mini-tutorial on mathematical aspects of computational ... · mini-tutorial on mathematical aspects...

Mini-tutorial on Mathematical Aspects of Computational Quantum ChemistrySIAM MS 18, July 2018

Benjamin Stamm Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany

Outline• Part 1: Introduction to Quantum Mechanics • Part 2: The Hartree-Fock theory • Part 3: Discretisation of the Hartree-Fock model • Part 4: Density Functional Theory

Goal• Give you an introduction to the big picture • Understanding the mathematical nature/structure of the problems • Derivation of the models (Hartree-Fock and DFT) • Difference between different errors: modelling, discretisation, non-linearity,

linear algebra • Balance between mathematical theory and practical usage of the methods in

chemistry

Based on a YouTube-lecture series• Available on YouTube

https://tinyurl.com/zbyjdho • Part 1: Introduction to quantum mechanics: a mathematical primer • Part 2: Hartree-Fock Theory • Part 3: Discretisation of the Hartree-Fock model • Part 4: Post Hartree-Fock methods: part 1 • Part 5: Post Hartree-Fock methods: part 2 (FCI, CI, Coupled cluster)

Available on the YouTube-channel of LJLL

Literature• Eric Cancès, Mireill Defranceschi, Werner Kutzelnigg, Claude Le Bris, Yvon

Maday: Computational Chemistry: a Primer • DFT-part: T. Helgaker, P. Jorgensen, and J. Olsen, “Principles of density-

functional theory”, (Wiley 2018)

Introduction to Quantum Mechanics(from a Mathematician for Mathematicians)

Recap the basics: Coulomb force/potential

Coulomb force acting on particle 1 due to the presence of particle 2:

! 2 charged particles located at x1, x2 of charge q1, q2

F = keq1q2

|x1 − x2 |2x1 − x2

|x1 − x2 |ke = (4πε0)− 1 q2q1 ↭x1 x2

! Coulomb potential and field created by particle 1:

E(x) = − ∇Φ(x) = q14πε0

x − x1

|x − x1 |3

Φ(x) = 14πε0

q1|x − x1 |

q1x1

Classical model of hydrogen atom: state• 1 proton fixed at the origin, charge • 1 electron at position with momentum (free to move) of mass • The state of the electron is defined by the position und the momentum

p+ x1

x2

x3

x p

Position of the electron:x = (x1, x2, x3)

Momentum of the electron:p = (p1, p2, p3)

p = mvMomentum/Velocity:

The Hydrogen Atom

qp = 1x p m = 1

x p

Classical model of hydrogen atom: force

• Force acting on electron due to proton:

• The state of the electron evolves according to Newton’s second law of motion

··x = ·v = ·p = F (m = 1)

• Equation of motion: second-order differential equation with initial conditions:

··x(t) = − x(t)|x(t) |3 , x(0) = x0, ·x(0) = p(0) = p0

F(x) = − x|x |3 (ignoring constants resp. setting them to 1 )

Classical model of hydrogen atom: reformulation• Important next step: rewriting the system as Hamiltonian system • Equivalent reformulation of classical mechanics • Mathematically: the second-order differential equation of Newton's second

law is reformulated as a system of first-order equations

H(p, x) = &total

H(p, x) = &total = &kin + &pot

• The Hamiltonian is equal to the total energy of the system:

• This energy is divided into two parts: the kinetic energy and the potential energy:

&kin(p) = |v |2

2 = |p |2

2

&pot(x) = − 1|x |

• Kinetic energy:

• Potential energy:

Classical model of hydrogen atom: Hamiltonian dynamics

• The Hamiltonian writes:

• Still: the state of the electron is described by x and p

H(p, x) = &kin(p) + &pot(x) = |p |2

2 − 1|x |

·x = ∂H(p, x)∂p

·p = − ∂H(p, x)∂x

x(0) = x0, p(0) = p0 .

• The dynamics is give by the first-order ODE:

∀t, &total = H(p(t), x(t)) = |p(t) |2

2 − 1|x(t) |

ddt

H(p, x) = ∂H∂p

·p + ∂H∂x

·x = ·x ·p − ·p ·x = 0

• The total energy is conserved:

• Proof:

Limitations of the classical model• Observation: this classical model can take any possible value of the energy

depending on the initial conditions:

Emission spectrum of hydrogen gaz

http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/bohr.html

∀t, &total = |p0 |2

2 − 1|x0 |

Limitations of the classical model• Looking at pur electrostatic forces is actually not entirely correct. A moving

charge in the presence of an electric field is also attracted to the nuclei and would collapse to it in very short time.

• Further: looking at critical points of the Hamiltonian:

We conclude that • No stationary point existing • The Hamiltonian is not bounded from below

∇H(x, p) = 0 ⇒ p = 0 and ∇&pot(x) = x|x |3 = 0.

inf(x,p)∈ℝ6

H(x, p) = − ∞ (realized as x → 0)

The quantum hydrogen atomClassical model: state: (x,p)

Quantum model:• State: the wave-function Ψ : (x, t) ↦ Ψ(x, t) ∈ℂ

p+ x1

x2

x3

p = mvMomentum/Velocity:

Probability of presenceof the electron

e�

The Hydrogen Atom

|Ψ(x, t) |2 : Probability distribution of finding the electron at position x at time t

| Ψ (p, t) |2 : Probability distribution of finding the electron with momentum p at time t

Ψ (p, t) is the Fourier transform of Ψ at time t

• Physical interpretation:

The quantum Hamiltonian• The energy is the expectation value of the classical Hamiltonian:

&(Ψ) = ∫ℝ3

|p |2

2 | Ψ (p, t) |2 dp − ∫ℝ3

1|x |

|Ψ(x, t) |2 dx

H(p, x) = |p |2

2 − 1x

= ∫ℝ3| ∇Ψ (p, t) |2 dp = ∫ℝ3

|∇Ψ(x, t) |2 d x

p Ψ (p) = − i ∇Ψ (p)• This is however not the usual form. Notice that

∫ℝ3|p |2 | Ψ (p, t) |2 dp = ∫ℝ3

|p Ψ (p, t) |2 dp = ∫ℝ3| − i ∇Ψ (p, t) |2 dp

• Then:

• Therefore

The quantum Hamiltonian

= 12 ∫ℝ3

|∇Ψ(x, t) |2 d x − ∫ℝ3

|Ψ(x, t) |2

|x |d x

= − 12 ∫ℝ3

Ψ(x, t)ΔΨ(x, t) d x − ∫ℝ3Ψ(x, t) 1

|x |Ψ(x, t) d x

= ∫ℝ3Ψ(x, t) (− 1

2 Δ − 1|x | ) Ψ(x, t) d x

= ∫ℝ3Ψ(x, t) H Ψ(x, t) d x,

&(Ψ) = ∫ℝ3

|p |2

2 | Ψ (p, t) |2 dp − ∫ℝ3

1|x |

|Ψ(x, t) |2 d x

H = − 12 Δ − 1

|x |is now an operator

Dynamics of quantum hydrogen• The dynamics is driven by the time-dependent Schrödinger equation

i∂Ψ∂t

= HΨ

Ψ(x, t) = Ψ0(x) e− iEt

• If H would be a multiplication by a constant E, i.e. H =E, then

HΨn = EnΨn

• This motivates to look at the spectrum of the operator H:

Ψ(x, t) = ∑n

αne− iEnt Ψn(x) where Ψ(x,0) = ∑n

αnΨn(x)

• Then, by linearity

Properties! The energy & is well defined and continuous on H1(ℝ3) .! The energy is bounded from below:

infΨ∈H1(ℝ3),∥Ψ∥= 1

&(Ψ) = − 12

In consequence, the quantum hydrogen atom model has a stable minimum.

! The minimum is obtained for functions of the form:

Ψ(x) = cπ

e− |x|, |c | = 1.

! These functions belong to H2(ℝ3) and satisfy:

− 12 ΔΨ(x) − Ψ(x)

|x |= − 1

2 Ψ(x), in L2(ℝ3) .

