1. Computational Physics (Lecture 23)

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 𝜌’, Ω 𝜌 ′ =𝑇𝑟[ 𝜌 ′ 𝐻−μ𝑁 + 1 β ln 𝜌 ′ ] , whose minimum is the equilibrium grand potential: Ω=Ω 𝜌0 ′ =− ln 𝑇𝑟 𝑒 −𝛽(𝐻−μ𝑁) ,
Where 𝜌0 ′ is the grand canonical density matrix
𝜌0 ′ = 𝑒 −𝛽(𝐻−μ𝑁) /Tr 𝑒 −𝛽(𝐻−μ𝑁) .

4. The proof is analogous to the Hohenberg-Kohn proofs and uses only the minimum property of Ω 𝜌 ′ 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 heat 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 current 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 electrons 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 led 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.

15. Great interests in improving the DFT approach
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.

16. 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.

17. 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.

18. The actual calculations are performed on the auxiliary independent-particle system defined by the auxiliary hamiltonian
𝐻 𝜎 𝑎𝑢𝑥=− 1 2 𝛻2+𝑉𝜎(𝒓).
The form of 𝑉𝜎(𝒓) is not specified and the expressions must apply for all 𝑉𝜎(𝒓) 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.

19. The independent-particle kinetic energy Ts is given by:
𝑇𝑠=− 1 2 𝜎 𝑖=1…𝑁𝜎 <Ψ𝑖|𝛻2|Ψ𝑖>= 1 2 𝜎 𝑖=1…𝑁𝜎 |𝛻Ψ𝑖|2
And we define the classical Coulomb interaction energy of the electron density n(r) interacting with itself (the Hartree energy): Ehartree [n] = 1/2 𝑑3𝑟 𝑑3 𝑟 ′ 𝑛 𝒓 𝑛 𝒓 ′ |𝒓− 𝒓 ′ |

20. 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] + 𝑑𝒓 𝑉𝑒𝑥𝑡 𝒓 𝑛 𝒓 +𝐸𝐻𝑎𝑟𝑡𝑟𝑒𝑒 𝑛 +𝐸𝐼𝐼+𝐸𝑥𝑐[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

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

22. 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.

23. 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:
𝛿𝐸𝐾𝑆 𝛿Ψ = 𝛿𝑇𝑠 𝛿Ψ + 𝛿𝐸𝑒𝑥𝑡 𝛿𝑛 + 𝛿𝐸𝐻𝑎𝑟𝑡𝑟𝑒𝑒 𝛿𝑛 + 𝛿𝐸𝑥𝑐 𝛿𝑛 𝛿𝑛 𝛿Ψ =0
Subject to the orthonormalization constraints.

24. 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 -𝜖𝑖)Ψ𝑖 (r)=0
HKs(r)=-1/2 𝛻2+VKS(r)
VKS = Vext + Vhartree + Vxc

25. 𝜖𝑖 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.

26. 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 approximated as a local or nearly local functional of the density.

27. The energy Exc can be expressed in the form
Exc[n] = 𝑑𝑟 𝑛 𝑟 𝜖𝑥𝑐(𝑛,𝑟)
𝜖𝑥𝑐(𝑛,𝑟) 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.

28. 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.

29. The change in energy is given by
Exc[n] = 0 𝑒2 𝑑λ<Ψ 𝑑𝑉𝑖𝑛𝑡 𝑑λ Ψ>−𝐸hartree 𝑛 =1/2 𝑑3𝑟𝑛(𝑟) 𝑑3 𝑟 ′ 𝑛𝑥𝑐(𝑟, 𝑟 ′ )/|𝑟−𝑟′|
Here 𝑛𝑥𝑐(𝑟, 𝑟 ′ )= 0 1 𝑑λ 𝑛𝑥𝑐λ 𝑟, 𝑟 ′ .
𝑛𝑥𝑐λ 𝑟, 𝑟 ′ is the hole summed over parallel and antiparallel spins.
The exchange correlation density can be written as
𝜖𝑥𝑐( 𝑛 ,𝑟)=1/2 𝑑3 𝑟 ′ 𝑛𝑥𝑐(𝑟, 𝑟 ′ )/|𝑟−𝑟′|

30. 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 e2 =0 to e2 =1.
For e2=0, the wavefunction is just the independent-particle K-S wavefunction.
Since the density everywhere is required to remain constant as λ 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).

