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FYS4411/FYS9411 Computational Physics IIUniversity of Oslo, Norway

Morten Hjorth-Jensen

1Department of Physics,University of Oslo, N-0316 Oslo, Norway

National Superconducting Cyclotron Laboratory and Department of Astronomyand Physics,

Michigan State University, East Lansing, Michigan, USA

Spring 2014

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Topics for Week 3, January 13-17

Introduction, discussion of project and definitions

I Presentation of topics to be covered and definitionsI Discussion of the project and first stepsI Reading assignment for next week: study chapter 4 of

Thijssen’s text and chapter 7 of lectures notes of MHJI Computational assignment: solve numerically the

one-body hydrogen-like problem (this week and nextweek).

I Form as soon as possible collaboration groups of 2-3participants.

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January 13 - May 31

Course overview, Aims and Computational aspects

I The aim is to use the Hartree-Fock method to study atomsand molecules

I The results can be used to provide potentials and chargedistributions for molecular dynamics simulations

I Eigenvalue solvers, efficient computations of integralsI Iterative methods for solutions of non-linear equationsI Object orientation and efficient parallelizationI The first part of the project (ready by March 7) deals with

Hartree-Fock calculations of closed-shell atoms like He, Beand Ne using standard hydrogen-like single-particle statefunctions.

I The second part includes optimized single-particle statefunctions and computations of both atoms and molecules

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January 13 - May 31

Projects, deadlines and oral exam

1. Deadline part 1: March 72. Deadline full project: 31 May3. Oral exam: To be settled

The oral exam is based on your presentation of the project.

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January 13 - May 31

More on projects

1. Keep a logbook, important for keeping track of all yourchanges etc etc.

2. The project should be written as a regular scientific article,with introduction, formalism, codes which have beendeveloped and discussion of results. Conclusions andreferences should also be included. An example can befound on the webpage of the course.

The methods are relevant for atomic, molecular,solid state,materials science, lief science, nanotechnology, quantumchemistry and nuclear physics.

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Lectures and ComputerLab

I Lectures: Thursday (14.15-16, room FV329)I Detailed lecture notes, all programs presented and

projects can be found at the homepage of the course.I Computerlab: 16-19 thursday, room FV329I Weekly plans and relevant information are on the official

webpage.I Chapters 7 and 16 of the FYS3150 lecture notes give a

good starting point. Chapter 16 will also be revised. Werecommend also J. M. Thijssen text ComputationalPhysics, chapters 3-6. For MPI we recommend Gropp,Lusk and Sjellum’s text.

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Thijssen’s text

J. M. Thijssen’s text

I ComputationalPhysics

I Chapters 3-6I see http://www.tn.tudelft.nl/tn/People/Staff/Thijssen/comphybook.html

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http://www.tn.tudelft.nl/tn/People/Staff/Thijssen/comphybook.htmlhttp://www.tn.tudelft.nl/tn/People/Staff/Thijssen/comphybook.htmlhttp://www.tn.tudelft.nl/tn/People/Staff/Thijssen/comphybook.htmlhttp://www.tn.tudelft.nl/tn/People/Staff/Thijssen/comphybook.htmlhttp://www.tn.tudelft.nl/tn/People/Staff/Thijssen/comphybook.html

MPI text

Gropp, Lusk andSjellum

I Using MPII Chapters 1-5I seehttp://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10761

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http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10761http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10761http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10761http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10761http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10761

Selected Texts and lectures on C/C++

J. J. Barton and L. R. Nackman,Scientific and Engineering C++, Addison Wesley,3rd edition 2000.

B. Stoustrup, The C++ programming language, Pearson, 1997.

George Em Karniadakis and Robert M. Kirby II, Parallel Scientific Computing inC++ and MPI http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521520805

D. Yang, C++ and Object-oriented Numeric Computing for Scientists andEngineers, Springer 2000.

More books reviewed at http:://www.accu.org/ andhttp://www.comeaucomputing.com/booklist/

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http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521520805http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521520805http:://www.accu.org/http://www.comeaucomputing.com/booklist/

Definitions

The quantum mechanical wave function of a given state withquantum numbers λ (encompassing all quantum numbersneeded to specify the system), ignoring time, is with N particles(electrons in our case)

Ψλ = Ψλ(x1, x2, . . . , xN),

with xi = (ri , σi) and the projection of σi takes the values{−1/2,+1/2} for fermions with spin 1/2. We will hereafteralways refer to Ψλ as the exact wave function, and if the groundstate is not degenerate we label it as

Ψ0 = Ψ0(x1, x2, . . . , xN).

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Definitions

Since the solution Ψλ seldomly can be found in closed form,approximations are sought. In these slides we define anapproximative wave function or an ansatz to the exact wavefunction as

Φλ = Φλ(x1, x2, . . . , xN),

withΦ0 = Φ0(x1, x2, . . . , xN),

being the ansatz to the ground state.

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Definitions

The wave function Ψλ is sought in the Hilbert space of eithersymmetric or anti-symmetric N-body functions, namely

Ψλ ∈ HN := H1 ⊕H1 ⊕ · · · ⊕ H1,

where the single-particle Hilbert space H1 is the space ofsquare integrable functions over ∈ Rd ⊕ (σ) resulting in

H1 := L2(Rd ⊕ (σ)).

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DefinitionsOur Hamiltonian is invariant under the permutation(interchange) of two fermions. Since we deal with fermionshowever, the total wave function is antisymmetric. Let P̂ be anoperator which interchanges two fermions. Due to thesymmetries we have ascribed to our Hamiltonian, this operatorcommutes with the total Hamiltonian,

[Ĥ, P̂] = 0,

meaning that Ψλ(x1, x2, . . . , xN) is an eigenfunction of P̂ as well,that is

P̂ijΨλ(x1, x2, . . . , xi , . . . , xj , . . . , xN) = βΨλ(x1, x2, . . . , xj , . . . , xi , . . . , xN),

where β is the eigenvalue of P̂. We have introduced the suffix ijin order to indicate that we permute fermions i and j . The Pauliprinciple tells us that the total wave function for a system offermions has to be antisymmetric, resulting in the eigenvalueβ = −1.