Comparison between modelsClassical model

• State described by position x(t) and momentum p(t)

• Hamiltonian:

• Energy:

• Evolution of state given by

·x = ∂H(p, x)∂p

·p = − ∂H(p, x)∂x

H(p, x) = |p |2

2 − 1x

&(t) = H(x(t), p(t))

Quantum model

• State described by wave-function Probability density for position and momentum

• Hamiltonian:

• Energy:

• Evolution of state given by

Ψ(t)

&(Ψ) = ∫ℝ3Ψ(x, t) H Ψ(x, t) dx,

H = − 12 Δ − 1

|x |

i∂Ψ∂t

= HΨ

The many-body problem

Many-body wave-function• M nuclei: • N electrons:

Z1, R1

Z2, R2

Z↵, R↵

M nuclei ofcharge Z↵,position R↵.

N electrons ofcharge -1,position xi.

x1

x2

x3

x4

xN

Wave-function of 3(N +M) variablesNuclei and electrons

charge Zα at position Rαcharge − 1 with position variable xi

Ψ(x1, …, xN; R1, …, RM)Ψ : (ℝ3 × {±12 })

(N+ M)→ ℂ• Wave-function:

Born-Oppenheimer approximation• The mass of a proton is about 1838 times the mass of the electron • The time-scales of the dynamics between the nuclei and the electrons can be

decoupled: electrons adapt instantaneously to new movement of nuclei • Due to large mass of the nuclei, they can be localised and modelled as

classical particles

• The potential of the M nuclei is given by:

• The state of the electrons is defined by the electronic wave-function:

• From now on, spin will be ignored

v(x) =M

∑α= 1

Zα

|Rα − x |

Ψ : (ℝ3 × {±12 })

N

→ ℂ

(x1, …, xN) ↦ Ψ(x1, …, xN)

The electronic many-body wave-function• Physical interpretation of the wave-function:

|Ψ(x1, …, xN) |2 : probability density function to find the

N electrons at position x1 to xN

∫ℝ3N|Ψ(x1, …, xN) |2 dx1…dxN = 1

• Since this is a probability density function, we require that

Z1, R1

Z2, R2

Z↵, R↵

M nuclei ofcharge Z↵,position R↵.

N electrons ofcharge -1,position xi.

x1

x2

x3

x4

xN

Wave-function of 3N variables

Electrons only

Born-Oppenheimer approximation:

Notation• Braket notation for high-dimensional integrals. For any operator

• we define

• In particular

⟨Φ |O |Ψ⟩ := ∫ℝ3NΦ(x1, …, xN) O Ψ(x1, …, xN) dx1…dxN

O : L2(ℝ3N, ℂ) → L2(ℝ3N, ℂ)

∥Φ∥2 = ⟨Φ |Φ⟩ = ∫ℝ3N|Φ(x1, …, xN) |2 dx1…dxN

O : L2(ℝ3, ℂ) → L2(ℝ3, ℂ)( f |O | g ) := ∫ℝ3f(x) O g (x) d x

( f | g ) := ∫ℝ3f(x) g (x) d x

• 3-dimensional integrals are denoted by

Anti-symmetry/Pauli principle• The fermionic nature of the electrons is expressed by the anti-symmetry of

the corresponding wave-function:

Ψ(x1, …, xi, …, xj, …, xN) = − Ψ(x1, …, xj, …, xi, …, xN)

Anti-symmetry/Pauli principle• The fermionic nature of the electrons is expressed by the anti-symmetry of

the corresponding wave-function:

Ψ(x1, …, xi, …, xj, …, xN) = − Ψ(x1, …, xj, …, xi, …, xN)

⇒ 0 = Ψ(x1, …, x, …, x, …, xN) = |Ψ(x1, …, x, …, x, …, xN) |2

Probability measure to find two electrons at same point is zero

Ψ(x1, …, x, …, x, …, xN) = − Ψ(x1, …, x, …, x, …, xN)• Consequence 1: Pauli exclusion principle:

|Ψ(x1, …, x, …, y, …, xN) |2 = |− 1 |2 |Ψ(x1, …, y, …, x, …, xN) |2

= |Ψ(x1, …, y, …, x, …, xN) |2

• Consequence 2: Indistinguishability of the particles:

The electronic density• Under the probabilistic interpretation, we can introduce the related and

extremely important notion of the electronic density: The probability density function associated with finding any one of the N

electrons at position x

Prob. Density of finding electron 1 at xρΨ(x) = ∫ℝ3(N− 1)|Ψ(x, x2, …, xN) |2 dx2…dxN

+ ∫ℝ3(N− 1)|Ψ(x1, x, …, xN) |2 dx1dx3…dxN

Prob. Density of finding electron N at x

+ ∫ℝ3(N− 1)|Ψ(x1, …, xN− 1, x) |2 dx1…dxN− 1

⋱

= N∫ℝ3(N− 1)|Ψ(x, x2, …, xN) |2 dx2…dxN (by indistinguishability)

Spaces• Square-integrable functions

L2(ℝ3N, ℂ) = {f : ℝ3N → ℂ ∫ℝ3N| f(x1, …, xN) |2 dx1…dxN < ∞}

L2as(ℝ3N, ℂ) = {f ∈L2(ℝ3N, ℂ) f is anti-symmetric}

N

⋀i= 1

Hs(ℝ3, ℝ) := L2as(ℝ3N, ℝ) ∩ (⊕N

i= 1 Hs(ℝ3, ℝ))

• Anti-symmetric tensor product spaces

WN :=N

⋀i= 1

H1(ℝ3, ℝ)

The electronic Hamiltonian• The Hamiltonian is a self-adjoint operator

with domain

given by

• The coordinate-specific Laplacian is given by

H : L2(ℝ3N, ℂ) → L2(ℝ3N, ℂ)

H = −N

∑j= 1 ( 1

2 Δxj+

M

∑α= 1

Zα

|Rα − xj | ) +N

∑i, j = 1

i < j

1|xi − xj |

.

= −N

∑j= 1

( 12 Δxj

− v(xj)) +N

∑i, j = 1

i < j

1|xi − xj |

.

Δxj=

3

∑d= 1

∂2

∂x2j,d

with xj = (xj,1, xj,2, xj,3)

N

⋀i= 1

H2(ℝ3, ℝ)

The ground-state energy• The energy of the system for a given wave-function writes

E(Ψ) = ⟨Ψ |H |Ψ⟩ = ∫ℝ3NΨ(x1, …, xN) H Ψ(x1, …, xN) dx1…dxN

&0 = infΨ ∈L2

as(ℝ3N, ℂ)∥Ψ∥= 1

&(Ψ) = infΨ ∈L2

as(ℝ3N, ℂ)∥Ψ∥= 1

⟨Ψ |H |Ψ⟩ = infΨ ∈L2

as(ℝ3N, ℂ)∥Ψ∥= 1

⟨Ψ |T + Vne + Vee |Ψ⟩ .

• The ground state energy is the lowest possible energy

Remarks• In case of external interactions, further contributions may be added to the

Hamiltonian. This is the case, for instance, for external electric or magnetic fields.

Ψ ∈L2as(ℝ3N, ℂ) : any complex-valued wave-function, there exist real-valued

functions Ψr, Ψi ∈L2

as(ℝ3N, ℝ) such that Ψ = Ψr+ iΨi .

⟨Ψ |H |Ψ⟩ = ⟨Ψr+ iΨi |H |Ψr+ iΨ⟩ = ⟨Ψr |H |Ψr⟩ + ⟨Ψi |H |Ψi⟩

And since the Hamiltonian operator H is real by hypothesis, it follows that

&0 = 2 infΨ ∈L2

as(ℝ3N, ℝ)∥Ψ∥= 1

⟨Ψ |H |Ψ⟩

Therefore, in order to solve for the ground state energy, it is sufficient to solve the problem minimisation problem

• Furthermore, under the assumption that each term in the Hamiltonian operator H including the external potential field is real, it is sufficient to consider only real-valued wave-functions.

