Computational Physics (Lecture 23)

1. Computational Physics(Lecture 23) PHY4370

2. Mermin finite temperature and ensemble density functional theory • The theorems of Hohenberg and Kohn for the ground state carry over to the equilibrium thermal distribution by constructing the density corresponding to the thermal ensemble. • For each of the conclusions of Hohenberg and Kohn for the ground state, there exists a corresponding argument for a system in thermal equilibrium, as was shown by Mermin.

3. To show this, Mermin constructed a grand potential functional of the trial density matrices ’, , whose minimum is the quilibrium grand potential: , • Where is the grand canonical density matrix • =/Tr .

4. The proof is analogous to the Hohenberg-Kohn proofs and uses only the minimum property of and the fact that the energy depends upon the external potential only through the Vext integral. • The Mermin theorem leads to even more powerful conclusions than the Hohenberg-Kohn theorems, • Namely that not only the energy, but also the entropy, specific heat, etc., are functionals of the equilibrium density.

5. However, the Mermin functional has not been widely applied. • The simple fact • it is much more difficult to construct useful, approximate functionals • for the entropy (which involves sums over excited states) than for the ground state energy. • For example, in the Fermi liquid description of a metal the specific head coefficient at low temperature is directly related to the effective mass at the Fermi surface. • Thus the Mermin functional for the free energy must correctly describe the effective mass (with all its many-body renormalization) as well as the ground state energy.

6. Current functionals • From the beginning, the HK theorems assumed that the hamiltonian is time reversal invariant. • If there is a magnetic field, the density is not enough • The properties of the system are a functional of both the density and the current density. • The theory is not as well developed as that for density.

7. Time dependent density functional theory • The Hohenberg-Kohn theorem have also been extended to the time domain, where it has been shown that, given the initial wavefunction at one time, the evolution at all later times is a unique functional of the time dependent density. • There has recently been considerable progress along these lines.

8. It has recently been pointed out that in an extended system, • i.e. one with no boundaries, the evolution is not a functional only of the density. • A simple counter example is a uniform ring of charge that can flow around the ring. • Since the density is always uniform, the state of the system is determined only if an extra condition, the current, is specified.

9. Electric fields and polarization • The issue of electric fields and polarization comes into play in extended systems. • In finite space, the potential due to an electric field V = Ex is unbounded; • There is no lower bound to the energy and therefore there is no ground state! • This is a famous problem in the theory of the dielectric properties of materials. • However, if the ground state does not exist, the HK theorems on the ground state do not apply.

10. Is there any way to include an electric field in DFT? • Very subtle problem and the answer is that in the presence of an electric field, one must apply some constraint, within which there is a stable ground state. • In the case of molecules, this is routinely done simply by constraining the electrons to remain near the molecules. • In a solid, the constraint is not so obvious. • All proposals involve constraining the eletrons to be in localized Wannier functions or equivalent conditions on Bloch functions.

11. Kohn-Sham ansants • If you don’t like the answer, change the question! • Replacing one problem with another. • The Kohn-Sham approach is to replace the difficult interacting many-body system obeying the hamiltonian with a different auxiliary system that can be solved more easily.

12. Since there is no unique prescription for choosing the simpler auxiliary system, this is an ansatz that rephrases the issues. • The ansatz of Kohn and Sham assumes that the ground state density of the original interacting system is equal to that of some chosen non-interacting system.

13. This leads to independent-particle equations for the non-interacting system that can be considered exactly soluble with all the difficult many-body terms incorporated into an exchange-correlation functional of the density. • By solving the equations one finds the ground state density and energy of the original interacting system with the accuracy limited only by the approximations in the exchange-correlation functional.

14. K-S approach has let to the very useful approximations that are now the basis of most calculations of “first-principles” or “ab initio” predictions for the properties of condensed matter or large molecular systems. • The local density approximation(LDA) or various generalized-gradient approximations (GGA) are remarkably accurate. • Semiconductors, sp-bonded metals, insulators, NaCl, molecules with covalent or ionic bonding. • Also appeared to be successful to transition metals with stronger effects of correlations. • Still fails in many strongly correlated cases. • Great interests in improving the DFT approach • To build upon the many successes of current approximations and overcome the failures in strongly correlated electrons systems.

15. K-S ansatz for the ground state. • By far the most widespread way in which the theory has been applied. • In the big picture this is only the first step. • The fundamental theorems of DFT show that in principle the ground state density determines everything. • A great challenge in present theoretical work is to develop methods for calculating excited state properties.