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Definitions and notations

The Schrödinger equation reads

Ĥ(x1, x2, . . . , xN )Ψλ(x1, x2, . . . , xN ) = EλΨλ(x1, x2, . . . , xN ), (1)

where the vector xi represents the coordinates (spatial and spin) of particle i , λ stands

for all the quantum numbers needed to classify a given N-particle state and Ψλ is the

pertaining eigenfunction. Throughout this course, Ψ refers to the exact eigenfunction,

unless otherwise stated.

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We write the Hamilton operator, or Hamiltonian, in a generic way

Ĥ = T̂ + V̂

where T̂ represents the kinetic energy of the system

T̂ =N∑

i=1

p2i2mi

=N∑

i=1

(−

~2

2mi∇i2)

=N∑

i=1

t(xi )

while the operator V̂ for the potential energy is given by

V̂ =N∑

i=1

ûext(xi ) +N∑

ji=1

v(xi , xj ) +N∑

ijk=1

v(xi , xj , xk ) + . . . (2)

Hereafter we use natural units, viz. ~ = c = e = 1, with e the elementary charge and c

the speed of light. This means that momenta and masses have dimension energy.

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If one does quantum chemistry or atomic physics as we do here, after havingintroduced the Born-Oppenheimer approximation which effectively freezes out thenucleonic degrees of freedom, the Hamiltonian for N = ne electrons takes the followingform

Ĥ =ne∑

i=1

t(xi )−ne∑

i=1

kZri

+

ne∑i


We can rewrite this as

Ĥ = Ĥ0 + ĤI =ne∑

i=1

ĥ0(xi ) +ne∑

i


The potential energy term due to the attraction of the nucleus defines the one-bodyfield ui = uext(xi ) of Eq. (2). We have moved this term into the Ĥ0 part of theHamiltonian, instead of keeping it in V̂ as in Eq. (2). The reason is that we willhereafter treat Ĥ0 as our non-interacting Hamiltonian. For a many-body wavefunctionΦλ defined by an appropriate single-particle basis, we may solve exactly thenon-interacting eigenvalue problem

Ĥ0Φλ = wλΦλ,

with wλ being the non-interacting energy. This energy is defined by the sum over

single-particle energies to be defined below. For atoms the single-particle energies

could be the hydrogen-like single-particle energies corrected for the charge Z . For

nuclei and quantum dots, these energies could be given by the harmonic oscillator in

three and two dimensions, respectively.

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We will assume that the interacting part of the Hamiltonian can be approximated by atwo-body interaction. This means that our Hamiltonian is written as

Ĥ = Ĥ0 + ĤI =N∑

i=1

ĥ0(xi ) +N∑

i


Our Hamiltonian is invariant under the permutation (interchange) of two fermions.Since we deal with fermions, the total wave function is antisymmetric. Let P̂ be anoperator which interchanges two fermions. Due to the symmetries we have ascribed toour Hamiltonian, this operator commutes with the total Hamiltonian,

[Ĥ, P̂] = 0,

meaning that Ψλ(x1, x2, . . . , xN ) is an eigenfunction of P̂ as well, that is

P̂ij Ψλ(x1, x2, . . . , xi , . . . , xj , . . . , xN ) = βΨλ(x1, x2, . . . , xi , . . . , xj , . . . , xN ),

where β is the eigenvalue of P̂. We have introduced the suffix ij in order to indicate that

we permute fermions i and j . The Pauli principle tells us that the total wave function for

a system of fermions has to be antisymmetric, resulting in the eigenvalue β = −1.

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In our case we assume that we can approximate the exact eigenfunction with a Slaterdeterminant

Φ(x1, x2, . . . , xN , α, β, . . . , σ) =1√

N!

∣∣∣∣∣∣∣∣∣ψα(x1) ψα(x2) . . . . . . ψα(xN )ψβ(x1) ψβ(x2) . . . . . . ψβ(xN ). . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

ψσ(x1) ψσ(x2) . . . . . . ψσ(xN )

∣∣∣∣∣∣∣∣∣ ,(7)

where xi stand for the coordinates and spin values of a particle i and α, β, . . . , γ are

quantum numbers needed to describe remaining quantum numbers.

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Let us denote the ground state energy by E0. According to the variational principle wehave

E0 ≤ E [Φ] =∫

Φ∗ĤΦdτ

where Φ is a trial function which we assume to be normalized∫Φ∗Φdτ = 1,

where we have used the shorthand dτ = dx1dx2 . . . dxN .

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In the Hartree-Fock method the trial function is the Slater determinant of Eq. (7) whichcan be rewritten as

Φ(x1, x2, . . . , xN , α, β, . . . , ν) =1√

N!

∑P

(−)P P̂ψα(x1)ψβ(x2) . . . ψν(xN ) =√

N!AΦH ,

(8)

where we have introduced the antisymmetrization operator A defined by thesummation over all possible permutations of two fermions.

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Definition and notations

Before we proceed, we need to discuss the single-particle states which enter thedefinition of the Slater determinant. We will start with hydrogen-like orbits and write asmall program which diagonalizes the hydrogen-like problem. To achieve this, we needto refresh our knowledge about the Schrödinger equation for the hydrogen atom.To solve the Schrödinger equation as a matrix diagonalization problem, let us study theradial part of the Schrödinger equation. The radial part of the wave function, R(r), is asolution to

−~2

2m

(1r2

ddr

r2ddr−

l(l + 1)r2

)R(r) + V (r)R(r) = ER(r).

In our case V (r) = −Z/r .

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Then we substitute R(r) = (1/r)u(r) and obtain

−~2

2md2

dr2u(r) +

(V (r) +

l(l + 1)r2

~2

2m

)u(r) = Eu(r).

We introduce a dimensionless variable ρ = (1/α)r where α is a constant withdimension length and get

−~2

2mα2d2

dρ2u(ρ) +

(V (ρ) +

l(l + 1)ρ2

~2

2mα2

)u(ρ) = Eu(ρ).

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Definition and notationsWe replace ρ with x , take away the centrifugal barrier (that is setting l = 0) term andset the potential equal to

V (x) = −kZxα

,

with k = 1.44 eVnm being a constant. We multiply with mα2/~2 and since α is just aconstant, we fix it by requiring that

kZmα~2

= 1,

giving

α =~2

kZm,

which is the Bohr radius and leads to a natural length scale. This leads to the equation

−12

d2

dx2u(x)−

1x

u(x) = λu(x),

with

λ =Emα2

~2.