• The operator H is a self-adjoint operator

with domain

The ground-state energy

H : L2(ℝ3N, ℂ) → L2(ℝ3N, ℂ)

• It is however not certain that a minimising wave-functions exists: the infimum is not always a minimum!

WN =N

⋀i= 1

H1(ℝ3, ℝ)

&0 = infΨ ∈WN∥Ψ∥= 1

⟨Ψ |H |Ψ⟩

• Square integrable spaces are not a natural space for these kind of problems: considering the problem in a weak setting, we define

and consider

N

⋀i= 1

H2(ℝ3, ℂ)

v ∈8• The operator H is bounded from below so that the ground-state energy exists for

potentials

Existence of minimiser• For an external Coulomb-type potential:

• Total nuclear charge:

• Considering N electrons

v(x) =M

∑α= 1

Zα

|Rα − x |

Z =M

∑α= 1

Zα

Definition: ΣN = inf σess(HN)

N

N − 1

ΣN

ΣN− 1&0,N− 1

HVZ Theorem: 1) σess(HN) = [ΣN, + ∞)2) ΣN = &0,N− 1

HVZ: Hunziker – van Winter – Zhislin [Zis60,vW64,Hun66]

Existence of minimiser• For an external Coulomb-type potential:

• Total nuclear charge:

• Considering N electrons

v(x) =M

∑α= 1

Zα

|Rα − x |

Z =M

∑α= 1

Zα

• Case N < Z + 1: has infinitely many eigenvalues of finite multiplicity such that

• Case N ≥ Z + 1: has at most a finite number of eigenvalues below its essential spectrum

• There exists M such that for all N ≥ M: has no eigenvalues below its essential spectrum

Theorem:HN &k,N

limk→∞

&k,N = ΣN

HN

HN

Collection of different results: [Zis60,ZS65,Jaf76,VZ77,Sig82,Rus82,Sig84]

Recap: 3 main ingredients• Wave-function of unitary norm: defines the state of the electrons in a

probabilistic setting. • Space of anti-symmetric functions: encapsulates fermionic nature of the

electrons • Hamiltonian: model of the interaction of the electrons including the

environment (if applicable).

Ψ :

&0 = infΨ ∈WN∥Ψ∥= 1

⟨Ψ |H |Ψ⟩

WN :

H :

Optimality conditions / Euler-Lagrange equations• Consider the definition of the ground-state energy:

• The mathematical structure is an constrained optimisation problem: • objective function: • constraint:

infΨ ∈WN∥Ψ∥= 1

⟨Ψ |H |Ψ⟩

&(Ψ) = ⟨Ψ |H |Ψ⟩∥Ψ∥2 = ⟨Ψ |Ψ⟩ = 1

• Construct the Lagrange multiplier by

• Next step: differentiate! • How to differentiate w.r.t. a function?

: : WN × ℝ → ℝ

:(Ψ, μ) := &(Ψ)− μ(∥Ψ∥2 − 1) = ⟨Ψ |H |Ψ⟩− μ(∥Ψ∥2 − 1)

Gâteaux derivative• Let X, Y be two Banach spaces. • let be an open set and let be an operator • Then we define the Gâteaux derivative of F at the point in the direction

as:

U ⊂ X < : X → Yu ∈U

ψ ∈X

d<(u ; ψ) = limϵ→0

<(u + ϵψ) − <(u )ϵ

= ddϵ

<(u + ϵψ)ϵ= 0

• Example: given by

Gâteaux derivative& : H1

0(Ω) → ℝ

&(v) = ∫Ω( 1

2 |∇v |2 − fv) dx

d&(u ; v) = ddϵ

&(u + ϵv)ϵ= 0

= ddϵ ∫Ω

( 12 |∇(u + εv) |2 − f(u + εv)) dx

ϵ= 0

= ∫Ω(∇u ⋅ ∇v − fv) dx

∀v ∈H10(Ω) : dE(u ; v) = 0

⇔ ∀v ∈H10(Ω) : ∫Ω

∇u ⋅ ∇v dx = ∫Ωfv dx

Weak formulation of the Laplace source problem

Optimality conditions / Euler-Lagrange equations• Consider the Lagrange multiplier function by: : WN × ℝ → ℝ

:(Ψ, μ) := &(Ψ)− μ(∥Ψ∥2 − 1) = ⟨Ψ |H |Ψ⟩− μ(⟨Ψ |Ψ⟩ − 1)

• This is the weak formulation of the time-independent Schrödinger equation

• The ground-state energy is the smallest eigenvalue of H

∇μ:(Ψ, μ) = 0 ⇔ ∥Ψ∥2 = 11)

∀Φ : d:(Ψ, μ; Φ) = 2⟨Φ |H |Ψ⟩ − 2μ⟨Φ |Ψ⟩ = 0

⇔ ∀Φ : ⟨Φ |H |Ψ⟩ = μ⟨Φ |Ψ⟩

2)

Curse of dimensionality• The only remaining issue is thus to solve the time-independent eigenvalue

problem

∀Φ : ⟨Φ |H |Ψn⟩ = En⟨Φ |Ψn⟩ ⇔ H |Ψn⟩ = En|Ψn⟩

• The wave-function depends on 3N variables. A brut-force discretisation uses a number of degrees of freedom that grows exponentially with the number of electrons N in the system.

• Example: CO2 has 22 electrons and would use 266 degrees of freedom if only 2 dofs would be use per dimension!

• Dirac stated in 1929(!):

“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore

becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems

without too much computation”

Intermediate conclusion• A model for the description of a molecule has been derived.

• It is based on 3 main ingredients: • Wave-function -> defines the state of the electron in a probabilistic manner • Space of anti-symmetric functions -> fermonic character of electrons • Hamiltonian -> operator acting on the wave-function

• Not solvable in practise due to curse of dimensionality.

• Next section: replacing the model (model error)

Hartree-Fock theory

The metaphor: Erwin Schrödinger’s cat

Schrödinger’s cat seen through two optical camera lenses: a high quality one (left) and a cheap and affordable one (right) .

(Spring) 2018 6 THE HARTREE-FOCK THEORY

Figure 6.1: The metaphor: Erwin Schrödinger’s cat seen through two optical camera lenses: a high quality one(left) and a cheap and a�ordable one (right).

6 The Hartree-Fock theory

We have seen in the last chapter that the classical model could not explain all the phenomena that were observedin experiments. Quantum mechanics then filled the gap as a complement to the classical theory. Unfortunately,the Schrödinger equation which is at the heart of quantum mechanics cannot be solved, even not approximatednumerically, in practice.In this chapter we will see how one can derive a simpler model, the Hartree-Fock model, which is tractable inpractice at the prize that we are introducing a modeling error and loose some accuracy.Let us recall the metaphor of Schrödinger’s cat: the Schrödinger equation was represented by the image of the catseen trough a photo-camera.Introducing the Hartree-Fock model is like taking a picture of the cat with a much cheaper camera, which has nota very good optical lens resulting in a bad-quality picture.However, this type of camera is a�ordable and allows us at least to take a picture. We see some details of the cat,but not all.Leaving the metaphor behind us, we saw in the last chapter the main hypotheses in the models used for describinga molecular system, leading to the electronic Schrödinger equation. In this chapter, we will see how we can simplifythis equation which is way too hard to be solved in practice. The main simplification will be the choice of aparticular form for the wave-function, much simpler than a general anti-symmetric function. This model is calledHartree-Fock model, and the equations are the Hartree-Fock equations.

44

Schrödinger model Hartree-Fock model

Our metaphor:cat: reality taking a picture: modelling optical lens of camera: model

Key-idea of the Hartree-Fock theory• Restrict the solution set in the variational principle. We will define some

subset

so that

VHF ⊂ WN

infΨ ∈WN∥Ψ∥= 1

⟨Ψ |H |Ψ⟩ ≤ infΨ ∈VHF∥Ψ∥= 1

⟨Ψ |H |Ψ⟩

• But what is a simple subset?

