Notes on Density-functional theory · on Density Functional Theory. DFT is really hard. It’s good to know 295b stu before learning the theory, since both are theories of interacting

Notes on Density-functional theory

Jimmy Qin

Winter 2018 - ?

These are notes on DFT from stuff recommended by Grigory Kolosev. Mostly Nogueira, A Primeron Density Functional Theory. DFT is really hard. It’s good to know 295b stuff before learningthe theory, since both are theories of interacting electrons.

I would like to learn more about TDDFT. I heard that you can incorporate Keldysh Green func-tions into TDDFT. A related technique is the MBPT technique in chapter 5 of Primer, which Ishould read. A reference on Keldysh in DFT is arxiv.org/abs/cond-mat/0506130.

I remember that I wrote these notes around the same time that I went to Disney World for thefirst time in perhaps ten years and that I was thinking about some DFT issue at Disney. I got totake pictures with Cinderella.

Chapter 0. What is DFT?

DFT is a reformulation of nonrelativistic quantum mechanics, based on the Hohenberg-Kohntheorems, which are given below. It is relevant for the ground state solution of the Schrodingerequation. Walter Kohn and John Pople (who wrote some code) won the Nobel prize in chemistryfor their work on this method in 1998.

1. The external potential, and hence ground-state energy, is a unique functional of the electronprobability density. There exists a 1:1 mapping between the ground-state wavefunction |Ψ〉and the ground-state electron density, n(~r).

2. The electron density n(~r) that minimizes E(n) is, in fact, the exact ground-state density.

This is good because it reduces the df in the system. A many-body system may have N electronsand hence be formulated in 3N -dimensional space, but the DFT formulation has only one function,n(~r), which exists in 3D space.

Kohn-Sham equations

The energy functional to be minimized begins life in terms of the single-electron wavefunctions,ψi:

E[ψi] = Eknown[ψi] + EXC[ψi].

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arxiv.org/abs/cond-mat/0506130

Jimmy Qin Chirality and Helicity

For some reason, Sholl writes these functionals in terms of the wavefunctions, but I think theyshould be written in terms of the density, n(~r). Eknown contains the kinetic term, external potential,and electron-electron interaction terms. EXC contains everything else. It is called the exchange-correlation functional.

Thus far, there has been no improvement on the many-body Schrodinger equation, since we havenot reduced the df in the above functionals. Kohn and Sham invented a method which involvessolving the Kohn-Sham equations

[~2

2m∇2 + V (~r) + VH(~r) + VXC(~r)]ψi = εiψi

for each electron i in the system. The good thing about this is that the wavefunctions are decou-pled. Well, not really, because the various potentials are functions of the density:

• V (~r) is the external (applied) potential.

• VH(~r) is the Hartree potential,

VH(~r) = e2

∫d3~r′

n(~r′)

|~r − ~r′|.

This actually contains a self-interaction contribution (electron i interacting with itself) whichis subtracted out in the exchange term.

• VXC = δEXC(~r)δn(~r)

is the exchange-correlation potential. It is defined as a functional derivativeof the exchange functional, with respect to the density.

Here’s the general form of the algorithm to solve the Kohn-Sham formulation of DFT:

1. Define initial trial density, n(~r).

2. Solve for ψi(~r) using KS equations.

3. Calculate an updated n(~r) = 2∑

i |ψi(~r)|2.

4. Again solve for ψi(~r).

Chapter 1. Introduction to DFT

Some topics covered in this chapter are the local spin-density functional and the generalizedgradient approximation.

Kohn-Sham spin-density functional theory

Often, the external potential is a function of spin, σ =↑, ↓ . So we have to split their densities.Suppose there are N total electrons and an external potential V ext(~r).

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The auxiliary one-body Schrodinger equation to be solved in the Kohn-Sham method is

(−∇2

2+ V ext + U + V σ

XC)ψασ(~r) = εασψασ(~r) .

Here, the Hartree energy U is a function of the total density n, but the exchange term is a functionof the separate spin densities n↑, n↓. α is the energy level for the particular spin σ. You need tosolve for the lowest levels of α (and correspondingly, the wavefunctions ψασ) because the totalN -particle wavefunction is antisymmetrized, so the individual wavefunctions must all be unique.

The density (for a particular spin) is

nσ(~r) =∑α

θ(µ− εασ)|ψασ(~r)|2 = N∑

σ2,··· ,σN

∫d3~r2 · · · d3~rN |Ψ(~rσ, ~r2σ2, · · · , ~rNσN)|2.

The former definition contains a Heaviside function that essentially acts as a Fermi-Dirac distri-bution - it cuts the occupation levels of the electrons off to preserve the number normalization,

N =∑σ

∫d3~rnσ(~r) .

The latter definition is obtained by fixing one of the electrons, with spin σ, at the position ~r andallowing the other N − 1 electrons to vary over position and spin.

Derivation of Kohn-Sham equations

Here is a very clear and constructive derivation of the Kohn-Sham equations, due to Levy. Thereis more physics in it than in the original by Hohenberg and Kohn.

We know E = minΨ〈Ψ|H|Ψ〉. We will separate this minimization into two steps, then use a thirdstep to constrain particle-number normalization.

1. Minimize E over all |Ψ〉 that give the same n(~r). This is useful because Vext depends onlyon n(~r). This part of the problem can be written

minΨ→n〈Ψ|T + Vee|Ψ〉+

∫d3~rV ext(~r)n(~r).

2. Minimize the result of step 1 over all possible densities, n(~r). If we define the functional

F [n] = minΨ→n〈Ψ|T + Vee|Ψ〉 ,

this step is equivalent to

minn

(F [n] +

∫d3~rV ext(~r)n(~r)).

The difficulty is that the passage from 〈Ψ|T + Vee|Ψ〉 → F [n] is generally not known. Doingso will involve the exchange functional, which we will meet later.

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3. Constrain particle number through a Lagrange multiplier:

δ

δn(~r)[F [n] +

∫d3~rV ext(~r)n(~r)− µ

∫d3~rn(~r)] = 0 =⇒ δF

δn(~r)+ V ext(~r) = µ .

µ is, in fact, the chemical potential introduced in the δ-functions from earlier to enforceparticle-number conservation. You tweak µ until

∫d3~rn(~r) = N is satisfied for the solution

n(~r) in the above equation.

This is typically generalized to n(~r) → nσ(~r) if V ext is spin-dependent, etc. The density n(~r)is called N-representable if it is obtainable from an antisymmetric N -particle wavefunction,and V ext-representable if it is obtainable from an antisymmetric N -particle wavefunction in thepresence of the external potential.

Now, we can derive the Kohn-Sham equations through functional differentiation. The point isto map this interacting system (1) onto an auxiliary non-interacting system (2). System (2) isdefined to have no electron-electron interactions, Vee = 0. All the interactions from system (1) aremapped onto the external potential of system (2). Thus,

F2[n] = minΨ→n〈Ψ|T |Ψ〉 non-interacting system only.