31. For solids like Si, the exchanges dominates over correlations.
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.

32. Functionals for exchange and correlation
The exchange and correlation functional can be reasonably approximated
As a local or nearly local functional of the density.
The exact functional must be very complex!

33. The local spin density approximation (LSDA)
Kohn and Sham showed in their seminal paper that the exchange and correlation function is generally local for solids, because solids can often be viewed as close to the limit of homogeneous electron gas.
Thus they proposed the local spin density approximation, so that the exchange correlation energy is an integral over all space with the exchange correlation energy density at the point assumed to be the same as in a homogeneous electron gas with the same density

34. Exc(LSDA) = 𝑑3𝑟 𝑛 𝑟 𝜖 hom 𝑒𝑐 (𝑛 𝑢𝑝 ,𝑛(𝑑𝑜𝑤𝑛)
The LSDA is the most general local approximation and is given explicitly for exchange (proportional to n1/3)and by approximate (or fitted) expressions for correlation.
In PZ, the exchange and correlation follow a similar form.
Read chapter 5 of Richard Martin’s book for detailed description.

35. This is not justified by a formal expansion in some small parameter.
The rationale for the local approximation is that for the densities typical of those found in solids
The range of effects of exchange and correlation is rather short.
This is not justified by a formal expansion in some small parameter.
It will be the best for solids close to a homogeneous gas (like a nearly free electron metal) and worst for very inhomogeneous cases.

36. The self interaction term can be cancelled by the non-local exchange interaction in Hartree-Fock.
However, in LDA, the cancellation is approximate and there remain self-interaction terms.

37. Generalized gradient approximations (GGA)
The success of the LSDA has led to the development of various generalized-gradient approximations
With improvement over LSDA.
In the chemistry community, GGA can provide the accuracy that has been accepted.

38. The first step beyond the local approximation is a functional of the magnitude of the gradient of the density as well as the value n at each point.
Which was suggested in K-S’s original paper.
Gradient expansion approximation doesn’t have to be better because it violates the sum rule and other relevant conditions.

39. The term GGA denotes a variety of ways proposed for functions that modify the behavior at large gradients in such a way as to preserve desired properties.
ExcGGA= 𝑑3𝑟 𝑛 𝑟 𝜖𝑥𝑐(𝑛𝑢𝑝, 𝑛𝑑𝑜𝑤𝑛, 𝛻𝑛𝑢𝑝 . 𝛻𝑛𝑑𝑜𝑤𝑛 )

40. Perdew and Wang (PW91), Perdew, Burke and Enzerhof (PBE) all proposed forms of the expansion of GGA.
Many GGA functionals that are used in quantitative calculations in chemistry.
Correlation is often treated using Lee Yang Parr (LYP).
Krieger and coworkers have constructed a functional KCIS based upon many-body calculations of an artificial jellium with a gap problem.

41. Hybrid functionals
The form of the coupling constant integration for the exchange-correlation energy is the basis for constructing a class of functionals called hybrid.
Because they are a combination of orbital-dependent Hartree-Fock and an explicit density functional.
The most accurate functionals available as far as energetics is concerned and the method of choice in the chemistry community.

42. The hybrid formulation arises by approximating the integral in terms of information at the end points and the dependence as a form of the coupling constant.
When the coupling constant is zero, it is Hartree-Fock exchange energy.
The potential part of the LDA and GGA functional is most appropriate at full coupling when the coupling constant is 1.

43. Therefore, it is possible to approximate the functional by assuming a linear dependence on the coupling constant
Leading to a half and half form:
Exc = ½(E_xHF + E_xcDFA)
DFA denotes an LDA or GGA functional.
Later Beck presented parameterized forms that are accurate for many molecules, such as B3P91, a three parameter functional that mixes HF exchange, the exchange functional of Becke and correlation from PW.