Symmetric functions• Let be given N functions: fi : ℝ3 → ℝ, i = 1,2,…, N

fs(x1, …, xN) =N

∑σ

fσ(1)(x1) ⋯ fσ(N)(xN)

sum over all N! permutations of {1,…, N}

• Symmetric tensor product:

f(x1, …, xN) = f1(x1) ⋯ fN(xN)• Tensor product:

f(x1, x2) = f1(x1)f2(x2) fs(x1, x2) = (f1(x1)f2(x2) + f2(x1)f1(x2))

• Example (N=2):

Anti-symmetric functions• Symmetric tensor product: fs(x1, …, xN) =

N

∑σ

fσ(1)(x1) ⋯ fσ(N)(xN)

fas(x1, …, xN) =N

∑σ

ε(σ) fσ(1)(x1) ⋯ fσ(N)(xN)

=

f1(x1) f2(x1) ⋯ fN(x1)f1(x2) f2(x2) ⋯ fN(x2)

⋮ ⋮ ⋱ ⋮f1(xN) f2(xN) ⋯ fN(xN)

• Anti-symmetric tensor product:

fs(x1, x2) = (f1(x1)f2(x2) + f2(x1)f1(x2)) fas(x1, x2) = (f1(x1)f2(x2) − f2(x1)f1(x2))

• Example (N=2):

Slater determinants• Let a set of N orthonormal functions be given:

ϕi : ℝ3 → ℝ, i = 1,…, N

(ϕi |ϕj) := ∫ℝ3ϕi(x) ϕj(x) dx = δij,

Φ(x1, …, xN) = 1N!

N

∑σ

ε(σ) ϕσ(1)(x1) ⋯ ϕσ(N)(xN)

= 1N!

ϕ1(x1) ϕ2(x1) ⋯ ϕN(x1)ϕ1(x2) ϕ2(x2) ⋯ ϕN(x2)

⋮ ⋮ ⋱ ⋮ϕ1(xN) ϕ2(xN) ⋯ ϕN(xN)

• The Slater determinant is given by the normalised anti-symmetric product

Φ = |ϕ1, … , ϕN⟩• We write

Duality • Each set of N orthonormal functions build a Slater determinant • Each Slater determinant is built by a set of N orthonormal functions • Mathematically, these are two separate things

Φ = |ϕ1, … , ϕN⟩ Φ = {ϕ1, … , ϕN}

high-dimensional function of “low rank”

ordered N-tuple of 3d-functions

Properties of Slater determinants• Anti-symmetric by construction (simplest “rank 1” anti-symmetric function) • Norm of a Slater-determinant:

∥Φ∥2 = ⟨Φ |Φ⟩ = 1N! ∑

σ,νε(σ) ε(ν)∫ℝ3N

(ϕσ(1)(x1) ⋯ ϕσ(N)(xN))(ϕν(1)(x1) ⋯ ϕν(N)(xN))

= 1N! ∑

σ,νε(σ) ε(ν)∫ℝ3

ϕσ(1)(x1) ϕν(1)(x1) dx1 × ⋯ × ∫ℝ3ϕσ(N)(xN) ϕν(N)(xN) dxN

= 1N! ∑

σ,νε(σ) ε(ν) δσ(1),ν(1) × ⋯ × δσ(N),ν(N)

= 1N! ∑

σ= 1

Slater determinants: electronic density• Similarly for the electronic density

ρΦ(x) = N∫ℝ3(N− 1)|Φ(x, x2, …, xN) |2 dx2…dxN

= NN! ∑

σ,νε(σ) ε(ν) ϕσ(1)(x) ϕν(1)(x) ∫ℝ3

ϕσ(2)(x2) ϕν(2)(x2) dx2 × ⋯

= NN! ∑

σ,νε(σ) ε(ν) ϕσ(1)(x) ϕν(1)(x) δσ(2),ν(2) × ⋯ × δσ(N),ν(N)

= NN! ∑

σ|ϕσ(1)(x) |2 =

N

∑i= 1

|ϕi(x) |2

Slater determinants: one-particle operators• One-body interaction

⟨Φ |A |Φ⟩ = 1N! ∑

σ,ν

N

∑j= 1

ε(σ) ε(ν)∫ℝ3N(ϕσ(1)(x1) ⋯ ϕσ(N)(xN)) a(xj) (ϕν(1)(x1) ⋯ ϕν(N)(xN))

=N

∑j= 1

∫ℝ3ϕj(x) a(x) ϕj(x) dx =

N

∑j= 1

(ϕj |a |ϕj)

A =N

∑j= 1

a(xj)

A = Vne =N

∑j= 1

v(xj)• Example 1: ⇒ ⟨Φ |Vne |Φ⟩ =N

∑j= 1

(ϕj |v |ϕj)

=N

∑j= 1

(v |ϕ2j ) = (v |ρΦ)

A = T = − 12

N

∑j= 1

Δxj• Example 2: ⇒ ⟨Φ |T |Φ⟩ =

N

∑j= 1

(ϕj |− 12 Δx |ϕj)

= 12

N

∑j= 1

∫ℝ3|∇ϕj(x) |2 dx = 1

2N

∑j= 1

∥∇ϕj∥2

Slater determinants: 2-body operators• 2-body interaction: W =

N

∑i, j = 1

i ≠j

w(xi, xj)

⟨Φ |W |Φ⟩ =N

∑i,j= 1

∫ℝ3 ∫ℝ3ϕi(x) ϕi(x) w(x, y) ϕj(y) ϕj(y) dy dx

−N

∑i,j= 1

∫ℝ3 ∫ℝ3ϕi(x) ϕj(x) w(x, y) ϕi(y) ϕj(y) dy dx

• Example: withVee =N

∑i, j = 1

i < j

1|xi − xj |

= 12

N

∑i, j = 1

i ≠j

1|xi − xj |

⟨Φ |Vee |Φ⟩ = 12

N

∑i,j= 1

∫ℝ3 ∫ℝ3

ϕi(x) ϕi(x) ϕj(y) ϕj(y)|x − y |

dy dx

− 12

N

∑i,j= 1

∫ℝ3 ∫ℝ3

ϕi(x) ϕj(x) ϕi(y) ϕj(y)|x − y |

dy dx

w(x, y) = 1|x − y |

The Hartree-term• The third term can be rewritten as

N

∑i,j= 1

∫ℝ3 ∫ℝ3

ϕi(x) ϕi(x) ϕj(y) ϕj(y)|x − y |

dy dx = ∫ℝ3 ∫ℝ3

∑Ni= 1 ϕi

2(x) ∑Nj= 1 ϕj

2(y)|x − y |

dy dx

= ∫ℝ3 ∫ℝ3

ρΦ(x) ρΦ(y)|x − y |

dy dx = ∫ℝ3ρΦ(x) JΦ(x) dx = (ρΦ |JΦ)

where

is the electrostatic potential generated by the electronic density.

JΦ(x) = ∫ℝ3

ρΦ(y)|x − y |

dy

• This term can be interpreted as the classical electrostatic (Coulomb) energy generated by the electronic density.