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Let us now see how we can rewrite this equation as a matrix eigenvalue problem. Firstwe need to compute the second derivative. We use here the following expression forthe second derivative of a function f

f ′′ =f (x + h)− 2f (x) + f (x − h)

h2+ O(h2),

where h is our step. Next we define minimum and maximum values for the variable x ,Rmin and Rmax, respectively. With a given number of steps, Nstep, we then define thestep h as

h =Rmax − Rmin

Nstep.

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We discretize x as

xi = Rmin + ih i = 1, 2, . . . ,Nstep − 1

we can rewrite the Schrödinger equation for xi as

−12

u(xk + h)− 2u(xk ) + u(xk − h)h2

−1xk

u(xk ) = λu(xk ),

or in a more compact way

−12

uk+1 − 2uk + uk−1h2

−1xk

uk = −12

uk+1 − 2uk + uk−1h2

+ Vk uk = λuk ,

where uk = u(xk ), uk±1 = u(xk ± h) and Vk = −1/xk , the given potential.

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Let us see how this recipe may lead to a matrix reformulation of the Schrödingerequation. Define first the diagonal matrix element

dk =1h2

+ Vk ,

and the non-diagonal matrix element

ek = −1

2h2.

In this case the non-diagonal matrix elements are given by a mere constant. Allnon-diagonal matrix elements are equal. With these definitions the Schrödingerequation takes the following form

dk uk + ek−1uk−1 + ek+1uk+1 = λuk ,

where uk is unknown.

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Since we have Nstep values of k we can write the latter equation as a matrix eigenvalueproblem

d1 e1 0 0 . . . 0 0e1 d2 e2 0 . . . 0 00 e2 d3 e3 0 . . . 0. . . . . . . . . . . . . . . . . . . . .0 . . . . . . . . . . . . dNstep−1 eNstep−10 . . . . . . . . . . . . eNstep−1 dNstep

u1u2. . .. . .. . .

uNstep

= λ

u1u2. . .. . .. . .

uNstep

(9)

When setting up the matrix, be careful with the endpoints. Our wave function has the

values u(0) = u(∞) = 0 and you don’t need to include these points in the equations.

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This is a matrix problem with a tridiagonal matrix of dimension Nstep × Nstep and willthus yield Nstep eigenvalues.The algorithm for solving this problem may take the following form

I Define values for Nstep, Rmin and Rmax. These values define in turn the step sizeh. Typical values for Rmax and Rmin could be 0 and 10 respectively for thelowest-lying states. The number of mesh points Nstep could be in the range 100to some thousands. You can check the stability of the results as functions of Nstepand Rmax and Rmin against the exact solutions.

I Construct then two one-dimensional arrays which contain all values of xk and thepotential Vk . For the latter it can be convenient to write a small function whichsets up the potential as function of xk . For the three-dimensional case you mayalso need to include the centrifugal potential.

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I Construct thereafter the one-dimensional vectors d and e, where d stands forthe diagonal matrix elements and e the non-diagonal ones. Be careful with theendpoints, since we know the wave function u at both ends of the chosen grid.

I We are now ready to obtain the eigenvalues by calling the function tqli whichcan be found on the web page of the course FYS3150, see fall 2013. Calling tqli,you have to transfer the vectors d and e and their dimension dimension n = Nstepand a matrix z of dimension Nstep1× Nstep which returns the eigenfunctions. Onreturn, the array d contains the eigenvalues. If z is given as the unity matrix oninput, it returns the eigenvectors. For a given eigenvalue k , the eigenvector isgiven by the column k in z, that is z[][k] in C, or z(:,k) in Fortran.

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I TQLI does however not return an ordered sequence of eigenvalues. You maythen need to sort them as e.g., an ascending series of numbers. The programwe provide includes a sorting function as well.

This program is discussed in chapter 7 of the FYS3150 Lecture notes.

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I Your task this week and next week is to set up a program which solves the aboveproblem.

I The program TQLI can be found in the lib.cpp (use also the lib.h file) at thewebpage of FYS3150, click on the program link of the Fall 2013 version of thecourse. Read also chapter 7 of the FYS3150 Lectures.

I The eigenvalue for the lowest state should be −1/2 for the hydrogen-likeproblem. You should also plot the eigenvector. Compute also the second l = 0state, both its energy and its eigenfuntion.

I Try also to include the centrifugal barrier and find the lowest eigenstate andeigenvector for l = 1 and comment your results.

I At the lab, we will also discuss how to use armadillo instead of the standardmemory allocation done in chapter 7.

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It is defined asA =

1N!

∑p

(−)pP̂, (10)

with p standing for the number of permutations. We have introduced for later use theso-called Hartree-function, defined by the simple product of all possible single-particlefunctions

ΦH (x1, x2, . . . , xN , α, β, . . . , ν) = ψα(x1)ψβ(x2) . . . ψν(xN ).

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Both Ĥ0 andˆ̂

IH are invariant under all possible permutations of any two fermions andhence commute with A

[H0,A] = [HI ,A] = 0. (11)

Furthermore, A satisfiesA2 = A, (12)

since every permutation of the Slater determinant reproduces it.

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The expectation value of Ĥ0∫Φ∗Ĥ0Φdτ = N!

∫Φ∗HAĤ0AΦHdτ

is readily reduced to ∫Φ∗Ĥ0Φdτ = N!

∫Φ∗H Ĥ0AΦHdτ,

where we have used eqs. (11) and (12). The next step is to replace theantisymmetrization operator by its definition Eq. (8) and to replace Ĥ0 with the sum ofone-body operators

∫Φ∗Ĥ0Φdτ =

N∑i=1

∑p

(−)p∫

Φ∗H ĥ0P̂ΦHdτ.

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The integral vanishes if two or more fermions are permuted in only one of theHartree-functions ΦH because the individual single-particle wave functions areorthogonal. We obtain then

∫Φ∗Ĥ0Φdτ =

N∑i=1

∫Φ∗H ĥ0ΦHdτ.