Now, define the exchange functional for system (1), EXC [n], by

EXC [n] := F [n]− T [n]− U [n] .

T is the kinetic energy and U is the Hartree self-energy, which is easily calculable. We see,therefore, that EXC contains the exchange energy, for example, of the Hartree-Fock method. Whilethe Hartree-Fock method expresses the exchange energy in terms of the individual wavefunctions|ψi〉, DFT expresses it in terms of the density, n. Additionally, the N -body wavefunction |Ψ〉that corresponds to the optimal density n may not be a simple Slater determinant of individualwavefunctions - it might also be a linear combination of Slater determinants, or something likethat.

We would like to find VS(~r) for the non-interacting system and force n(~r) for this auxiliary systemto be exactly the same as the optimal n(~r) for the original, interacting system. We may absorb anydifference in the Lagrange multiplier µ for the two systems into VS(~r), so the Lagrange equationsbecome

System (1):δF

δn+ V ext = µ. System (2):

δT

δn+ VS = µ.

Subtracting one from the other and using the definition of exchange functional gives

VS = V ext +δU

δn+δEXCδn

.

The above equation is the central equation of Kohn-Sham theory. Note that VS and V ext arefunctions of position and U and EXC are functionals of the density n(~r). Basically, the wholepoint is to try to find approximations for EXC . Then, you just take a functional derivative to getthe Kohn-Sham potential, which you can use to numerically solve the one-particle Schrodingerequation.

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Exchange and correlation energies

We split the exchange-correlation energy functional into two parts, the exchange and the corre-lation part:

EXC [n] = EX [n] + EC [n] .

Let’s try to get some intuition for what the exchange and correlation energies actually are. Thisincludes stuff from the coupling-constant integration method, to be covered a little later. Themain points are this (heuristic only):

• EX [n] removes the spurious self-interaction energy in the Hartree energy, U [n]. It alsoaccounts for the Pauli exclusion “force,” which tends to push electrons with the same spinaway from each other (like the exchange term in the Hartree-Fock method). It is present ineven in systems with zero coupling constant, λ = 0, so it is present even in a Kohn-Shamnoninteracting (Vee = 0) system. For 1 electron, EXC [n] = −U [n].

• EC [n] is always negative or zero. For 1 electron, EC [n] = 0. The correlation functional is,heuristically, how much better the Ψ wavefunction, which is optimized for the energy T+Vee,is compared to the Φ wavefunction, which is optimized for the energy T only, at minimizingT + Vee. So, EC is the effect of the interaction Vee. EC accounts for the Coulomb repulsionforce between electrons. It is not present for λ = 0.

Now, let’s define the exchange and correlation functionals rigorously. Let Φminn minimize 〈T 〉 for

the density n(~r); let Ψminn minimize 〈T+Vee〉 for the same density n(~r). (So obviously, Ψmin

n = Φminn

in the case of just 1 electron, since there is no interaction Vee.) The functionals are defined as

EX [n] = 〈Φminn |Vee|Φmin

n 〉 − U [n] ,

EC [n] = F [n]− (TS[n] + U [n] + EX [n]) = 〈Ψminn |T + Vee|Ψmin

n 〉 − 〈Φminn |T + Vee|Φmin

n 〉 .

The facts above in the bullet points are obvious from the above definitions. For 1 electron,EXC [n] = −U [n] and cancels the spurious self-interaction in the Hartree functional.

Coupling-constant integration

This still leaves open the question: how to actually obtain the functionals EX [n], EC [n]? To do so,we will use a method that involves integrating over an artificial parameter λ, through the followingtheorem. You can prove it easily.

Feynman-Hellmann Theorem: Suppose Hλ depends on a parameter λ. Obviously, the corre-sponding eigenfunction |Ψλ〉 will depend on the same parameter. Letting Eλ = 〈Ψλ|Hλ|Ψλ〉, wehave the “force” w.r.t. λ,

dEλdλ

= 〈Ψλ|∂Hλ

∂λ|Ψλ〉 .

The ingenious set-up is this: define Ψλn as the normalized, antisymmetric wavefunction giving n(~r),

which minimizes 〈T + λVee〉. Obviously, the boundary cases are

Ψ1n = Ψmin

n ,Ψ0n = Φmin

n .

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Therefore, λ = 1 corresponds to the physical system and λ = 0 corresponds to the Kohn-Shamauxiliary (non-interacting) system. In systems with λ = 0, the correct many-body wavefunctionis typically a simple Slater determinant.

Because F [n] = 〈T + λVee〉λ=1 and T [n] = 〈T + λVee〉λ=0, we have

EXC [n] = (

∫ 1

0

dλd

dλ〈T + λVee〉λ)− U [n] =⇒ EXC [n] = (

∫ 1

0

dλ〈Vee〉)− U [n] .

The last step follows from Feynman-Hellman (i.e. T has no λ-dependence).

We can also recast this in the language of holes!

Hole interpretation: A hole is a local deficit of electron density, i.e. where n(~r) is relativelylower. This is an idea of conditional probability. Basically, if you know an electron is located at~r, the other N − 1 electrons are pushed away (both by Coulomb and exclusion principle, and bythe self-interaction correction, i.e. you need to take away some local density to correct for theself-interaction) and therefore there is a local deficit of density around ~r.

To make this precise, introduce the one- and two-electron reduced density matrices, ρ1(~r′σ,~rσ)and ρ2(~r′, ~r). In ρ1, we do not sum over the spins; in ρ2, we do. These are just mathematicaldefinitions. The physical interpretations will come later, when we combine with other things.Explicitly,

ρ1(~r′σ,~rσ) = N∑

σ2,··· ,σN

∫d3~r2 · · · d3~rNΨ∗(~r′σ,~r2σ2, · · · , ~rNσN)Ψ(~rσ, ~r2σ2, · · · , ~rNσN).

ρ2(~r′, ~r) = N(N − 1)∑

σ1,··· ,σN

∫d3~r3 · · · d3~rN |Ψ(~r′σ1, ~rσ2, · · · , ~rNσN)|2.

How can we interpret these reduced density matrices? Note that ρ2(~r′, ~r) is the joint density offinding an electron at ~r′ and another one at ~r. If n2(~r, ~r′) is the electron density at ~r′ given thatthere is an electron at ~r (i.e. a conditional probability, kind of), then

ρ2(~r′, ~r) = n(~r)n2(~r, ~r′).

Now, define the density of an exchange-correlation hole at ~r′ about an electron at ~r,nλXC(~r, ~r′), by

nλXC(~r, ~r′) = n2(~r, ~r′)− n(~r′) =⇒∫d3~r′nλXC(~r, ~r′) = −1 ⇐⇒

∫d3~r′nλ2(~r, ~r′) = N − 1.