=N

∑j= 1

(ϕj |JΦ |ϕj)or

The exchange-term• The fourth term can be rewritten as

N

∑i,j= 1

∫ℝ3ϕi(x) ϕj(x) ∫ℝ3

ϕi(y) ϕj(y)|x − y |

dy dx

with the non-local operator:

KΨ ϕ(x) =N

∑i= 1

∫ℝ3

ψi(y) ϕ(y)|x − y |

dy ψi(x)

=N

∑j= 1

∫ℝ3ϕj(x) KΦ ϕj(x) dx =

N

∑j= 1

(ϕj |KΦ |ϕj)

The Hartree-Fock energy• For each Slater determinant

define

&HF(Φ) := &(Φ) = ⟨Φ |H |Φ⟩ = ⟨Φ |T + Vne + Vee |Φ⟩

=N

∑j= 1

(ϕj |− 12 Δ + v |ϕj) + 1

2N

∑j= 1

(ϕj |JΦ − KΦ |ϕj)

Φ = |ϕ1, … , ϕN⟩ Φ = {ϕ1, … , ϕN}

is short notation for

The Hartree-Fock energy

EHF(Φ) =N

∑j= 1

∫ℝ3 [ 12 |∇ϕj(x) |2 + v(x) |ϕj(x) |2 ] dx

+ 12

N

∑i,j= 1

∫ℝ3 ∫ℝ3 [ϕi(x) ϕi(x) ϕj(y) ϕj(y)

|x − y |−

ϕi(x) ϕj(x) ϕi(y) ϕj(y)|x − y | ] dy dx

&HF(Φ) =N

∑j= 1

(ϕj |− 12 Δ + v |ϕj) + 1

2N

∑j= 1


• Contains quadratic and quartic terms (single-/double-particle operators)

Invariance of the HF-energy wrt to unitary transforms• The Hartree-Fock energy is invariant under a unitary transform. Let

be a set of orbitals and

a unitary matrix. Then define a new set of (rotated) orbitals

Then, there holds

Φ = {ϕ1, …, ϕN}

&HF(Φ) = &HF(Φ) = &HF(UΦ)

U ∈ℝN×N, UTU = UUT = IN

Φ = {ϕ1, …, ϕN}, ϕi =N

∑j= 1

Uij ϕj .

• Define the set of Slater determinants (no vector space!)

Back to the Hartree-Fock model

VHF = {Φ = |ϕ1, …, ϕN⟩ ϕi ∈H1(ℝ3)} ⊂ WN

VHF = {Φ = {ϕ1, …, ϕN} ϕi ∈H1(ℝ3)} = [H1(ℝ3)]N

&0 = &(Ψ0) = infΨ ∈WN

s.t. ∥Ψ∥= 1

⟨Ψ |H |Ψ⟩ ≤ infΦ ∈VHF

s.t. (ϕi |ϕj) = δij

⟨Φ |H |Φ⟩ = &(Φ0) = &HF

• There naturally holds

• Note that

&HF := &HF(Φ0) = &(Φ0)

Euler-Lagrange equations for HF• We consider therefore the constrained minimization problem

infΦ ∈VHF

s.t. (ϕi |ϕj) = δij

&HF(Φ)

• Construct the Lagrange multiplier function

by

:(Φ, Λ) := &HF(Φ) −N

∑i,j= 1

λij((ϕi |ϕj) − δij)

: : [H1(ℝ3)]N × ℝN×Nsym → ℝ

∇λij:(Φ, Λ) = 0 ⇔ (ϕi |ϕj) = δij

• There holds

Euler-Lagrange equations for HF• The derivative of the energy is far less trivial. Consider the test function

and the Gâteaux derivative

Ψj = {0,…, ψ, …,0}

d:(Φ, Λ; Ψj) = ddε

:(Φ+ εΨj, Λ) |ε= 0

Φ = {ϕ1, …, ϕN}• The final result is given by: find such that

KΦ ϕj (x) =N

∑i= 1

∫ℝ3


dy ϕi(x)JΦ(x) = 12 ∫ℝ3

ρΦ(y)|x − y |

dy

∀i, j : (ϕi |ϕj) = δij

∀j, ∀ψ : (ψ |− 12 Δ + v |ϕj) + (ψ |JΦ − KΦ |ϕj) =

N

∑i= 1

λji (ψ |ϕi)

Euler-Lagrange equations for HF• The derivative of the energy is far less trivial. Consider the test function

and the Gâteaux derivative

Ψj = {0,…, ψ, …,0}

d:(Φ, Λ; Ψj) = ddε

:(Φ+ εΨj, Λ) |ε= 0

Φ = {ϕ1, …, ϕN}• The final result is given by: find such that


∀j, ∀ψ : (ψ |− 12 Δ + v |ϕj) + (ψ |JΦ − KΦ |ϕj) =

N

∑i= 1

λji (ψ |ϕi)

&HF(Φ) =N

∑j= 1

(ϕj |− 12 Δ + v |ϕj) + 1

2N

∑j= 1


Euler-Lagrange equations• Recall

• If a set of orbitals satisfies the Euler-Lagrange equations, then any unitary transform of this set provides the same HF-energy.

Φ = {ϕ1, …, ϕN}


∀j, ∀ψ : (ψ |− 12 Δ + v |ϕj) + (ψ |JΦ − KΦ |ϕj) =

N

∑i= 1

λji (ψ |ϕi)

∀j, ∀ψ : (ψ |− 12 Δ + v |ϕj) + (ψ |JΦ − KΦ |ϕj) = εj (ψ |ϕj)


• One can find the unitary transform U such that

Weak versus strong formulation• This weak formulation

corresponds to the strong formulation

with Fock-operator

∀j : FΦ ϕj = εj ϕj, in ℝ3

FΨ ϕ = (− 12 Δ + v)ϕ + (JΨ − KΨ)ϕ

non-local operator!

∀j, ∀ψ : (ψ |− 12 Δ + v |ϕj) + (ψ |JΦ − KΦ |ϕj) = εj (ψ |ϕj)



linear part non-linear part

Strong form all together without neat notation

∀j : [− 12 Δ+ v(x)] ϕj(x) + ∫ℝ3

∑Ni= 1 |ϕi(y) |2

|x − y |dy ϕj(x) −

N

∑i= 1

∫ℝ3


dy ϕi(x) = εj ϕj, in ℝ3

∀i, j : ∫ℝ3ϕi ϕj = δij


Schrödinger’s cat seen through two optical camera lenses: a high quality one (left) and a cheap and affordable one (right).





44


<HF(ΨHF) = 0<(Ψ) = 0

modelling

<(ΨHF) ≠0 model error

Discretisation of HF

Representation of orbitals• Basis functions (basis set): {χμ}Q

μ= 1

ϕj ≈ ϕδj :=

Q

∑μ= 1

Cμj χμ, Cμj ∈ℝ• Discrete approximation:

VQ =Q

∑μ= 1

cμ χμ cμ ∈ℝ, ∀μ = 1,…, Q• Orbital approximation space:

C ∈ℝQ×N

VδHF = {Φδ = {ϕδ

1, …, ϕδN} ϕδ

i ∈VQ} ⊂ [H1(ℝ3)]N ⊂ VHF

• The set of all (discrete) orbitals are represented by a matrix

Density-matrix

ρΦ(x) =N

∑i= 1

|ϕi(x) |2 =N

∑i= 1

Q

∑μ= 1

Q

∑ν= 1

Cμi χμ(x) χν(x) Cνi

=Q

∑μ= 1

Q

∑ν= 1

χμ(x) χν(x)N

∑i= 1

Cμi Cνi =Q

∑μ= 1

Q

∑ν= 1

χμ(x) Dμν χν(x)

• Write the electronic density in terms of the basis functions:

• The density matrix:

D = C C⊤

Discrete Euler-Lagrange equations

Repeat the same procedure but in the discrete setting to obtain the following discrete variational problem:

• One-particle operator: (χμ |− 12 Δ + v |ϕδ

j )

{εδ1, …, εδ

N}Φδ = {ϕδ1, …, ϕδ

N} ∈[VQ]N

∀j, ∀μ : (χμ |− 12 Δ + v |ϕδ

j ) + (χμ |VΦδ − KΦδ |ϕδj ) = εδ

j (χμ |ϕδj )

∀i, j : (ϕδi |ϕδ

j ) = δij

Find and such that

(χμ |VΦδ − KΦδ |ϕδj )• 2-body operators:

(χμ |ϕδj ) resp. (ϕδ

i |ϕδj ) = δij• Scalar product:

&HF(Φδ) =N

∑j= 1

(ϕδj |− 1

2 Δ + v |ϕδj ) + 1

2N

∑j= 1

(ϕδj |JΦδ − KΦδ |ϕδ

j )• Energy:

