For a fixed initial state and given an analytic time dependent potential, the mapping to the time dependent probability density is injective. That is, for the same initial state, two different external potentials can not give the same probability density function ρ(r, t). {Ψ(0), v} {Ψ(0), v } ρ = ρ Impossible Proof. This proof follows the one in chapter one of the book Time-dependent Density Functional Theory, Marques et. al. The equation of motion for the probability current is t j = in r..n t (Ψ Ψ Ψ Ψ ) = in ( ψ r..n t Ψ + Ψ Ψ t Ψ ) t Ψ Ψ Ψ t = N r..n ((HΨ ) Ψ Ψ (HΨ) + (HΨ) Ψ Ψ (HΨ )) We now study the difference of currents corresponding to potentials V = 71

5 j v(j) and V = j v (j). We get j(r, t) j t( (r, t) ) t=0 = N r..n ((V V )Ψ Ψ Ψ ((V V )Ψ) ) + (V V )Ψ Ψ Ψ ((V V )Ψ ) = N ( r..n Ψ ( (V V ))Ψ + Ψ( (V V ))Ψ ) t=0 = N r..n Ψ (V V ) = ρ(r, 0) (v(r, 0) v (r, 0)). In the last step we use V = 1 j v(j) = 1v(1), because v(r j 1 ) is a constant with respect to the r 1 coordinates. The conclusion is that if the potentials differ by more than a time dependent function at t = 0, the first derivatives of the probability currents j and j are different, and therefore the probability currents will become different. We need the analytic assumption to rule out the case of the potentials becoming different at a later time. Using this assumption, this implies that some derivative is different. Now, applying the equation of motion repeatedly one obtains after some algebra (see also the proof of van Leeuwen s theorem later) ( ) k+1 ( j(r, t) j (r, t) ) = ρ(r, 0) t t=0 t=0 ( ( ) k (v(r, 0) v (r, 0))). t That is, some derivative of j will be different, and therefore the current themselves will become different. Define the function ( ) k w k (r) = (v(r, 0) v (r, 0)). t In order to extend the proof to the probability densities we use the continuity equation ( ) k+ ( ρ(r, t) ρ (r, t) ) = (ρ(r, 0) w k (r)). t t=0 It suffices to show that the expression on the right hand side is not zero. Now (w k ρ w k ) = ρ ( w k ) + w k (ρ w k ). The left hand side is ds (w k ρ w k ) = 0, 7

6 assuming that ρ( ) = 0 or that we integrate far enough and the potential dies fast enough (as 1/r is enough). In the right side, by assumption (the potentials are different) we get ρ ( w k ) > 0. Therefore, w k (ρ w k ) 0 (ρ w k ) 0. Then the derivatives of the probability densities are not zero, and the densities themselves will become different. This concludes the proof..5.4 van Leeuwen s Theorem As mentioned at the beginning of the lecture, van Leeuwen s theorem establishes the non-interacting ν-representability for the density matrix corresponding to any analytical potential (see Fig. ). The proof is constructive. Theorem.. For given initial state Ψ and analytic potential v(r) there exists initial state Φ and non-interacting Kohn-Sham potential v KS such that ρ KS = ρ for all t. This theorem is actually more general in that one can choose different interactions for the target system. Proof. This proof follows the one in chapter two of the book Time-dependent Density Functional Theory, Marques et. al. We start by writing the time evolution of a component α of the probability current. Here α stands for one Cartesian coordinate, r x, r y of r z. t j α = in ( ) dr..n t Ψ α Ψ + Ψ t α Ψ + c.c. = N dr..n (HΨ ) α Ψ Ψ H α Ψ + c.c. We expand the Hamiltonian term by term. We start with the kinetic term j j /. We have, for the first two terms, up to the 1/ factor, j jψ α Ψ = j j ( j Ψ ) α Ψ jψ j α Ψ Ψ j j α = j j Ψ j α Ψ + jψ j α Ψ. The product of divergences cancels. The integral of the divergence of j 1 gives a surface integral at infinity, which is 0 (we used the same manipulation 73