Orthogonality of the single-particle functions allows us to further simplify the integral,and we arrive at the following expression for the expectation values of the sum ofone-body Hamiltonians

∫Φ∗Ĥ0Φdτ =

N∑µ=1

∫ψ∗µ(x)ĥ0ψµ(x)dx. (13)

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We introduce the following shorthand for the above integral

〈µ|ĥ0|µ〉 =∫ψ∗µ(x)ĥ0ψµ(x)dx.,

and rewrite Eq. (13) as ∫Φ∗Ĥ0Φdτ =

N∑µ=1

〈µ|ĥ0|µ〉. (14)

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The expectation value of the two-body part of the Hamiltonian (assuming a two-bodyHamiltonian at most) is obtained in a similar manner. We have∫

Φ∗ĤIΦdτ = N!∫

Φ∗HAĤIAΦHdτ,

which reduces to

∫Φ∗ĤIΦdτ =

N∑i≤j=1

∑p

(−)p∫

Φ∗H v̂(xij )P̂ΦHdτ,

by following the same arguments as for the one-body Hamiltonian.

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Because of the dependence on the inter-particle distance rij , permutations of any twofermions no longer vanish, and we get

∫Φ∗ĤIΦdτ =

N∑i


We obtain

∫Φ∗ĤIΦdτ =

12

N∑µ=1

N∑ν=1

[∫ψ∗µ(xi )ψ

∗ν(xj )v̂(xij )ψµ(xi )ψν(xj )dxi dxj

−∫ψ∗µ(xi )ψ

∗ν(xj )v̂(xij )ψν(xi )ψµ(xj )dxi dxj

].

(15)

The first term is the so-called direct term. In Hartree-Fock theory it leads to the

so-called Hartree term, while the second is due to the Pauli principle and is called the

exchange term and in Hartree-Fock theory it defines the so-called Fock term. The

factor 1/2 is introduced because we now run over all pairs twice.

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The last equation allows us to introduce some further definitions. The single-particlewave functions ψµ(x), defined by the quantum numbers µ and x (recall that x alsoincludes spin degree, later we will also add isospin) are defined as the overlap

ψα(x) = 〈x |α〉.

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We introduce the following shorthands for the above two integrals

〈µν|V |µν〉 =∫ψ∗µ(xi )ψ

∗ν(xj )v̂(xij )ψµ(xi )ψν(xj )dxi dxj ,

and〈µν|V |νµ〉 =

∫ψ∗µ(xi )ψ

∗ν(xj )v̂(xij )ψν(xi )ψµ(xj )dxi dxj .

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The direct and exchange matrix elements can be brought together if we define theantisymmetrized matrix element

〈µν|V |µν〉AS = 〈µν|V |µν〉 − 〈µν|V |νµ〉,

or for a general matrix element

〈µν|V |στ〉AS = 〈µν|V |στ〉 − 〈µν|V |τσ〉.

It has the symmetry property

〈µν|V |στ〉AS = −〈µν|V |τσ〉AS = −〈νµ|V |στ〉AS.

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The antisymmetric matrix element is also hermitian, implying

〈µν|V |στ〉AS = 〈στ |V |µν〉AS.

With these notations we rewrite Eq. (15) as

∫Φ∗ĤIΦdτ =

12

N∑µ=1

N∑ν=1

〈µν|V |µν〉AS. (16)

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Combining Eqs. (14) and (24) we obtain the energy functional

E [Φ] =N∑µ=1

〈µ|ĥ0|µ〉+12

N∑µ=1

N∑ν=1

〈µν|V |µν〉AS. (17)

This equation is very useful, in particular if we only look at the unperturbed part H0.

This part can be represented by closed form expressions that can be used to check our

algorithms.

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Hartree-Fock theory

I Repetition from last weekI Derivation of the Hartree-Fock equations by varying the

single-particle functions or the coefficients of thesingle-particle equations (to be continued next week)

I Reading assignment for next week: chapter 3(background) and chapter 4.1-4.4 of Thijssen. The link tothe text has been sent to you by email.

See also suggested exercises this week.

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Hartree-Fock: our many-body approach in this course

HF theory is an algorithm for a finding an approximative expression for the ground stateof a given Hamiltonian. The basic ingredients are

I Define a single-particle basis {ψα} so that

ĥHFψα = εαψα

withĥHF = t̂ + ûext + ûHF

I where ûHF is a single-particle potential to be determined by the HF algorithm.

I The HF algorithm means to choose ûHF in order to have

〈Ĥ〉 = EHF = 〈Φ0|Ĥ|Φ0〉

a local minimum with Φ0 being the SD ansatz for the ground state.

I The variational principle ensures that EHF ≥ Ẽ0, Ẽ0 the exact ground stateenergy.

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How to solve Schrödinger’s equation for single-particlepotentials

Last week we showed that we rewrite the SE as an eigenvalue problem

d1 e1 0 0 . . . 0 0e1 d2 e2 0 . . . 0 00 e2 d3 e3 0 . . . 0. . . . . . . . . . . . . . . . . . . . .0 . . . . . . . . . . . . dNstep−1 eNstep−10 . . . . . . . . . . . . eNstep−1 dNstep

u1u2. . .. . .. . .

uNstep

= λ

u1u2. . .. . .. . .

uNstep

(18)

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This is a matrix problem with a tridiagonal matrix of dimension Nstep × Nstep and willthus yield Nstep eigenvalues.The algorithm for solving this problem may take the following form

I Define values for Nstep, Rmin and Rmax. These values define in turn the step sizeh. Typical values for Rmax and Rmin could be 0 and 10 respectively for thelowest-lying states. The number of mesh points Nstep could be in the range 100to some thousands. You can check the stability of the results as functions of Nstepand Rmax and Rmin against the exact solutions.

I Construct then two one-dimensional arrays which contain all values of xk and thepotential Vk . For the latter it can be convenient to write a small function whichsets up the potential as function of xk . For the three-dimensional case you mayalso need to include the centrifugal potential.

51 / 234


I Construct thereafter the one-dimensional vectors d and e, where d stands forthe diagonal matrix elements and e the non-diagonal ones. Be careful with theendpoints, since we know the wave function u at both ends of the chosen grid.

I We are now ready to obtain the eigenvalues by calling the function tqli , see thecode examples at the webpage. Calling tqli, you have to transfer the vectors dand e and their dimension dimension n = Nstep and a matrix z of dimensionNstep1× Nstep which returns the eigenfunctions. On return, the array d containsthe eigenvalues. If z is given as the unity matrix on input, it returns theeigenvectors. For a given eigenvalue k , the eigenvector is given by the column kin z, that is z[][k] in C, or z(:,k) in Fortran. Alternatively, if you use armadillo, seebelow, you would call one of the armadillo functions or lapack functions.