The hole has opposite charge, as evidenced by its normalization. Above, everything is implicitly afunction of λ, because that determines the many-particle wavefunction, and hence n(~r) and n2(~r).

Okay, now we can use this stuff to re-express EXC [n]. We know that∫ 1

0

dλ〈Ψλn|Vee|Ψλ

n〉 =1

2

∫ 1

0

dλ

∫d3~rd3~r′

n(~r)n2(~r, ~r′)

|~r − ~r′|and U [n] =

1

2

∫d3~rd3~r′

n(~r)n(~r′)

|~r − ~r′|,

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so the definition ρ2(~r, ~r′) = n(~r)n2(~r, ~r′) gives

EXC [n] =1

2

∫d3~rd3~r′

n(~r)nXC(~r, ~r′)

|~r − ~r′|, where nXC(~r, ~r′) =

∫ 1

0

dλnλXC(~r, ~r′).

We sometimes split the exchange-correlation hole into an exchange hole and a correlation hole,as

nXC(~r, ~r′) = nX(~r, ~r′) + nC(~r, ~r′), where nX(~r, ~r′) := nλ=0XC (~r, ~r′).

Note that although this gave us (more of) a physical interpretation, I think it still didn’t makecomputational life that much easier. Now we have to analytically determine Ψλ

n or something! Didit make life any easier?

Some properties of DFT functionals

Let’s get a better feel for the various functionals, like U [n], T [n], EX [n], EC [n] that appear in DFT.

Uniform coordinate rescaling: Consider the following rescalings of the many-body wavefunc-tion Ψ(~r1, · · · , ~rN) and the corresponding density, n(~r).

Ψγ(~r1, · · · , ~rN) = γ3N/2Ψ(γ~r1, · · · , γ~rN) and nγ(~r) = γ3n(γ~r).

Both of these rescalings preserve particle-number normalization, and 〈Ψγ|Ψγ〉 = 〈Ψ|Ψ〉. To thinkabout this, note that the density and wavefunction are smaller in volume, but more concentrated,for γ > 1. So, γ > 1 means you have the same number of electrons but proportionally squish themcloser together.

Recall the “constrained search,” where we wanted to find the best Ψ for a given n; namely, thesearch is constrained to those many-body wavefunctions Ψ that yield the given n(~r). The goodthing about this rescaling is that Ψγ yields nγ(~r). It’s easy to show.

It’s easy to show that the Hartree energy, non-interacting kinetic energy, and exchange energyscale as

U [nγ] = γU [n], TS[nγ] = γ2TS[n], EX [nγ] = γEX [n].

We had to use the non-interacting kinetic energy above, rather than the interacting kinetic energy,because minimizing 〈T 〉 gives the correct scaling, but minimizing 〈T+Vee〉mixes two different kindsof scaling. More on this in the next paragraph, about EC [n]. It’s no surprise that U [n] and EX [n]scale in the same way, since they are the same kind of energy (electron-electron interaction).Kinetic energy dominates in the high-density, γ → ∞ limit; interactions dominate in the low-density, γ → 0 limit.

EC [n] is not so simple. That is because (refer to earlier definition) it is based on the minimizationof 〈T + Vee〉. T scales as γ2 and Vee scales as γ, so they are not compatible. However, 〈T + γVee〉scales as γ2. This is just like modifying the problem to contain a coupling constant; the result is

EC [nγ] = γ2E1/γC [n].

To foreshadow the LDAs we will study later, suppose

F [n] =

∫d3~rf(n(~r)) and F [nλ] = λpF [n].

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You can show this implies f(λ3n) = λp+3f(n) =⇒ f(n) ∝ n1+p/3. This kind of LDA canbe applied, for instance, to the kinetic energy; it basically returns the Thomas-Fermi theorythat foreshadowed DFT. It cannot be applied to something manifestly nonlocal like U [n] =12

∫d3~rd3~r′ n(~r)n(~r′)

|~r−~r′| . Obviously there are two locations (~r and ~r′) involved in U [n]!

Spin-density functionals: certain functionals can be separated into spin-density functionals,like

TS[n↑, n↓] = TS[n↑] + TS[n↓].

This kind of decomposition also holds for the exchange functional, EX [n], but not for the corre-lation functional, EC [n]. That is because the exchange functional has to do with self-interactionsand the Pauli exclusion principle; the Pauli exclusion “force” is non-existent between electrons ofdifferent spin. However, the correlation functional has to do with Coulomb repulsion, which ispresent between all electrons regardless of spin.

Derivative discontinuity

This is kind of a gnarly little topic. I’ll do my best.

DFT as we have studied up to now has incorporated a fixed particle number, N =∫d3~rn(~r).

Now we extend the “constrained search” to systems that may not have integer particle number.It turns out that: the ground-state energy E0(N) varies linearly between two adjacent integersand has a derivative discontinuity at each integer.

Because the chemical potential is µ = ∂E∂N

, the chemical potential (and hence, where the electronoccupations get cut off) changes discontinuously at every integer, N = Z.

Now I follow the original paper by Perdew, Parr, Levy, and Balduz. Phys. Rev. Lett. 49, no.23, “Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of theEnergy.”

Generally, we constrain particle number with a Lagrange multiplier µ that turns out to be thechemical potential. For example, demand

0 = E0[n]− µ∫d3~rn(~r) =⇒ δE0[n]

δn= µ.

What about E0[n] for general, non-integer values of particle number, N? We perform the followingthought experiment.

Claim: E0 cannot be differentiable at integer values of N .Reasoning: Suppose X and Y are different neutral atoms at different chemical potentials, µY <µX . Shift a small amount of electron density, δN , from X to Y . Then δEX+Y = (µY −µX)δN < 0,so it is energetically favorable to have Y carry a net negative charge and X carry a net positivecharge. Contradiction.

Generally, fractional electron number can arise as a time-average in an open system. To accountfor this, we introduce an ensemble or statistical mixture Γ, which is a collection of pure states(i.e. fixed-particle-number states) and their probabilities:

Γ = Ψi, pi =⇒ 〈O〉 =∑i

pi〈Ψi|O|Ψi〉.

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Now, suppose∫d3~rn(~r) = N = M + ω, where M = bNc and ω = N −M . As before, define the

constrained search functional F over the “allowed ensembles” Γ which give the (perhaps fractionalparticle number) density n(~r):

F [n] = minΓ→n〈T + Vee〉Γ.

Claim: If the pure-state energies En are concave-up in n for n ∈ Z, then the above constrainedsearch will yield

EN = (1− ω)EM + ωEM+1 for N = M + ω.

Proof : Central to this theorem is the fact that the pure-state energies (i.e. ground-state energieswhen the particle number is fixed) are concave-up in particle number. The result of the theoremis that for fractional particle number, you just linearly interpolate between the nearest two integerparticle numbers. Neat!