16. Two assumptions of K-S ansatz: • The exact ground state density can be represented by the ground state density of an auxiliary system of non-interacting particles. • The non-interacting V representability. • No rigorous proofs for real systems of interest. • The auxiliary hamiltonian is chosen to have the usual kinetic operator and an effective local potential Veff(r) acting on an electron of spin at r. The local form is not essential, but it is an extremely useful simplification that is often taken as the defining characteristic of the K-S approach. • Veff(r) should depend upon spin to give the correct density for each spin.

17. The actual calculations are performed on the auxiliary independent-particle system defined by the auxiliary hamiltonian • . • The form of is not specified and the expressions must apply for all in some range to define functions for a range of densities. • For a system of N = N(up) + N(down) independent electrons obeying this hamiltonian, the ground state has one electron in each of the orbitals with the lowest eigenvalues of the hamiltonian.

18. The independent-particle kinetic energy Ts is given by: • And we define the classical Coulomb interaction energy of the electron density n(r) interacting with itself (the Hartree energy): Ehartree [n] = 1/2

19. The K-S approach to the full interacting many body problem is to rewrite the H-K expression for the ground state energy functional in the form • EKS = Ts[n] + . • Here Vext(r) is the external potential due to the nuclei and any other external fields. EII is the interaction between the nuclei. • The sum of the terms form a neutral grouping

20. The independent-particle kinetic energy Ts is given explicitly as a functional of the orbitals. • Ts for each spin must be a qunique functional of the density n® by application of the H-K arguments. • All many-body effects of exchange and correlation are grouped into the exchange correlation energy Exc.

21. Exc can be written in terms of H-K functional • Exc[n]=FHK[n]-(Ts[n]+Ehartree[n]). • Exc must be a functional because the right hand side is a functional. • If Exc is known, the exact ground state energy and density of the many-body electron problem could be found by solving the K-S equations for independent-particles. • K-S method provides a feasible approach to calculating the ground state properties of the manybody electron systems.

22. The K-S variational equations • Solution of the K-S auxiliary system for the ground state can be viewed as the problem of minimization with respect to either the density expressed as a functional of the orbitals but all other terms are considered to be functionals of the density. • The variational equation is: • Subject to the orthonormalization constraints.

23. This is equivalent to the Rayleigh-Ritz principle and the general derivation of the Schrodinger equation, except for the explicit dependence of Ehartree and Exc on n. • Use the earlier expression for n and Ts and the Lagrange multiplier method, the K-S Schrodinger-like equations: • (HKs -)Ψ(r)=0 • HKs(r)=-1/2 +VKS(r) • VKS = Vext + Vhartree + Vxc

24. are the eigenvalues and HKS is the effective hamiltonian. • The equations with a potential that must be found self-consistently with the resulting density. • These equations are independent of any approximation to the functional and would lead to the exact ground state density and energy for interacting system if Exc is known. • It follows from the H-K theorems that the ground state density uniquely determines the potential at the minimum, so that there is a unique K-S potential associated with any given interacting electron system.

25. Exc, Vxc and the exchange-correlation hole • The genius of the K-S approach is • By explicitly separating out the independent particle kinetic energy and the long range Hartree terms, • The remaining exchange-correlation functional can be reasonably be approximated as a local or nearly local functional of the density.

26. The energy Exc can be expressed in the form • Exc[n] = • is an energy per electron at point r that depends only upon the density in some neighborhood of point r. • Only the total density appears because the Coulomb interaction is independent of spin. • In a spin polarized system, this term incorporates in the information on the spin densities.

27. Although the energy density is not uniquely defined by the integral, a physically motivated definition follows from the analysis of the exchange correlation hole. • An informative relation can be found using the “coupling constant integration formula”. • Adiabatic connection. • For more information on the exchange correlation hole, read Richard Martin’s book.

28. The change in energy is given by • Exc[n] = • Here = • is the hole summed over parallel and antiparallel spins. • The exchange correlation density can be written as

29. This is an important result • The exact exchange-correlation energy can be understood in terms of the potential energy due to the exchange-correlation hole averaged over the interaction from e^2 =1 to e^2 =1. • For e^2=0, the wavefunction is just the independent-particle K-S wavefunction. • Since the density everywhere is required to remain constant as is varied, the exchange correlation density is implicitly a functional of the density in all space. • Exc[n] can be considered as an interpolation between the exchange only and the full correlated energies at the given density n(r).

30. The exchange-correlation hole obeys a sum rule that its integral must be unity. • For solids like Si, the exchanges dominates over correlations. • Remove the self-interaction term in the Hartree interactions.