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Example of c++ code for solving Schrödinger’sequation and read and write to file

We are using armadillo here. For standard c++, see the code example on thewebpage. Idem for the Fortran code.

/∗Solves the one−p a r t i c l e Schrodinger equat ionf o r a p o t e n t i a l s p e c i f i e d i n f u n c t i o np o t e n t i a l ( ) . This example i s f o r the harmonic

o s c i l l a t o rcompile as c++ −O3 −o ho3d . x ho3d . cpp − l a r m a d i l l o

− l l apack − l b l a s∗ /#include #include #include #include #include using namespace s td ;using namespace arma ;/ / ou tput f i l e as g loba l v a r i a b l eofstream o f i l e ;

53 / 234


/ / f u n c t i o n dec la r a t i onsdouble P o t e n t i a l ( double ) ;/ / Main program s t a r t s herei n t main ( i n t argc , char∗ argv [ ] ){

i n t i , j , MaxStep , OrbL ;double Rmax, Step , Const1 , Const2 ,

O r b i t a l F a c t o r ;char ∗ out f i lename ;

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/ / Read i n output f i l e , abor t i f there are toofew command− l i n e arguments

i f ( argc


/ / i n i t i a l i s e constants assuming Rmin = 0 . 0 ;Step = Rmax / ( MaxStep+1) ;Const2 = −0.5 / ( Step ∗ Step ) ; Const1 = 1.0 / (

Step ∗ Step ) ;O r b i t a l F a c t o r = OrbL ∗ ( OrbL + 1) ;/ / se t up r , d iagonal and non−diagonal elementsvec r ( MaxStep+1) , d ( MaxStep−1) , e ( MaxStep−1) ;for ( i = 0 ; i < MaxStep−1; i ++) {

r ( i ) = ( i +1) ∗ Step ; / / avoid r = 0 and i n f t y, known values

d ( i ) = Const1+ P o t e n t i a l ( r ( i ) ) + 0.5∗O r b i t a l F a c t o r / ( r ( i ) ∗ r ( i ) ) ;

}

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e = Const2 ;mat A( MaxStep−1,MaxStep−1) ;/ / Set up mat r i xfor ( i = 0 ; i < MaxStep−1; i ++) {

/ / Diagonal elementsA( i , i ) = d ( i ) ;/ / Non−diagonal elementsi f ( i < MaxStep − 2) {

A( i , i + 1) = Const2 ;A( i + 1 , i ) = Const2 ;

}}

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vec e i g v a l ( MaxStep−1) ;mat eigvec ( MaxStep−1,MaxStep−1) ;/ / The e igenvectors are normal izedeig sym ( e igva l , eigvec , A) ;

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/ / p r i n t r e s u l t so f i l e


/∗The f u n c t i o n p o t e n t i a l ( )c a l c u l a t e s and r e t u r n the value o f thep o t e n t i a l f o r a given argument x .The p o t e n t i a l here i s the harmonic o s c i l l a t o r

∗ /double P o t e n t i a l ( double x ){

return 0.5∗ x∗x ;

} / / End : f u n c t i o n p o t e n t i a l ( )

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Programming exercise week 4

Hydrogen-like SEThis exercise is meant to get you started with programming andyou are to solve the above described single-particle problemnumerically. You can use the codes on the webpage asexamples or simply write you own code.

I Compute the lowest eigenvalue for hydrogen-like atoms(replace the harmonic oscillator potential with an attractiveCoulomb interaction −Z/r . Compare with the exact values

I Add also a centrifugal barrier and compute the eigenvaluesfor 2p state.

I Plot the wave functions for the 1s, 2s and 2p states andcomment your results.

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Paper and pencil exercises week 4 and 5

Slater determinantsFor a two-electron system, the wave function can be written as

Ψ(x1, x2) = Φ(r1, r2) · χ(s1, s2) (19)

Because the wave function Ψ is antisymmetric under particle exchange, we may take Φsymmetric in 1 and 2 and χ antisymmetric, or vice versa.We construct the wave function Φ and χ from the orthonormal spatial orbitals φ1(r),φ2(r) and the spin-up and -down functions α(s) and β(s) respectively.

1. Write down the antisymmetric wave functions which can be constructed in thisway.

2. Write down all possible Slater determinants which can be built from theone-electron spin-orbitals consisting of a product of one of the spatial orbitals φ1and φ2 and a spin-up or -down spinor.

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Paper and pencil exercises week 4 and 5

Slater determinantsConsider a Slater determinant;

ΨAS =1√

N!

∣∣∣∣∣∣∣∣∣ψ1(x1) ψ1(x2) . . . . . . ψ1(xN )ψ2(x1) ψ2(x2) . . . . . . ψ2(xN ). . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

ψN (x1) ψN (x2) . . . . . . ψN (xN )

∣∣∣∣∣∣∣∣∣ , (20)where ψk (x) are orthonormal single-particle states.

1. Show that the Slater determinant is normalized.

2. Show that the density of electrons with coordinates x, is given by

n(x) = N∫

dx2 . . . dxN |ΨAS(x, x2, . . . , xN )|2 (21)

can be written in terms of the ψk as

n(x) =∑

k

|ψk (x)|2. (22)

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Background material: Variational Calculus andLagrangian Multiplier

The calculus of variations involves problems where the quantity to be minimized ormaximized is an integral.In the general case we have an integral of the type

E [Φ] =∫ b

af (Φ(x),

∂Φ

∂x, x)dx ,

where E is the quantity which is sought minimized or maximized. The problem is that

although f is a function of the variables Φ, ∂Φ/∂x and x , the exact dependence of Φ

on x is not known. This means again that even though the integral has fixed limits a

and b, the path of integration is not known. In our case the unknown quantities are the

single-particle wave functions and we wish to choose an integration path which makes

the functional E [Φ] stationary. This means that we want to find minima, or maxima or

saddle points. In physics we search normally for minima. Our task is therefore to find

the minimum of E [Φ] so that its variation δE is zero subject to specific constraints. In

our case the constraints appear as the integral which expresses the orthogonality of

the single-particle wave functions. The constraints can be treated via the technique of

Lagrangian multipliers

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Background material: Euler-Lagrange equations

We assume the existence of an optimum path, that is a path for which E [Φ] isstationary. There are infinitely many such paths. The difference between two paths δΦis called the variation of Φ.We call the variation η(x) and it is scaled by a factor α. The function η(x) is arbitraryexcept for

η(a) = η(b) = 0,

and we assume that we can model the change in Φ as

Φ(x , α) = Φ(x , 0) + αη(x),

andδΦ = Φ(x , α)− Φ(x , 0) = αη(x).