• Define for each self-adjoint one-particle operator the symmetric matrix

One-particle operators

Aμν = (χμ |a | χν)

and

(χμ |a |ϕδj ) =

Q

∑ν= 1

(χμ |a | χν) Cνj =Q

∑ν= 1

Aμν Cνj = (AC)μj

N

∑i= 1

(ϕδi |a |ϕδ

i ) =N

∑i= 1

(C⊤AC)ii = Tr(C⊤AC) = Tr(AD)D = C C⊤

• Then

Eigenvalue problem: non-linear terms• The non-linear terms are not so easy… • Any arbitrary defined by

is represented by some matrix and build the density matrix

Ψδ = {ψ δ1, …, ψ δ

N} ∈[VQ]N

ψδj :=

Q

∑μ= 1

Cμj χμ

C D = CC⊤

(χμ |VΨδ − KΨδ | χν) = ∑μ′�ν′�

[(μν |μ′�ν′�) − (μν′�|μ′�ν)] Dν′�μ′ �

• Then, there holds

• The bi-electronic integrals are given by

(μν |μ′�ν′�) = ∫ℝ3 ∫ℝ3

χμ(x) χν(x) χμ′�(y) χν′ �(y)|x − y |

dx dy

• 4-dimensional tensor with symmetries: • each entry is a 6-dimensional integral

(μν |μ′�ν′�) = (νμ |μ′�ν′�) = (ν′�μ′�|μν)

Outline• Part 1: Introduction to Quantum Mechanics • Part 2: The Hartree-Fock theory • Part 3: Discretisation of the Hartree-Fock model • Part 4: Density Functional Theory

The discrete non-linear eigenvalue problem• The eigenvalue problem writes

F(D)μν := (χμ |− 12 Δ + v + JΨδ − KΨδ | χν)

F(D) C = SCEC⊤SC = IN

D = CC⊤

= hμν+ ∑μ′�ν′�

[(μν |μ′�ν′�) − (μν′�|μ′�ν)] Dν′�μ′ �

with discrete Fock operator:

= hμν + G(D)μν

• Then, the discrete Hartree-Fock energy can be written as

&HF(D) = Tr(hD) + 12 Tr(G(D)D)

Solving the non-linearity• Self-consistent field iterations: for k=1,2,…

F(D[k− 1]) C[k] = SC[k]E[k]

(C[k])⊤SC[k] = IN

D[k] = C[k](C[k])⊤

• At each iteration k, one needs to solve a (linear) eigenvalue problem • On top of that, it is common to use some sort of acceleration technique: DIIS,

charge mixing, Pulay mixing, Anderson acceleration, …

Basis functions

Choice of basis functions• Many possible basis functions can be used

• finite elements • wavelets • plane waves (for periodic boundary conditions) • Linear Combinations of Atomic Orbitals (LCAO)

• Consequence of choice of basis functions: • Discretisation error • Stability/properties of the involved matrices • Sparsity/structure of the involved matrices • Complexity of matrix-assembly resp. matrix-vector multiplication

Linear Combination of Atomic Orbitals - LCAO• Idea: use basis functions centered around atoms

ϕδj (x) =

Q

∑μ= 1

Cμj χμ(x) =M

∑α= 1

Qα

∑μ= 1

C(α,μ)j ξαμ (x − Rα)

Slater-type orbitals (STO)• First idea: use eigenstates of the hydrogen atom (or similar functions)

ξ(r, θ, φ) = rℓe− Zrn Ym

ℓ (θ, φ) (up to normalization) .

• Drawback: • bi-electronic integrals expensive to compute

• not a complete basis (hydrogen Hamiltonian has a continuous spectrum)

(μν |μ′�ν′�) → J(Q4),

• Advantage: Correct asymptotic behaviour near nuclei (cusp) and far away (fall-off)(Spring) 2018 7 DISCRETIZATION OF THE HARTREE-FOCK MODEL

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1l = 0

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1l > 0

(a) Radial dependency of Slater-type orbitals (STO).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

(b) Radial dependency of Gaussian-type orbitals (GTO).

Figure 7.1

Then, the orbitals are sought as linear combination of atomic orbitals, in short LCAO.

Ï”

j(x) =Mÿ

–=1

N–ÿ

µ=1

C–

µj ›–

µ (x ≠ R–).

Slater-type orbitals (STO) The first idea that came up to design atomic orbitals was a very natural one:using hydrogen-like orbitals as basis functions. A simplified version thereof is called Slater type orbitals and thebasis functions are given by

›(r, ◊, Ï) = r¸e≠ Zr

n Y m

¸ (◊, Ï) (up to normalization).

These functions are written in terms of spherical coordinates and have a specific dependency in the radial part.The angular dependency is described by means of spherical harmonics which are denoted by Y m

¸.

Figure 7.1 (left) illustrates some examples of the dependency on the radial part of such basis functions.The disadvantage of this Ansatz is that the bielectronic 6-dimensional integrals

(µ‹|µÕ‹ Õ) =⁄

R3

⁄

R3

‰µ(x)‰‹(x)‰µÕ(y)‰‹Õ(y)|x ≠ y| dx dy æ O(N”

4),

are expensive to compute without further simplifications and the number of entries of this rank 4 tensor scales asN”

4.

Gaussian-type orbitals (GTO) A fundamental idea regarding the computational complexity was the introduc-tion of Gaussian-type basis functions. The reasons is that integrals of Gaussians, products thereof and polynomialstimes Gaussians can be computed analytically, which is not possible for general basis functions. This reduces thecomputational complexity of the 6-dimensional bielectronic integrals: the complexity is reduced to a one-dimensionalintegral.However, the number of entries in this rank 4 tensor scales still as O(N”

4). In practice, the complexity is reduced toO(N”

2.7) since the overlap of gaussians corresponding to far-away atoms is negligible. This type of basis functions

63

(Spring) 2018 7 DISCRETIZATION OF THE HARTREE-FOCK MODEL

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1l = 0

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1l > 0


0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1


Figure 7.1


Ï”

j(x) =Mÿ

–=1

N–ÿ

µ=1

C–

µj ›–

µ (x ≠ R–).


›(r, ◊, Ï) = r¸e≠ Zr

n Y m



¸.


(µ‹|µÕ‹ Õ) =⁄

R3

⁄

R3


4),


4.




63

Gaussian-type orbitals (GTO)• Idea: use Gaussians to describe radial decay

ξαμ (x, y, z) = xnx yny znze− αμr2

(up to normalization)

(Spring) 2018 7 DISCRETIZATION OF THE HARTREE-FOCK MODEL

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1l = 0

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1l > 0


0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1


Figure 7.1


Ï”

j(x) =Mÿ

–=1

N–ÿ

µ=1

C–

µj ›–

µ (x ≠ R–).


›(r, ◊, Ï) = r¸e≠ Zr

n Y m



¸.


(µ‹|µÕ‹ Õ) =⁄

R3

⁄

R3


4),


4.




63

• Drawback: Correct asymptotic behaviour near nuclei (cusp) and far away (fall-off)

• Advantage: bi-electronic integrals easy to compute due to analytic formulae available for Gaussians.

Difference• Example of s-type orbitals:

Contracted Gaussians• Idea: particular linear combinations of Gaussians are sufficient for an

accurate representation of the orbitals: find linear combinations of Gaussians that approximate well STO’s

ξαμ (x, y, z) = ∑

kCk xnk

x ynky znk

z e− αkr2(up to normalization)

≈=++

Contracted Gaussians• Idea: particular linear combinations of Gaussians are sufficient for an

accurate representation of the orbitals: find linear combinations of Gaussians that approximate well STO’s

ξαμ (x, y, z) = ∑

kCk xnk

x ynky znk

z e− αkr2(up to normalization)

Error cascade• Sources of errors:

• Model error • Discretisation error • Tolerance of stopping criterion of SCF-iterations • Tolerance of solving the linear eigenvalue problem at each SCF-iteration

<HF(ΨHF) = 0

<(Ψ) = 0

modelling

<(ΨHF) ≠0 model error

<δHF(Ψδ

HF) = 0

<δHF( Ψ δ

HF) ≠0

discretisation error

<HF(ΨδHF) ≠0

discretisation

actual “solution”

<( Ψ δHF) = (<( Ψ δ

HF) − <HF( Ψ δHF)) + (<HF( Ψ δ

HF) − <δHF( Ψ δ

HF)) + <δHF( Ψ δ

HF) model error discretisation error rest


Schrödinger’s cat seen through two optical camera lenses: a high quality one (left) and a cheap and affordable one (right) .