7 in the previous theorem). We are left with in 1 dr..n ( Ψ ) α Ψ Ψ α Ψ + c.c.. Now we look at the components of the divergence one by one. (( ) β β Ψ α Ψ ) Ψ βα Ψ + c.c. ( ( ) = β β Ψ ) α Ψ + 1 (Ψ βα + ( ( = ) β β Ψ α Ψ + = β T βα. ( β Ψ α Ψ ( ) ( β Ψ α Ψ + ( ) β Ψ ) ) ) β Ψ α Ψ + Ψ βα Ψ ) α Ψ 1 βα (Ψ Ψ) The last equation defines the momentum-stress tensor T βα. We now look at the electron-electron interaction. We have N dr..n 1 1 r i r j Ψ α Ψ Ψ 1 1 α r i r j Ψ + c.c. j k j k = N dr..n Ψ 1 1 α r i r j. j k The complex conjugate cancels the factor of 1/ and taking the derivative of the product on the second term only the factor above survives. We can simplify this expression further by noticing that α stands for one of the three coordinates of r 1. Therefore, either r i or r j must be r 1. We can choose k = 1, and cancel another factor of 1/ to get N dr..n Ψ 1 α r j>1 1 r j. Finally, because Ψ is antisymmetric, we can write this expression as N(N 1) V ee,α = dr..n Ψ 1 α r 1 r The remaining term in H, the one with the external potential, was treated in the proof of the previous theorem. Putting it all together we write t j α = ρ α v β β T βα V ee,α. 74

8 Taking the divergence and using the continuity equation of the probability current we write where ρ = (ρ v) + q (1) t q = βα T βα + α,β α α V ee,α. This equation is critical to van Leeuwen s theorem because it establishes the mathematical relation between the probability density and the external potential. The goal of this theorem is to define an external potential which gives the same probability density than the original interacting problem, but with a new Hamiltonian H = T + V ee. That is, the interaction is now different, and we will be particularly interested in the non-interacting Kohn-Sham system. The first step is to choose an initial state Ψ for the new system. It only needs two conditions. The first one is that at time 0 it gives the target probability density ρ (0) = ρ(0). In the Kohn-Sham system where the initial state is a Slater determinant of wave functions ψ j we just need The second condition is j ψ j(0) = ρ(0). t ρ t=0 = t ρ t=0 Using again the continuity equation of the probability current we rewrite this condition as j = j. The probability current is defined taking gradients of the wave function, independently of the Hamiltonian. Therefore, this condition can also be obeyed by an appropriate choice of phases of the initial wave function Ψ (0), without reference to v (0). At this point it is convenient to introduce the shorter standard notation t = t. Evaluating Eq. (1) at t = 0 we get t ρ t=0 = ρ v + q t=0. Because we want, for all t, ρ(t) = ρ (t), we will impose t ρ t=0 = t ρ t=0. Further, the definition of q (0) does not involve the evaluation of v, only the 75

9 momentum-stress tensor (itself a combination of derivatives of ψ (0), and an expression involving the new electron-electron interaction V ee,α. In the Kohn- Sham case, this term will be zero. In conclusion, we can write the equation (ρ(0) v (0)) = t ρ q (0). Only the left hand side depends on v, and this is a Sturm-Liouville equation for v which, by standard theory of differential equations, has a unique solution. Having defined the derivative of the external potential at t = 0, we define the potential recursively using the analytic assumption. That is, we will define all the derivatives at t = 0. Taking the derivative of Eq (1) we obtain 3 t ρ = (ρ ( t v )) + (( t ρ ) v ) + t q. Equating again the derivatives of the probability densities of both systems, 3 t ρ = 3 t ρ, we get (ρ ( t v )) = 3 t ρ (( t ρ) v ) t q. The quantity t q t=0 is calculated form v (0), which has already been defined. Therefore, this is again a differential equation which gives t v t=0. Continuing in the same fashion, we get k+ t ρ = k t q + k l=0 ( ) k (t k l ρ l l tv ). Equating again the two probability densities this gives the final recursive equation (ρ k t v ) = k+ t k 1 ρ t k q l=0 ( ) k (t k l ρ l l tv ). Lower derivatives are already fixed recursively, so this is an equation for k t v. Finally, from the analytic assumption v is defined from its Taylor s series at t = 0 v (t) = k=0 1 k! k t v (0)t k. 76