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We choose Φ(x , α = 0) as the unkonwn path that will minimize E . The valueΦ(x , α 6= 0) describes a neighbouring path.We have

E [Φ(α)] =∫ b

af (Φ(x , α),

∂Φ(x , α)∂x

, x)dx .

In the slides I will use the shorthand

Φx (x , α) =∂Φ(x , α)∂x

.

In our case a = 0 and b =∞ and we know the value of the wave function.

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The condition for an extreme of

E [Φ(α)] =∫ b

af (Φ(x , α),Φx (x , α), x)dx ,

is [∂E [Φ(α)]

∂x

]α=0

= 0.

The α dependence is contained in Φ(x , α) and Φx (x , α) meaning that[∂E [Φ(α)]

∂α

]=

∫ ba

(∂f∂Φ

∂Φ

∂α+

∂f∂Φx

∂Φx

∂α

)dx .

We have defined∂Φ(x , α)∂α

= η(x)

and thereby∂Φx (x , α)

∂α=

d(η(x))dx

.

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Using∂Φ(x , α)∂α

= η(x),

and∂Φx (x , α)

∂α=

d(η(x))dx

,

in the integral gives[∂E [Φ(α)]

∂α

]=

∫ ba

(∂f∂Φ

η(x) +∂f∂Φx

d(η(x))dx

)dx .

Integrate the second term by parts∫ ba

∂f∂Φx

d(η(x))dx

dx = η(x)∂f∂Φx|ba −

∫ baη(x)

ddx

∂f∂Φx

dx ,

and since the first term dissappears due to η(a) = η(b) = 0, we obtain[∂E [Φ(α)]

∂α

]=

∫ ba

(∂f∂Φ−

ddx

∂f∂Φx

)η(x)dx = 0.

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[∂E [Φ(α)]

∂α

]=

∫ ba

(∂f∂Φ−

ddx

∂f∂Φx

)η(x)dx = 0,

can also be written as

α

[∂E [Φ(α)]

∂α

]α=0

=

∫ ba

(∂f∂Φ−

ddx

∂f∂Φx

)δΦ(x)dx = δE = 0.

The condition for a stationary value is thus a partial differential equation

∂f∂Φ−

ddx

∂f∂Φx

= 0,

known as Euler’s equation. Can easily be generalized to more variables.

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Background material: Lagrangian Multipliers

Consider a function of three independent variables f (x , y , z) . For the function f to bean extreme we have

df = 0.

A necessary and sufficient condition is

∂f∂x

=∂f∂y

=∂f∂z

= 0,

due to

df =∂f∂x

dx +∂f∂y

dy +∂f∂z

dz.

In physical problems the variables x , y , z are often subject to constraints (in our case Φ

and the orthogonality constraint) so that they are no longer all independent. It is

possible at least in principle to use each constraint to eliminate one variable and to

proceed with a new and smaller set of independent varables.

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The use of so-called Lagrangian multipliers is an alternative technique when theelimination of of variables is incovenient or undesirable. Assume that we have anequation of constraint on the variables x , y , z

φ(x , y , z) = 0,

resulting in

dφ =∂φ

∂xdx +

∂φ

∂ydy +

∂φ

∂zdz = 0.

Now we cannot set anymore

∂f∂x

=∂f∂y

=∂f∂z

= 0,

if df = 0 is wanted because there are now only two independent variables! Assume x

and y are the independent variables. Then dz is no longer arbitrary.

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However, we can add to

df =∂f∂x

dx +∂f∂y

dy +∂f∂z

dz,

a multiplum of dφ, viz. λdφ, resulting in

df + λdφ = (∂f∂z

+ λ∂φ

∂x)dx + (

∂f∂y

+ λ∂φ

∂y)dy + (

∂f∂z

+ λ∂φ

∂z)dz = 0.

Our multiplier is chosen so that

∂f∂z

+ λ∂φ

∂z= 0.

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However, we took dx and dy as to be arbitrary and thus we must have

∂f∂x

+ λ∂φ

∂x= 0,

and∂f∂y

+ λ∂φ

∂y= 0.

When all these equations are satisfied, df = 0. We have four unknowns, x , y , z and λ.Actually we want only x , y , z, λ need not to be determined, it is therefore often calledLagrange’s undetermined multiplier. If we have a set of constraints φk we have theequations

∂f∂xi

+∑

k

λk∂φk∂xi

= 0.

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Background material: Variational Calculus andLagrangian Multipliers

Let us specialize to the expectation value of the energy for one particle inthree-dimensions. This expectation value reads

E =∫

dxdydzψ∗(x , y , z)Ĥψ(x , y , z),

with the constraint ∫dxdydzψ∗(x , y , z)ψ(x , y , z) = 1,

and a HamiltonianĤ = −

12∇2 + v̂(x , y , z).

I will skip the variables x , y , z below, and write for example v̂(x , y , z) = V .

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The integral involving the kinetic energy can be written as, if we assume periodicboundary conditions or that the function ψ vanishes strongly for large values of x , y , z,∫

dxdydzψ∗(−

12∇2)ψdxdydz = −ψ∗∇ψ|+

∫dxdydz

12∇ψ∗∇ψ.

Inserting this expression into the expectation value for the energy and taking thevariational minimum we obtain

δE = δ{∫

dxdydz(

12∇ψ∗∇ψ + Vψ∗ψ

)}= 0.

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The constraint appears in integral form as∫dxdydzψ∗ψ = constant,

and multiplying with a Lagrangian multiplier λ and taking the variational minimum weobtain the final variational equation

δ

{∫dxdydz

(12∇ψ∗∇ψ + Vψ∗ψ − λψ∗ψ

)}= 0.