44


Our metaphor:cat: reality taking a picture: modelling optical lens of camera: model saving the picture with pixels: discretisation

The metaphor: basis set 1 (coarse)

Schrodinger model Hartree-Fock model

Discretized Schrodinger model Discretized Hartree-Fock model

The metaphor: basis set 2



Summary so far• The Hartree-Fock model was discretised. • All theory is known to be implemented • Choice of basis function important for

• accuracy • efficiency • sparsity/structure of involved matrices

• Depending on the purpose of the computation, the Hartree-Fock model does not always yield accurate results (even for very fine basis sets).

• “Electron correlation” not dealt with properly -> post Hartree-Fock methods (watch video 4,5)

Density Functional Theory (DFT)(alternative model to HF)

Recap and existence of minimisers• Hamiltonian

with

for some

HN(v) = T + Vne(v) + Vee

T = − 12

N

∑j= 1

ΔxjVne(v) =

N

∑j= 1

v(xj) Vee =N

∑i, j = 1

i < j

1|xi − xj |

v ∈8 := L 32(ℝ3) + L∞(ℝ3)

• Then, the Hamiltonian is bounded from below and this set of potentials contains Coulomb-like potentials

• But the existence of the minimiser is not guaranteed

E0,N(v) = infΨ ∈WN∥Ψ∥= 1

⟨Ψ |HN(v) |Ψ⟩ = infΨ ∈WN∥Ψ∥= 1

⟨Ψ |T + Vne(v) + Vee |Ψ⟩

• Example of the oxygen atom with N electrons:

The electron in excess is “anywhere infinitely far away”: no interaction with the potential.

Example

v(x) = − 8|x |

&0,9(v) = infΨ ∈W9∥Ψ∥= 1

⟨Ψ |H9(v) |Ψ⟩ = ⟨Ψ9 |H9(v) |Ψ9⟩

= &0,10(v) = infΨ ∈W10∥Ψ∥= 1

⟨Ψ |H10(v) |Ψ⟩

• a ground-state exists for N=9, but not for N>9:

Admissible potentials• Introduce the set of admissible potentials for N-electron systems:

VN = {v ∈8 | HN(v) has a ground-state}

= {v ∈8 | infΨ∈WN

⟨Ψ |HN(v) |Ψ⟩ admits a minimizer}

• No assumption on the uniqueness of the ground-state wave-function! • Depends on the number of electrons in the system (electronegativity)!

• Density-based reformulation of the electronic Schrödinger equation

Key-idea of DFT

• In the following, we will investigate the relations between potential, wave-function and electronic density:

• Many properties of a molecule can be derived from the density only.

low dimensional

Ψ ∈WN

v ∈VN\ℝ ρ ∈AN

ρΨΨv

ρv

Ψ ↦ ρΨ(x) = N∫ℝ3(N− 1)|Ψ(x, x2, …, xN) |2 dx2…dxN

• For given wave-function, the electronic density is given by

• No explicit characterisation. The expression

is only of theoretical interest. Not useful to design numerical methods.

• For densities , we say that is v-representable, i.e. it is the ground-state density of some .

Ground-state densities

• Goal: Density-based reformulation of the electronic Schrödinger equation.

ρ ∈AN

infρ∈AN

ρv ∈VN

Ψ ∈WN


ρΨΨv

ρv

AN = {ρ | ∃v ∈VN, Ψ ∈WN s.t. ρ = ρΨ and Ψ is a ground-state of HN(v)}

The relation between wave-function and potentialLemma: Let be two admissible potentials. Then

have the same ground-state wave-functions if and only if .

v1, v2 ∈VN

H(v1), H(v2)

v1 − v2 ∈ℝ

Ψv1,1, Ψv1,2 ∈WNv1 ∈VN \ℝ

ΨvSolution map

v2 ∈VN \ℝ Ψv2,1 ∈WN

H(v1)

H(v2)

Uniqueness is not addressed. The family of ground-state wave-functions is considered.

The Hohenberg-Kohn theorem (1964)Theorem: A ground-state density determines the potential uniquely up to a constant.

Ψ ∈WN


ρΨΨv

ρv

v1 − v2 ∉ ℝ ⇒ ρ1 − ρ2 ≠0

ρ1 − ρ2 = 0 ⇒ v1 − v2 ∈ℝ

Let and the corresponding ground-state densities. Then, there holds:

v1, v2 ∈VN ρ1, ρ2 ∈AN

We can define an inverse mapping v : AN → VN \ℝρ ↦ vρ

The Hohenberg-Kohn universal density functional

• Define the Hohenberg-Kohn universal density functional:

given by

• Consider the inverse mapping:

v : AN → VN \ℝρ ↦ vρ

<HK : AN → ℝ

<HK(ρ) = &0(vρ) − (vρ |ρ)

= ⟨Ψρ |HN(vρ) |Ψρ⟩ − (vρ |ρ)

Ψ ∈WN


ρΨΨv

ρv

Shifting the potential by a constantLemma:

• Any minimising ground-state wave-function of v is also minimising ground-state wave-function of v+c.

• No assumption on the uniqueness of the ground-state wave-function!

∀v ∈VN, ∀c ∈ℝ : &0,N(v + c) = &0,N(v)+ cN

HN(v + c) = T + Vne(v + c) + Vee = T + Vne(v) + cN+ Vee = HN(v) + cN .

Vne(v + c) =N

∑j= 1

(v(xj)+ c) =N

∑j= 1

v(xj)+ cN

&0(v + c, N ) = infΨ∈WN

⟨Ψ |HN(v + c) |Ψ⟩ = infΨ∈WN

⟨Ψ |HN(v) + cN |Ψ⟩

= infΨ∈WN

⟨Ψ |HN(v) |Ψ⟩+ cN ⟨Ψ |Ψ⟩ = &0(v, N )+ cN

The Hohenberg-Kohn universal density functional

&0(vρ+ c) − (vρ+ c |ρ) = &0(vρ)+ cN − (vρ |ρ)− cN = &0(vρ) − (vρ |ρ)

• Well-defined since

• Define the Hohenberg-Kohn universal density functional:

given by

• Consider the inverse mapping:

v : AN → VN \ℝρ ↦ vρ

<HK : AN → ℝ

<HK(ρ) = &0(vρ) − (vρ |ρ)

= ⟨Ψρ |HN(vρ) |Ψρ⟩ − (vρ |ρ)

The Hohenberg-Kohn universal density functional• Hohenberg-Kohn universal density functional:

• Alternative characterisation:

&0(vρ) = ⟨Ψρ |HN(vρ) |Ψρ⟩ ≤ ⟨Ψ |HN(vρ) |Ψ⟩ = ⟨Ψ |T + Vne(vρ) + Vee |Ψ⟩= ⟨Ψ |T + Vee |Ψ⟩ + (vρ |ρΨ)

<HK(ρ) = minΨ ∈WNρΨ = ρ

⟨Ψ |T + Vee |Ψ⟩

<HK(ρ) = ⟨Ψρ |HN(vρ) |Ψρ⟩ − (vρ |ρ) = ⟨Ψρ |T + Vne(vρ) + Vee |Ψρ⟩ − (vρ |ρ)= ⟨Ψρ |T + Vee |Ψρ⟩

∀Ψ ∈WN s.t. ρΨ = ρ : <HK(ρ) ≤ ⟨Ψ |T + Vee |Ψ⟩

FHK(ρ) = &0(vρ) − (vρ |ρ)

Lemma: let , be arbitrary. Then, there holds

• Equality holds if and only if is a ground-state density for .