44. Perdew, Ernzerhof and Burke proposed the form:
Exc=E_xcLDA+1/4(E_xHF-E_xcDFA)
The ¼ was found by fitting.
Hybrid can be used in different ways.
Not strictly within the usual K-S approach if the H_F equations are solved with a non-local exchange operator.

45. Solving K-S equations
A straightforward way to solve the Kohn-Sham equations is to follow the condition that the effective potential and the density are consistent
i.e. to obtain a self-consistent solution.

46. The first step is to make an initial guess of the density first;
then calculate the corresponding effective potential and solve the Kohn-Sham equations;
from the results, the new electron density can be calculated;
if the density and the effective potential are consistent, the energy, force, stresses, eigenvalues etc can be achieved and the calculation is done;
otherwise, the effective potential can be recalculated based on a linear combination of previous densities and the process is repeated.
In this step, a so called steepest decent method can be applied to find the minimized energy. (Also, CG is widely used in many packages.)

48. To solve the Kohn-Sham equations, a basis set need to be chosen.
There are two types of basis sets:
one is based on plane waves;
the other is based on localized orbits.
Plane waves and grids are two major methodologies to solve differential equations.
Plane waves are especially appropriate for periodic crystals calculation
and localized orbits (grids) are suitable for finite systems. In modern electronic structure calculations, both methods are extensively applied with fast Fourier transformation.

49. Pseudopotential.
To achieve accurate electronic calculation results, the Coulomb potential of the nucleus and the effects of the core electrons need to be included.
In this sense, an all electron calculated potential is necessary.
However, the Coulomb potential of the nucleus is usually strong and the core electrons are usually tightly bound, which usually results in a high kinetic energy and a lot of terms in Fourier transform which affect the calculation efficiency.
Also, the core electrons have a small effect in valence electron bonding.

50. One way to improve the calculation efficiency is to replace all electron calculated potential with a so-called “pseudopotential”.
Instead of solving the difficult problem of the Coulomb potential of the nucleus, an effective ionic potential acting on the valence electrons can be calculated.

51. This potential is called pseudopotential.
A pseudopotential can be generated in an atomic calculation and the calculation result can be the input of calculations of properties of valence electrons in solids or molecules, because the core states almost remain unchanged in these situations
except for some extreme conditions, such as very high pressures.

52. Pseudo-potentials can usually be constructed in two parts, the local (l-independent) part plus the non-local (l-dependent) part (l is the orbit quantum number)
To preserve the right physics of the all electron case, pseudo-potentials usually satisfy a so-called norm conserving condition: the eigenvalues and the orbits are required to be the same for the pseudo and the all electron case for r>Rc,
each potential Vl(r) equals to the local (l-independent) all electron potential, and for r->∞.

53. Schematic illustration of a pseudo-potential.

54. The ab-initio norm-conserving and ultra-soft pseudopotentials are the basis of the accurate electronic calculations.
One goal of creating potential is to make the potential as smooth as possible, because in plane wave calculations, the potential is expanded in Fourier components and the performance of the calculation is to the power of the number of Fourier components.

55. One way to achieve the smoothness is to introduce the concept of “ultra-soft pseudopotentials”. “Ultra-soft pseudopotential” is a practical approach for solving equations beyond the applicability of those formulations

56. The concept of ultra-soft pseudopotential was proposed by Blochl and Vanderbilt
They rewrote a non-local potential in a form involving a smooth function that is not norm conserving

57. Schematic illustration of an all electron calculated wave function (solid line), a norm conserving pseudo-potential calculated one (dotted) and an ultra-soft pseudo-potential calculated one(dashed)
As we can see, the ultra-soft pseudo-potential yields the smoothest wave function among the three and is the most efficient in calculation.

58. Projector augmented waves (PAW)
The projector augmented wave method is a general approach to solution of the electronic structure problem
That formulates the orthogonalized plane wave method.
Like the ultrasoft PP mthod, it introduces projectors and auxiliary locallized functions.
The difference is that PAW approach keeps the full all-electron wave function in a form similar to the general OPW expression.