Introducing the function f

f =12∇ψ∗∇ψ + Vψ∗ψ − λψ∗ψ =

12

(ψ∗xψx + ψ∗yψy + ψ

∗zψz ) + Vψ

∗ψ − λψ∗ψ,

where we have skipped the dependence on x , y , z and introduced the shorthand ψx ,

ψy and ψz for the various derivatives.

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For ψ∗ the Euler equation results in

∂f∂ψ∗

−∂

∂x∂f∂ψ∗x

−∂

∂y∂f∂ψ∗y

−∂

∂z∂f∂ψ∗z

= 0,

which yields

−12

(ψxx + ψyy + ψzz ) + Vψ = λψ.

We can then identify the Lagrangian multiplier as the energy of the system. Then thelast equation is nothing but the standard Schrödinger equation and the variationalapproach discussed here provides a powerful method for obtaining approximatesolutions of the wave function.

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Hartree-Fock theory

I Repetition from last weekI Derivation of the Hartree-Fock equations by varying the

single-particle functions or the coefficients of thesingle-particle equations (continued from last week)

I See updated project and discussion of a Hartree-Fockcode

I Next week we will discuss how to introduce classes andobject orientation for the Hartree-Fock project.

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Finding the Hartree-Fock functional E [Φ]

We have the Hamiltonian

Ĥ = Ĥ0 + ĤI =A∑

i=1

ĥ0(xi ) +A∑

i


Let us denote the ground state energy by E0. According to the variational principle wehave

E0 ≤ E [Φ] =∫

Φ∗ĤΦdτ

where Φ is a trial function which we assume to be normalized∫Φ∗Φdτ = 1,

where we have used the shorthand dτ = dx1dx2 . . . dxN .

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In the Hartree-Fock method the trial function is the Slater determinant which can berewritten as

Ψ(x1, x2, . . . , xN , α, β, . . . , ν) =1√

N!

∑P

(−)PPψα(x1)ψβ(x2) . . . ψν(xN ) =√

N!AΦH ,

where we have introduced the anti-symmetrization operator A defined by thesummation over all possible permutations of two fermions. It is defined as

A =1

N!

∑P

(−)PP,

with the the Hartree-function given by the simple product of all possible single-particlefunction

ΦH (x1, x2, . . . , xN , α, β, . . . , ν) = ψα(x1)ψβ(x2) . . . ψν(xN ).

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Both Ĥ0 and ĤI are invariant under permutations of fermions, and hence commute withA

[H0,A] = [HI ,A] = 0.

Furthermore, A satisfiesA2 = A,

since every permutation of the Slater determinant reproduces it.

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Variational Calculus and Lagrangian Multiplier, back toHartree-Fock

Our functional for N electrons is then

E [Φ] =N∑µ=1

∫ψ∗µ(xi )ĥ0(xi )ψµ(xi )dxi +

12

N∑µ=1

N∑ν=1

[∫ψ∗µ(xi )ψ

∗ν(xj )v̂(rij )ψµ(xi )ψν(xj )dxi dxj

−∫ψ∗µ(xi )ψ

∗ν(xj )v̂(rij )ψν(xi )ψµ(xj )dxi dxj

]The more compact version is

E [Φ] =N∑µ=1

〈µ|ĥ0|µ〉+12

N∑µ=1

N∑ν=1

[〈µν|v̂(rij )|µν〉 − 〈µν|v̂(rij )|νµ〉

].

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Hartree-Fock: Variational Calculus and LagrangianMultiplier

If we generalize the Euler-Lagrange equations to more variables and introduce A2Lagrange multipliers which we denote by �µν , we can write the variational equation forthe functional of E

δE −N∑µ=1

N∑ν=1

�µνδ

∫ψ∗µψν = 0.

For the orthogonal wave functions ψµ this reduces to

δE −N∑µ=1

�µδ

∫ψ∗µψµ = 0.

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Variation with respect to the single-particle wave functions ψµ yields then

N∑µ=1

∫δψ∗µĥ0(xi )ψµdxi +

N∑µ=1

N∑ν=1

[∫δψ∗µψ

∗ν v̂(rij )ψµψνdxi dxj −

∫δψ∗µψ

∗ν

1rijψνψµdxi dxj

]

+N∑µ=1

∫ψ∗µĥ0(xi )δψµdxi +

N∑µ=1

N∑ν=1

[∫ψ∗µψ

∗ν

1rijδψµψνdxi dxj −

∫ψ∗µψ

∗ν v̂(rij )ψνδψµdxi dxj

]

−N∑µ=1

Eµ∫δψ∗µψµdxi −

N∑µ=1

Eµ∫ψ∗µδψµdxi = 0.

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Although the variations δψ and δψ∗ are not independent, they may in fact be treated assuch, so that the terms dependent on either δψ and δψ∗ individually may be set equalto zero. To see this, simply replace the arbitrary variation δψ by iδψ, so that δψ∗ isreplaced by −iδψ∗, and combine the two equations. We thus arrive at theHartree-Fock equations−1

2∇2i − uext(xi ) +

N∑ν=1

∫ψ∗ν(xj )v̂(rij )ψν(xj )dxj

ψµ(xi )−

N∑ν=1

∫ψ∗ν(xj )v̂(rij )ψµ(xj )dxj

ψν(xi ) = �µψµ(xi ).Notice that the integration

∫dxj implies an integration over the spatial coordinates rj

and a summation over the spin-coordinate of fermion j .

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The two first terms are the expectation value of the one-body operator. The third or

direct term is the averaged field set up by all other electrons. As written, the term

includes the ’self-interaction’ of electrons when i = j . The self-interaction is cancelled

in the fourth term, or the exchange term. The exchange term results from our inclusion

of the Pauli principle and the assumed determinantal form of the wave-function. The

effect of exchange is for electrons of equal single-particle quantum numbers to avoid

each other.

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A theoretically convenient form of the Hartree-Fock equation is to regard the direct andexchange operator defined through

V dµ(xi ) =∫ψ∗µ(xj )ψµ(xj )v̂(rij )dxj

and

V exµ (xi )g(xi ) =(∫

ψ∗µ(xj )v̂(rij )g(xj )dxj

)ψµ(xi ),

respectively.