Variational principles

• It follows the following variational principles

v ∈VN ρ ∈AN

&0(v) ≤ <HK(ρ) + (v |ρ)

vρ

<HK(ρ) ≥ &0(v) − (v |ρ)Fenchel’s inequality

&0(v) = minρ∈AN

(<HK(ρ) + (v |ρ))

<HK(ρ) = maxv∈VN

(&0(v) − (v |ρ))

(Hohenberg-Kohn variational principle)

(Lieb variational principle)

Hohenberg-Kohn theory• Despite this wonderful theory, the two sets and are unknown in practice

because they are defined implicitly

i.e. using the condition that a ground-state wave-function and density exists.

AN = {ρ | ∃v ∈VN, Ψ ∈WN s.t. ρ = ρΨ and Ψ is a ground-state of HN(v)}VN = {v ∈8 | inf

Ψ∈WN

⟨Ψ |HN(v) |Ψ⟩ admits a minimizer}

Ψ ∈WN


ρΨΨv

ρv

VN AN

• Unfortunately, this implicit definition is highly impractical from the point of view of constructing numerical methods and algorithms to solve the problem.

Levy-Lieb constrained-search functional• Introduce

IN = {ρ | ∃Ψ ∈WN s.t. ρΨ = ρ} no assumption on minimiser

= {ρ ≥ 0, ρ ∈H1(ℝ3), (ρ |1) = N} explicit characterisation

• Minimization of energy over densities:

&0(v) = infΨ∈WN

(⟨Ψ |T + Vee + Vne(v) |Ψ⟩ ) = infρ∈IN

infΨ ∈WNρΨ = ρ

(⟨Ψ |T + Vee + Vne(v) |Ψ⟩ )= inf

ρ∈IN

infΨ ∈WNρΨ = ρ

(⟨Ψ |T + Vee |Ψ⟩ + (ρΨ |v) )= inf

ρ∈IN( inf

Ψ ∈WNρΨ = ρ

⟨Ψ |T + Vee |Ψ⟩ + (ρ |v) )

&0(v) = infρ∈IN

(<LL(ρ) + (ρ |v) )<LL(ρ) = inf

Ψ ∈WNρΨ = ρ

⟨Ψ |T + Vee |Ψ⟩<LL : IN → ℝ

Kohn-Sham DFT-model• Key-idea: considering a reference-system of N non-interacting particles • Solve this system to obtain an approximation for the kinetic part of the density

functionalinf

Ψ ∈WNρΨ = ρ

⟨Ψ |T |Ψ⟩

• The nature of N non-interacting particles suggest that the solution can be written as a Slater determinant.

Φ = |ϕ1, … , ϕN⟩

⟨Φ |T |Φ⟩ =N

∑j= 1

(ϕj |− 12 Δx |ϕj) = 1

2N

∑j= 1

∥∇ϕj∥2• Then

NKS(ρ) := inf {12

N

∑j= 1

∫ℝ3|∇ϕj |

2 ϕj ∈H1(ℝ3), (ϕi |ϕj) = δij, ρΦ = ρ}• Unfortunately, a minimizing wave-function of (1) for a general density has is not

always in the form of a Slater determinant (there are known counterexamples). • However: insist on the above definition and use it as a good approximation of the

kinetic energy.

(1)

Building up the pieces for the Levy-Lieb functional• The Coulomb energy associated to the charge density given by

seems an obvious part that needs to be accounted for in the density functional as it represents the classical "mean-field" interaction of the electrons.

ρ

O(ρ) = 12 ∫ℝ3 ∫ℝ3

ρ(x) ρ(y)|x − y |

d x dy

• Existence of the exchange-correlation functional is provided but it is not known.

• We therefore make the following Ansatz:

where

<LL(ρ) = infΨ ∈WNρΨ = ρ

⟨Ψ |T + Vee |Ψ⟩ = NKS(ρ) + O(ρ) + &xc(ρ)

&xc(ρ) := <LL(ρ) − NKS(ρ) − O(ρ) Exchange-correlation functional

• One uses approximations in practise.• Is a "correction" of the errors that have been done in the approximation of the

kinetic energy (non-interacting electrons) and the electron-electron interaction using the classical Coulomb energy.

From DFT to Kohn-Sham DFT• The DFT-problem writes:

with

&0(v) = infρ∈IN

(<LL(ρ) + (ρ |v) )

<LL(ρ) = infΨ ∈WNρΨ = ρ

⟨Ψ |T + Vee |Ψ⟩ = NKS(ρ) + O(ρ) + &xc(ρ)

• Note that

<LL(ρ) + (ρ |v) = NKS(ρ) + O(ρ) + &xc(ρ) + (ρ |v)

= infΦ

ρΦ = ρ{1

2N

∑j= 1

∫ℝ3|∇ϕj |

2 ϕj ∈H1(ℝ3), (ϕi |ϕj) = δij} + O(ρ) + &xc(ρ) + (ρ |v)

= infΦ

ρΦ = ρ{1

2N

∑j= 1

∫ℝ3|∇ϕj |

2 + O(ρ) + &xc(ρ) + (ρ |v) ϕj ∈H1(ℝ3), (ϕi |ϕj) = δij}= inf

ΦρΦ = ρ

{12

N

∑j= 1

∫ℝ3|∇φj |

2 + O(ρΦ) + &xc(ρΦ) + (ρΦ |v) φj ∈H1(ℝ3), (φi |φj) = δij}

From DFT to Kohn-Sham DFT• Thus

with<LL(ρ) + (ρ |v) = inf

ΦρΦ = ρ

{&KS(Φ) ϕj ∈H1(ℝ3), (ϕi |ϕj) = δij}

&KS(Φ) = 12

N

∑j= 1

∫ℝ3|∇φj |

2 + O(ρΦ) + &xc(ρΦ) + (ρΦ |v)

&0,KS(v) = infΦ {&KS(Φ) ϕj ∈H1(ℝ3), (ϕi |ϕj) = δij}

&0(v) = infρ∈IN

(<LL(ρ) + (ρ |v) ) = infρ∈IN

infΦ

ρΦ = ρ{&KS(Φ) ϕj ∈H1(ℝ3), (ϕi |ϕj) = δij}

= infΦ {&KS(Φ) ϕj ∈H1(ℝ3), (ϕi |ϕj) = δij}

and

The exchange-correlation functional• A simple approximation is indeed a local approximation of the form

• This is known as Local Density Approximation. • Note this is a local operator (compare with the HF exchange operator).

ELDAxc (ρ) = ∫ℝ3

ρ(x) exc(ρ(x)) dx .

• An alternative is the Generalised Gradient Approximation (GGA):

EGGAxc (ρ) = ∫ℝ3

h(ρ(x), ∇ρ(x)) dx .

• A very popular choice is a mixing of the Hartree-Fock exchange functional, the LDA-functional and the GGA-functional: B3LYP

First-order optimality condition• Construct the Lagrange multiplier function by

:(Φ, Λ) := &KS(Φ) −N

∑i,j= 1

λij((φi |φj) − δij)

• The final result is given by: find such that

vxc,Φ (x) = exc(ρΦ(x)) + ρΦ(x) e′�xc(ρΦ(x))JΦ(x) = 12 ∫ℝ3

ρΦ(y)|x − y |

dy


Φ = {ϕ1, …, ϕN}

∀j, ∀ψ : (ψ |− 12 Δ + v |ϕj) + (ψ |JΦ + vxc,Φ |ϕj) = εj (ψ |ϕj)

• Different derivation than Hartree-Fock, but very similar structure at the end • Discretization and resolution: similar as Hartree-Fock

Summary of DFT• The DFT-model was introduced: unknown quantity is the electronic density

(low dimensional) instead of the wave-function. • Subtle details on the relation between potentials and densities. • Reformulation in terms of an explicitly-known space for densities: Levy-Lieb

constrained functional: is not known in general. • Idea of Kohn-Sham: consider a reference system of independent particles

plus Coulomb energy. • Fix the rest through the exchange-correlation functional. • Efficient models but not systematic improvable. • Euler-Lagrange functionals and general structure similar to the case of

Hartree-Fock

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