Suppose the pure-state wavefunctions and their corresponding probabilities are Γ = Ψi, pi. Iclaim that to minimize the energy, pi = 0 for all i 6= M,M + 1 and that pM = 1 − ω, pM+1 = ω.This is due to the constraints ∑

i

pi = 1 and∞∑

j=−∞

jpM+j = ω.

Basically, there are very few constraints and hence a lot of “floating” parameters. The fact that12(EM−j+EM+j) > EM means that it is energetically favorable to put all the “floating” probability

in pM . Another way to restate this is that∑k

kpM+k = 0,∑k

pM+k = const. =⇒∑k

kEM+k ≥ EM , with equality iff pM+k = 0 for all k 6= 0.

This follows from Jensen’s inequality, f(E(X)) ≤ E(f(X)). The concave-up condition on integer-particle-number energies is a very weak requirement and almost always satisfied.

Uniform electron gas

The uniform electron gas is a paradigm for the application of DFT to interacting electrons. Theuniform electron gas has n(~r) = const. over ~r =⇒ N = ∞. You can assume a positive uniformbackground, which ensures overall charge neutrality. (Some simple metals, like Na, act like anelectron gas.)

We study all of the important functionals for the uniform electron gas.

• Kohn-Sham potential: The external potential vs(~r) is uniform, by symmetry. We can set

vs(~r) = 0 .

• Kohn-Sham orbitals: Instead of infinite volume, we assume finite volume V , just for sim-plicity. The Kohn-Sham orbitals are

ψ~k =1√Vei~k·~r, N = nV =

k3F

3π2V, kF = Fermi momentum.

The Fermi momentum is equivalent to the imposition of chemical potential, N = 2∑

~k θ(µ−E(~k)). It’s a classic calculation.

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• Kohn-Sham non-interacting kinetic energy: The energy density (i.e. per electron), assumingeach has kinetic energy k2/2, is

tS(n) :=T [n]

N=

3

5

k2F

2.

• Kohn-Sham exchange energy: For the Kohn-Sham non-interacting system, Φminn is a Slater

determinant. Then, the one- and two-electron density matrices are

ρ1(~r′σ,~rσ) =∑α

θ(µ− εασ)ψ∗ασ(~r′)ψασ(~r)

ρ2(~r′, ~r) = n(~r)n(~r′) + n(~r)nX(~r, ~r′) , where nX(~r, ~r′) = −∑σ

|ρ1(~r′σ,~rσ)|2

n(~r).

The exact exchange energy is

EX [n] =1

2

∫d3~rd3~r′

n(~r)nX(~r, ~r′)

|~r − ~r′|.

Proofs of the above: If you assume Φminn is a Slater determinant, the above things aren’t so

hard to get. They even make sense physically.

First, ρ1. The many-body wavefunction is Ψ(~rσ,~r2σ2, · · · , ~rNσN ) = 1√N !

detij ψiσi(~rj). Basically,

when you insert this expression into the definition of ρ1, you’ll get some integrals that look like,for example, ∑

σi,σj

∫d3~r2ψiσi(~r2)ψjσj (~r2),

which vanishes for i 6= j by orthogonality of the individual wavefunctions. Thus, the density matrixρ1(~r′σ,~rσ) completely vanishes unless you “line things up” correctly. For example, the followingorder of the wavefunctions, where the ordering corresponds to their position ~r, is fine (here, wewould identify σ4 → σ:

Ψ∗ ⊃ ψ∗4σ4(~r′)ψ∗3σ3

(~r2)ψ∗8σ8(~r3) · · · ⇐⇒ Ψ ⊃ ψ4σ4(~r)ψ3σ3(~r2)ψ8σ8(~r3) · · · .

There is an overall factor of 1N ! from the many-body wavefunction. Then, there is a factor of

(N − 1)! from the ordering of the terms being integrated over. When you multiply by N as in thedefinition of ρ1, there is no more prefactor. Check!

Now for ρ2. Let’s expand some relevant expressions - this makes it immediately obvious.

ρ2(~r′, ~r) =∑i 6=j|ψi(~r′)|2|ψj(~r)|2 − ψ∗i (~r′)ψj(~r′)ψ∗i (~r)ψj(~r).

n(~r′)n(~r) =∑i

|ψi(~r′)|2|ψi(~r)|2 +∑i 6=j|ψi(~r′)|2|ψj(~r)|2.

nX(~r, ~r′) = − 1

n(~r)[∑i

|ψi(~r′)|2|ψi(~r)|2 +∑i 6=j

ψ∗i (~r′)ψj(~r

′)ψ∗i (~r)ψj(~r)].

Now for the exchange energy. This follows because the correlation energy is zero for λ = 0.

The calculation of nX is in the Primer. You can calculate the exchange energy per electron,

eX [n] =1

2N(

∫d3~rn)(

∫d3~u

nX(~r, ~r + ~u)

u) = − 3

4πkF .

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• Correlation energy: Exact analytic expressions for ec(n), the correlation energy per electron,are known only in extreme (n→∞ and n→ 0) limits. Introducing the Seitz radius

rs := (4πn/3)1/3,

the low-density limit, for example, is

ec(n)→ −d0

rs+

d1

r3/2s

+ · · · as rs →∞.

• Equilibrium: The uniform electron gas is in equilibrium when

∂

∂n(ts(n) + ex(n) + ec(n)) = 0.

This gives rs ≈ 4.1, which is close to the observed valence electron density of sodium.

Linear response to external potential

We would like to study the linear response of the density to a small perturbation in the externalpotential. In other words, given δv(~r′), what is δn(~r)? To first order, the perturbation looks like

δn(~r) =

∫d3~r′χ(~r − ~r′)δv(~r′) to first order.

For a plane wave, define the following (with major abuse of notation):

δn(~r) = ei~q·~rδn(~q), δv(~r) = ei~q·~rδv(~q), χ(~q) =

∫d3~xe−i~q·~xχ(~x) =⇒ δn(~q) = χ(~q)δv(~q).

For any non-interacting system, including the Kohn-Sham auxiliary system,

χs(~q) =−kFπ2

F (q

2kF), where F (x) = Lindhard function =

1

2+

1− x2

4xln |1 + x

1− x|.

However, the external potential δv(~r) isn’t the only contribution to the Kohn-Sham potential:

δvs(~r) = δv(~r) + δ(δU

δn(~r)) + δ(

δEXCδn(~r)

).

You can check in the Primer that the Fourier transform is

δvs(~q) = δv(~q) +4π

q2(1−GXC(q))δn(~q),

where GXC(q) = γXC(q)( q2kF

)2 is called the local-field factor. Defining εs(~q) = 1 − 4πq2 (1 −

GXC(q))χs(q), the density response function is

χ(q) =χs(q)

εs(q).

11


There are some interesting low-q and high-q limits, etc. that you can take.

The second-order change, δE, in the total energy, follows from the Hellmann-Feynman theorem.Let λ ∈ [0, 1] and let

vλ(~r) = λδv(~r) and nλ(~r) = n+ λδn(~r).

We then have

δE =

∫ 1

0

dλ

∫d3~rnλ(~r)

d

dλvλ(~r) =

1

2δn(−~q)δv(~q).

The above hinges on the vanishing of∫ 1

0dλ

∫d3~rnδv(~r). I guess that’s true if v(~r) has no v(~q = 0)

component. In other words, the spatial average of v(~r) must be zero and there is no constant“shift.” Otherwise, there would be a first-order change in the total energy (of course!)

Derivation of the Lindhard function:

LSD Approximation

The local spin density approximation (LSD) for the exchange-correlation energy was pro-posed in the original work of Kohn and Sham. We get to use the method of separating T [n] andEX [n], etc. into the up- and down- spin components that we covered earlier, and also the densityt[n] and ec[n], etc. that we just covered for the uniform electron gas. For any energy componentG, the LSD approximation is

GLSD[n↑, n↓] =

∫d3~rn(~r)g(n↑(~r), n↓(~r)).

Here, g(n↑(~r), n↓(~r)) is the energy per particle in an electron gas with uniform spin densities n↑(~r)and n↓(~r), and n(~r)d3~r = (n↑ + n↓)d

3~r is the average number of electrons in that region. This isexact for uniform density and good for slowly-varying density. It is not usable for densities thatvary strongly with ~r.

According to the classic (original) paper: Hohenberg and Kohn, Phys. Rev. B 136, 864 (1964),“Inhomogeneous Electron Gas,” this is because the energy density is generally a functional of the(spatially-varying) density n(~r). Rotational symmetry constrains the expansion to be of the form

g[n] = g(n) + g2(n)∇n · ∇n+ g3(n)∇2n+ · · · .

So, the LSD approximation amounts to taking the leading term in this gradient expansion, whichis good for slowly-varying densities. The full analysis of LSD’s strengths and weaknesses is quiteinvolved. See the Primer.

GEA Approximation

Now, let’s meet the gradient expansion approximation and see why it doesn’t work very well.The upshot is that, if ~u is the distance from an electron to its corresponding exchange-correlationhole, taking terms from the gradient expansion of g[n] above improves behavior at small u butmakes stuff worse at large u - in fact, it destroys hole-number normalization, which is a problemwe can’t tolerate.

12


However, the GEA approximation for TS[n] is just fine, because there is no electron-hole interactionto worry about; TS[n] is manifestly local. For noninteracting kinetic energy, “GEA is its ownGGA.”

Just to see what this looks like, define the reduced density gradient s := 12kF

|∇n|n. Because

sγ(~r) = s(γ~r) under a uniform rescaling, we get stuff like

TS[n] = AS

∫d3~rn5/3(1 + αs2 + · · · ), EX [n] = AX

∫d3~rn4/3(1 + µs2 + · · · ).

GGD Approximation

The generalized gradient density approximation, in its most general form, is

EGGAXC [n↑, n↓] =

∫d3~rf(n↑, n↓,∇n↑,∇n↓).

The form of f is tailored to match certain imposed boundary conditions, and is expressed in termsof a gradient parameter t and the average polarization

ζ =n↑ − n↓n↑ + n↓

.

You can look at the form in the Primer, but it’s really quite complicated and there is not too muchphysics. The relevant paper is https://journals-aps-org.ezp-prod1.hul.harvard.edu/prl/pdf/10.1103/PhysRevLett.77.3865.

Ch 4. Time-Dependent Density Functional Theory

TDDFT can be viewed as an alternative formulation of quantum mechanics in 3 dimensionsrather than 3N dimensions. For weak t-dependent potentials, we may use linear-response theory;for strong t-dependent potentials, we must solve the full Kohn-Sham equations.

Regular DFT was about finding the ground state. TDDFT is about propagating a many-bodysystem forward in time, regardless of what state you started in. It is hence much more general.

Let R = (~r1, · · · , ~rN). We may then write

H = T (R) +W (R) + Vext(R, t),

where W is the Coulomb electron-electron interaction and the external potential may be time-dependent. For a laser interacting with matter, for example, there are two terms. First is forelectron-nucleus interaction and the second is for the laser, treated as a classical perturbation.

Vext = Uen + Vlaser, Uen = −∑i,j

Zj

|~ri − ~Rj(t)|, Vlaser = Ef(t) sin(ωt)

∑i

~ri · ~α.

13

https://journals-aps-org.ezp-prod1.hul.harvard.edu/prl/pdf/10.1103/PhysRevLett.77.3865

https://journals-aps-org.ezp-prod1.hul.harvard.edu/prl/pdf/10.1103/PhysRevLett.77.3865


Runge-Gross theorem

The Runge-Gross theorem generalizes Hohenberg-Kohn theorem to time-dependent potentials.The proof is more complicated and makes use of Taylor expansions in time. It’s a little long butnot very interesting. You can read it in the Primer.

Runge-Gross theorem: Let v(~r, t) and v′(~r, t) be two time-dependent potentials correspondingto systems 1 and 2, respectively. Let both systems start with the same density, n(~r, t = 0) =n′(~r, t = 0).

The claim is that if v and v′ differ by more than a purely time-dependent function, they producedifferent densities for t > 0:

v(~r, t) 6= v′(~r, t) + c(t) =⇒ n(~r, t) 6= n′(~r, t) for t > 0.

This theorem guarantees a 1:1 correspondence between the time-dependent potential and thedensity. Classically, you can understand it like this: the freedom to choose c(t) is just the typicalgauge freedom of Newtonian mechanics, in which you can add an arbitrary constant to the potentialand get the same force.

Kohn-Sham auxiliary system

By Runge-Gross theorem, the external potential vS(~r, t) that gives the same density in the Kohn-Sham system as the density of the original interacting system is unique. The time-dependentSchrodinger equation, and corresponding density, is

(−∇2

2+ vS(~r, t))ψi(~r, t) = i∂tψ(~r, t) =⇒ n(~r, t) =

∑i

|ψi(~r, t)|2.

As in time-independent DFT,vS = vext + vH + vXC ,

where vext is the external potential andvH(~r, t) = δUδn(~r,t)

=∫d3~xn(~x,t)

|~x−~r| is the Hartree potential.However, in TDDFT vXC cannot easily be obtained from EXC because of a problem with causality.It turns out that, as first discovered in the paper Phys. Rev. Lett. 80 1280 “Causality andSymmetry in Time-Dependent Density Functional Theory,” by van Leeuwen, that there exists afunctional AXC [n] such that

vXC(~r, t) =δAXCn(~r, τ)

|n(~r,t), where τ = Keldysh pseudo-time.

ALDA Approximation: We need to know how to approximate vXC . The adiabatic localdensity approximation is the simplest and most important approximation. It borrows from theLDA in regular DFT. The approximation is simply

vALDAXC (~r, t) = vHEG

XC (n)|n=n(~r,t).

The RHS is the XC-potential of the (static) uniform electron gas of density n(~r, t). It is as if,at each point in spacetime, we assume the gas has uniform density across all of spacetime. Ofcourse, this is very crude, but it turns out to give quite good results. There are more generalapproximations, of course. ALDA works well for potentials that are only weakly dependent on t.

14


Linear response theory

Suppose that vTD = 0 for t < t0, where TD means “time-dependent.” Also, assume the system isoriginally in the ground state, with density n. Let the t-dependent perturbation be δv, so to firstorder,

δn(~r, ω) =

∫d3~r′χ(~r, ~r′, ω)δv(~r′, ω).

χ is generally incalculable for interacting electrons. But for the noninteracting Kohn-Sham system(i.e. jellium), χKS is known. Because the density of the Kohn-Sham auxiliary system is the sameas the density of the original interacting system, we have

δn(~r, ω) =

∫d3~r′χKS(~r, ~r′, ω)δvKS(~r′, ω).

χKS is known:

χKS(~r, ~r′, ω) = limε→0+

∑jk

(fk − fj)ψ∗j (~r

′)ψk(~r′)ψ∗k(~r)ψj(~r)

ω − (εj − εk) + iε.

We would like to find δvKS in terms of δv, which will allow us to use χKS instead of the unknownχ. (Well, this is tantamount to expressing χ in terms of χKS.) This is not hard; just write outvKS:

δvKS = δv + δvH + δvXC , where δvH =

∫d3~r′

δn(~r′)

|~r − ~r′|, δvXC =

∫dt′d3~r′

δvXC(~r, t)

δn(~r′, t′)δn(~r′, t′).

We call the functional derivative above the exchange kernel or “local field correction,”

fXC(~rt, ~r′t′) =δvXC(~r, t)

δn(~r′, t′).

Rearranging things implicitly defines χ in terms of χXC :

χ(~r, ~r′, ω) = χKS(~r, ~r′, ω) +

∫d3~xd3~x′χ(~r, ~x, ω)[

1

|~x− ~x′|+ fXC(~x, ~x′, ω)]χKS(~x′, ~r′, ω) .

The above equation is exact. fXC is the only unknown term, if we are solving for χ. How can weapproximate it?

There is another ALDA approximation for fXC . Basically, it’s the same ALDA approximation asabove, and assumes locality in spacetime:

fALDAXC (~rt, ~r′t′) = δ(~r − ~r′)δ(t− t′)∂v

HEGXC (n)

∂n|n=n(~r,t).

There are, of course, other better approximations. But this (completely local!) ALDA does areasonably good job. It corresponds to the low-frequency limit.

15


Excitation energies

The density response function χ encodes information about the excitation energies, Em. How? Inthe “Lehmann representation,”

χ(~r, ~r′, ω) =∑m

〈0|n(~r)|m〉〈m|n(~r′)|0〉ω − (Em − E0) + iε

− 〈0|n(~r′)|m〉〈m|n(~r)|0〉ω + (Em − E0) + iε

.

The response function χ thus, has poles corresponding to the excitation energies,

ω → Ωm = Em − E0.

Because the change in potential, δv, doesn’t have poles, this means the change in density, δn(~r, ω),also has poles at ω = Ωm. χKS has poles at excitations of the non-interacting system, which aregenerally different.

Unfortunately, the states |m〉 are generally not known, so neither is χ. How can we massage outthe locations of the poles, Ωm? We apply the operator I =

∫d3~r′δv(~r′, ω) to the boxed equation

above. The result is∫d3~r′(δ(~r − ~r′)− Ξ(~r, ~r′, ω))δn(~r′, ω) =

∫d3~r′χKS(~r, ~r′, ω)δv(~r′, ω), where

Ξ(~r, ~r′, ω) =

∫d3~r′′χKS(~r, ~r′′, ω)(

1

|~r′′ − ~r′|+ fXC(~r′′, ~r′, ω)).

(Clarification: the term which gives Ξ is confusing to derive. It comes from the form∫d3~sδv(~s, ω)

∫d3~r′d3~r′′χ(~s, ~r′, ω)(

1

|~r′′ − ~r′|+ fXC(~r′′, ~r′, ω))χKS(~r′′, ~r′, ω).)

We are interested only in the poles of χ, at ω = Ωm. The RHS does not have poles at Ωm, sothe operator O =

∫d3~r′(δ(~r− ~r′)− Ξ(~r, ~r′, ω)) must have eigenvalue 0 at ω = Ωm. Therefore, the

frequency dependent eigenvalue λ(ω) must satisfy λ(ω)→ 1 as ω → Ωm, where λ is defined as theeigenvalue ∫

d3~r′Ξ(~r, ~r′, ω)ξ(~r′, ω) = λ(ω)ξ(~r, ω).

Because the above determines where λ(ω) = 1, we can determine the excitation energies if fXC isknown. χKS, of course, is already known.

In fact, we may recast this as an eigenvalue equation that has Ωm as the eigenvalues. See thePrimer for the derivation, which is not particularly illuminating. The result is

[(εj − εk) +∑j′k′

Mjkj′k′(Ω)]βj′k′ = Ωβjk.

You can check what all the funny symbols are in the Primer.

Ch 5. DFT and Self-Energy Approaches

This is a good review article to read because it gives you lots of practice with Green functions.Fun, right?

16


Ch 6. A Tutorial on Density Functional Theory

This is a good review paper. It is less theoretical and focused more on actually solving things (i.e.writing programs) for various systems (atoms, plane-waves for periodic solids, etc.).

Solving Kohn-Sham equations

The Kohn-Sham equations are said to be “easy” to solve. That means they are easier to solvethan the Hartree-Fock equations, etc. To review, the Kohn-Sham equations are

(−∇2

2+ vKS[n](~r))ψi(~r) = εiψi(~r), where

vKS[n](~r) = vext(~r) + vH [n](~r) + vXC [n](~r).

The starting “initial guess” for the density, n0(~r), is very important. Often, for molecules we start

with n0(~r) =∑

α nα(~r − ~Rα), where α indexes the atom of the molecule and nα is the atomicdensity if that atom was not close to any other atoms. For single atoms, we use the Thomas-Fermidensity.

Let’s study the different kinds of potentials, and how to find them.

External potential: Most often, this looks like

vext(~r) =∑α

vα(~r − ~Rα).

Sometimes, the Coulomb potential is unusable due to its long range, so we use pseudopotentialsinstead.

Hartree potential: We can either integrate vH(~r) =∫d3~r′ n(~r′)

|~r−~r′| or solve the Poisson equation

∇2vH(~r) = −4πn(~r).

Exchange potential: We often use the LDA approximation, which treats the electrons as locallylike a free electron gas, vLDA

XC (~r) = ∂∂nεHEG(n)|n=n(~r). ε

HEG is a function of n, not a functional ofn(~r). It is well known for various ranges of n via Monte-Carlo methods.

Eigenstates: We would like the N/2 lowest eigenstates of HKS. If the Kohn-Sham equations arereducible to one dimension, like for an atom, there is an efficient integration method that we willcover later. Other methods, like basis set, plane wave, and real-discretized-space method, requirediagonalizing the matrix H. Here, the multiple dimensions refer to the coefficient multiplying basisfunction, the momentum of the plane wave, and the position in real space, respectively. This isan O(N3) process, where N is the dimension of the matrix. N is approximately proportional tothe number of atoms in the system, by a simple scaling argument.

Claim: We can relate the Kohn-Sham energy eigenvalues to the total energy of the true interactingsystem. The relation is

Etot =occ∑i

εi −∫d3~r(

1

2vH(~r) + vXC(~r))n(~r) + EXC .

17


Proof : In the original interacting system,

Etot = −1

2

∫d3~r∇2n(~r) +

∫d3~rvext(~r)n(~r) +

1

2

∫d3~rd3~r′

n(~r)n(~r′)

|~r − ~r′|+ EXC .

Note thatocc∑i

εi =occ∑i

∫d3~rψ∗i (~r)(−

∇2

2+ vext(~r) + vH [n](~r) + vXC [n](~r))ψi(~r)

= −1

2

∫d3~r∇2n(~r) +

∫d3~rvext(~r)n(~r) +

∫d3~rd3~r′

n(~r)n(~r′)

|~r − ~r′|+

∫d3~rvXC [n](~r))n(~r).

Atoms

To solve the Kohn-Sham equations for a single atom, perform a spherical averaging of the density.Typically, the true many-body potential is actually spherically symmetric, so this is a reasonablething to do.

According to the Shell theorems, the Hartree potential becomes

vH(r) =4π

r

∫ r

0

dr′r′2n(r′) + 4π

∫ ∞r

dr′r′n(r′).

The Kohn-Sham orbitals are thus ψi(~r) = Rnl(r)Ylm(θ, φ). The radial equation for R can be recastas two coupled first-order differential equations:

∂rfnl(r) = gnl(r), ∂rgnl +2

rgnl −

l(l + 1)

r2fnl + 2(εnl − vKS)fnl = 0.

Above, fnl corresponds to Rnl. To integrate, just fix a behavior near r = 0 and integrate to generalr. You can see the Primer for more details. If εnl is not an eigenvalue, then fnl diverges as r →∞.There is a nice algorithm to simultaneously solve for ε and f at once. See the Primer.

Periodic solids

The plane-wave Bloch expansion of the Kohn-Sham wavefunctions simplifies things here. Let thepseudopotential, which is the effective potential felt by the valence electrons due to the nucleus+ core electrons, be wα(~r, ~r′). The variables ~r and ~r′ are how far away the electrons are from thenucleus, and we need both ~r and ~r′ because the pseudopotential is generally nonlocal:∫

d3~rvext(~r)n(~r)→N∑i=1

∫d3~rd3~r′ψ∗i (~r)w(~r, ~r′)ψi(~r

′).

If atom α is located at position ~τα in the unit cell, we need to sum over all atoms and all positionsof their relatives in other cells (here, ~Rj is the position of the jth unit cell):

w(~r, ~r′) =∑j,α

wα(~r − ~Rj − ~τα, ~r′ − ~Rj − ~τα).

The Kohn-Sham Bloch wavefunctions and corresponding matrix Schrodinger equation are

ψ~k,n(~r) = ei~k·~r

∑~G

c~k,n(~G)ei~G·~r =⇒

∑~G′

H ~G, ~G′(~k)c~k,n(~G′) = ε~k,nc~k,n(~G).

You can see expressions for n, T, EH etc. in the Bloch expansion in the Primer.

18


Pseudopotentials

The many-body Schrodinger equation is simpler if we treat only the valence electrons, and pretendthe core electrons are tightly bound to the nucleus.

Let the exact solutions of the Schrodinger equation by |ψc〉 for the core electrons and |ψv〉 for thevalence electrons. Introduce the pseudowavefunction |φv〉 for each valence electron v, and write

|ψv〉 = |φv〉+∑c

αcv|ψc〉, where αcv = −〈ψc|φv〉.

Note that |φv〉 is not arbitrary ; it is determined by |ψv〉. This decomposition allows us to write aSchrodinger-like equation,

Hpseudo|φv〉 = Ev|φv〉, where Hpseudo = H−∑c

(Ec − E)|ψc〉〈ψc|.

From the above, we can read off the pseudopotential, which is manifestly nonlocal because itcontains a matrix:

wpseudo = v −∑c

(Ec − E)|ψc〉〈ψc| , where v is the original potential.

There exist good approximations for the pseudopotential, so we can solve for the eigenvalue ofthe above equation, E; see the Primer for some examples. Common approximations look likewl(r, r

′) = vl(r)vl(r′) and wl(r, r

′) = wl(r)δ(r − r′). Note that this is a self-consistent equationbecause Hpseudo itself depends on E.

Near the core, the two parts of the pseudopotential, wpseudo, tend to cancel each other. Basically,v is often Coulombic and the other part is the shielding from the core electrons. |φv〉 tends to besmooth in the core and does not oscillate like |ψv〉 does. The fact that w → 0 for r > Rc and thatw is not too much different from 0 for r < Rc is why the uniform electron gas approximation is adecent approximation for many molecules and metals. Many band structures for valence electronsin metals and semiconductors are not too much different from that for the free electron.

Grigory’s Paper

Typically, when people do DFT for molecules, they use the Born-Oppenheimer approximation,which separates the motion of ions from the motion of electrons. This method avoids the Born-Oppenheimer approximation and leads to more exact quantum-mechanical results. It exploitscertain properties of the Feynman path integral formalism.

Method

Our method has three parts.

1. Propagation in imaginary time, τ , using the DFT Hamiltonian

19


2. Path integral in the language of DFT

3. Extraction of the ground state.

We cover each of these parts of the method in turn. In the following, we describe the ions bytheir positions ~Ri and use ~R to refer to all the ions together. We describe the electrons by theirKohn-Sham orbitals, |ψl(t)〉.

DFT imaginary-time propagation

Real-time TDDFT time propagation involves propagating the Kohn-Sham orbitals, |ψl(t)〉, intime, via the Schrodinger equation H~R|ψl(t)〉 = i~∂t|ψl(t)〉. The time-propagation, of course, isdone via an exponential, which is unitary.

We now Wick-rotate to imaginary time, τ = it. The Schrodinger equation becomes

HKS~R[n(~r, τ)]|ψl(τ)〉 = −∂τ |ψl(τ)〉,

and the propagation is

|ψl(τ)〉 = T e−∫ τ0 H

KS~R

[n(τ ′),~r]dτ ′|ψl(τ = 0)〉.

Here, T is the time-ordering. This means when you expand the exponential in a power series,the H’s need to be time-ordered, later on left. The Hamiltonians evaluated at different times τmay not commute. The coordinates of the ions, ~R, are assumed to be constant throughout thepropagation.

Claim: If |ψl(0)〉 is a complete, orthonormal basis of initial states, propagation for τ → ∞ensures |ψl(τ →∞)〉 are the correct DFT stationary eigenstates (we may have to renormalize).Proof : Suppose |φj〉 are the stationary eigenstates. Due to the factor e−τH from the time-propagation, the largest component of |ψl(τ → ∞)〉 is that |φj〉, with lowest j, such that〈φj|ψl〉 6= 0. If you time propagate all the states l, eventually you will be able to isolate all thestationary states, j.

Path integral

The trace,

〈O〉 =tr(ρO)

tr(ρ), ρ = e−βH,

must sum over all ionic configurations ~R and over a complete basis of electronic states. We cancompute ρ = e−βH be mapping it to K propagations in Euclidean time, where the length of eachinterval is ∆τ = β/K:

e−βH = e−∆τH · · · e−∆τH.

To compute this, insert the identity

1 =

∫ ∏I

d3 ~RI |~R(j)I 〉〈~R

(j)I |

∑nj

|Ψnj〉〈Ψnj |.

20


between each propagation through ∆τ . Here, j ranges from 1 to K and tells us which interval weare in, and I indexes which ion we are talking about. In fact, we will only need the complete basisof electronic states for the first propagation, j = 1.

The important thing about the above is that the |Ψnj〉 are ion-independent states : Ne×Ne Slaterdeterminants of all possible selections of Ne orbitals from the Hilbert space H , where H refersto the single-particle Kohn-Sham system, i.e. |Ψnj〉 = 1√

Ne!detmn |ψm(~rn)〉. This is the essence of

Grigory’s method - the propagation is done in the interacting system for the ions I, but in theauxiliary Kohn-Sham system for the electrons! That was the point of part 1 of this method - itpropagates electrons in the Kohn-Sham system.

The result is that

〈O〉 =1

Z

∫D ~Re−SE

∑n1

〈Ψn1|O∏j

e−∆τH~Rj|Ψn1〉, where

D ~R =∏j,I

d3 ~R(j)I and SE =

1

2~2∆τ

∑j,I

MI |~R(j+1)I − ~R

(j)I |

2.

The path integral is over all paths ~R(j) starting and ending at ~R(j). We know this because ofhow the identity is inserted to the left of all the e−∆τH.

How can we actually perform the propagation? Note that the Hamiltonian currently in use,H~Rj, is for the interacting system; specifically, all the energies in the real system except thekinetic terms for the ions, which are included in SE. This, as we know, can be mapped via DFTto the Kohn-Sham Hamiltonian, HKS

~Rj[n(τ)]. The result is

H~Rj =Ne∑i=1

HKS~Rj

[n(τ), ~ri] + ∆E~R[n], where

∆E~R[n] =∑I<J

Z2e2

|~RI − ~RJ |− 1

2

∫d3~rVH [n,~r]n(~r) + EXC [n]−

∫d3~rVXC [n,~r]n(~r).

We actually derived this already, in Ch 6. of the Primer. The sum over i refers to a sum over theelectrons. Basically,

∑Nei=1HKS

~Rj[n(τ), ~ri] =

∑Nei=1 ψ

∗i (~ri)HKS

~Rj[n(τ), ~ri]ψ

∗i (~ri).

Now, something magical is going to happen: it turns out that ∆E doesn’t affect the propagationof the Kohn-Sham electrons, and we may ignore it. That is because ∆E is a function only of thetotal density n(~r) and not the individual electrons, ψi or ~ri. Because it is not a function of ~ri,it can be considered just a “constant shift” that is τ -dependent but not spatially dependent, justlike shifting the global potential profile by a constant, which does not change the force. This isthe same logic of the Runge-Gross theorem.

Propagation

Now, let’s tie things together. There will be a lot of large τ limits involved, which turns out tocorrespond to low-T .

1. Select a set of ionic configurations, ~R = ~R(j), from the Gaussian distribution e−SE .

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2. Note that the small-time propagation becomes

e−∆τH~R(j) → T (j) = Te−

∫ j∆τ(j−1)∆τ

HKS~R

[n(τ ′),~r1,~r2,··· ]dτ ′.

We have ignored ∆E~R[n] for the reasons listed above. (We are only propagating the

electrons here).

3. For the selected ionic configuration ~R, suppose the eigenstates and eigenvalues of the totaltime-propagation operator T (K)( ~R), where

T (K)( ~R) =K∏j=1

T (j),

are |Ψ(K)k ( ~R)〉 and Λk( ~R).

We may thus rephrase Z and 〈O〉 as

Z =

∫D ~Re−SE

∑k

Λk( ~R),

〈O〉 =1

Z

∫D ~Re−SE

∑k

Λk( ~R)〈Ψ(K)k ( ~R)|O|Ψ(K)

k ( ~R)〉.

4. Now, we take the large-τ or equivalently the small-T limit. In this limit, only the largesteigenvalue Λm contributes, which you can understand from the fact that all terms pick upΛ ∼ e−

∫dτH = e−Se(

~R), where Se is the electronic action. This is essentially a saddle-pointapproximation. You can find Λm by repeated action on an initial state. Just apply theoperator T (K)( ~R) a lot of times to something,

limL→∞

[T (K)( ~R)]L|Ψ〉 ∝ ΛLm for large enough L.

Basically, we apply the operator enough times until we approximate a geometric series. Thedesired eigenvalue is the common ratio of this series.

The result for 〈O〉 is

〈O〉 =1

Z

∫D ~Re−SEΛm( ~R)〈Ψ(K)

m ( ~R)|O|Ψ(K)m ( ~R)〉 .

Above, the eigenvector |Ψ(K)m ( ~R)〉 corresponds to the maximum eigenvalue, Λm.

Grigory says in the paper that in practice, the path integral is computed by first selecting anionic configuration ~R from the Gaussian distribution e−SE and then accepting/rejecting the ionic

configuration according to the eigenvalue Λm( ~R). The total path integral is then turned into a sumover ionic configurations which have been accepted. This works because Λ ∼ e−

∫dτH is inherently

between 0 and 1.

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Notes on Density-functional theory · on Density Functional Theory. DFT is really hard. It’s good to know 295b stu before learning the theory, since both are theories of interacting