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The function g(xi ) is an arbitrary function, and by the substitution g(xi ) = ψν(xi ) weget

V exµ (xi )ψν(xi ) =(∫

ψ∗µ(xj )v̂(rij )ψν(xj )dxj

)ψµ(xi ).

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We may then rewrite the Hartree-Fock equations as

ĥHF (xi )ψν(xi ) = �νψν(xi ),

with

ĥHF (xi ) = ĥ0(xi ) +N∑µ=1

V dµ(xi )−N∑µ=1

V exµ (xi ),

and where ĥ0(i) is the one-body part. The latter is normally chosen as a part whichyields solutions in closed form. The harmonic oscilltor is a classical problem thereof.We normally rewrite the last equation as

ĥHF (xi ) = ĥ0(xi ) + ûHF (xi ).

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Hartree-Fock by varying the coefficients of a wavefunction expansion

Another possibility is to expand the single-particle functions in a known basis and varythe coefficients, that is, the new single-particle wave function is written as a linearexpansion in terms of a fixed chosen orthogonal basis (for example harmonic oscillator,Laguerre polynomials etc)

ψHFp =∑λ

Cpλψλ. (23)

In this case we vary the coefficients Caλ. If the basis has infinitely many solutions, weneed to truncate the above sum. In all our equations we assume a truncation has beenmade.The single-particle wave functions ψλ(r), defined by the quantum numbers λ and r aredefined as the overlap

ψλ(r) = 〈r|λ〉.

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We will omit the radial dependence of the wave functions and introduce first thefollowing shorthands for the direct and exchange part

〈µν|V |µν〉 =∫ψ∗µ(ri )ψ

∗ν(rj )V (rij )ψµ(ri )ψν(rj )dri drj ,

and〈µν|V |νµ〉 =

∫ψ∗µ(ri )ψ

∗ν(rj )V (rij )ψν(ri )ψµ(ri )dri drj .

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Since the interaction is invariant under the interchange of two particles it means forexample that we have

〈µν|V |µν〉 = 〈νµ|V |νµ〉,

or in the more general case

〈µν|V |στ〉 = 〈νµ|V |τσ〉.

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The direct and exchange matrix elements can be brought together if we define theantisymmetrized matrix element

〈µν|V |µν〉AS = 〈µν|V |µν〉 − 〈µν|V |νµ〉,

or for a general matrix element

〈µν|V |στ〉AS = 〈µν|V |στ〉 − 〈µν|V |τσ〉.

It has the symmetry property

〈µν|V |στ〉AS = −〈µν|V |τσ〉AS = −〈νµ|V |στ〉AS .

The antisymmetric matrix element is also hermitian, implying

〈µν|V |στ〉AS = 〈στ |V |µν〉AS .

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We have for the interaction part

∫Φ∗Ĥ1Φdτ =

12

N∑µ=1

N∑ν=1

〈µν|V |µν〉AS . (24)

Combining Eqs. (14) and (24) we obtain the energy functional

E [Φ] =N∑µ=1

〈µ|ĥ0|µ〉+12

N∑µ=1

N∑ν=1

〈µν|V |µν〉AS . (25)

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If we vary the above energy functional with respect to the basis functions |µ〉, thiscorresponds to what was done in the previous case. We are however interested indefining a new basis defined in terms of a chosen basis as defined in Eq. (23). We canthen rewrite the energy functional as

E [ΦHF] =N∑

i=1

〈i|ĥ0|i〉+12

N∑ij=1

〈ij|V |ij〉AS , (26)

where ΦHF is the new Slater determinant defined by the new basis of Eq. (23).

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Using Eq. (23) we can rewrite Eq. (26) as

E [ΦHF] =N∑

i=1

∑αβ

C∗iαCiβ〈α|h|β〉+12

N∑ij=1

∑αβγδ

C∗iαC∗jβCiγCjδ〈αβ|V |γδ〉AS . (27)

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We wish now to minimize the above functional. We introduce again a set of Lagrangemultipliers, noting that since 〈i|j〉 = δi,j and 〈α|β〉 = δα,β , the coefficients Ciγ obey therelation

〈i|j〉 = δi,j =∑αβ

C∗iαCiβ〈α|β〉 =∑α

C∗iαCiα,

which allows us to define a functional to be minimized that reads

F [ΦHF] = E [ΦHF]−N∑

i=1

�i∑α

C∗iαCiα. (28)

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Minimizing with respect to C∗iα, remembering that C∗iα and Ciα are independent, we

obtaind

dC∗iα

E [ΦHF]− A∑j

�j∑α

C∗jαCjα

= 0, (29)which yields for every single-particle state i the following Hartree-Fock equations

∑γ

Ciγ〈α|ĥ0|γ〉+N∑

j=1

∑βγδ

C∗jβCjδCiγ〈αβ|V |γδ〉AS = �k Ciα. (30)

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We can rewrite this equation as

∑β

〈α|ĥ0|β〉+N∑j

∑γδ

C∗jγCjδ〈αγ|V |βδ〉AS

Ciβ = �HFi Ciα. (31)Note that the sums over greek indices run over the number of basis set functions (in

principle an infinite number).

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Defining

hHFαβ = 〈α|ĥ0|β〉+N∑

j=1

∑γδ

C∗jγCjδ〈αγ|V |βδ〉AS ,

we can rewrite the new equations as∑β

hHFαβCiβ = �HFi Ciα. (32)

Note again that the sums over greek indices run over the number of basis set functions

(in principle an infinite number).

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Rewriting the energy functional

The equationĥHF (xi ) = ĥ0(xi ) + ûHF (xi ),

allows us to rewrite the ground state energy (adding and subtracting ûHF (xi ))

EHF0 = 〈Φ0|Ĥ|Φ0〉 =N∑

i≤F〈i|ĥ0 + ûHF |i〉+

12

N∑i≤F

N∑j≤F

[〈ij|v̂ |ij〉 − 〈ij|v̂ |ji〉]−N∑

i≤F〈i|ûHF |i〉,

as

EHF0 =N∑

i≤Fεi +

12

N∑i≤F

N∑j≤F

[〈ij|v̂ |ij〉 − 〈ij|v̂ |ji〉]−N∑

i≤F〈i|ûHF |i〉,

which is nothing but

EHF0 =N∑

i≤Fεi −

12

N∑i≤F

N∑j≤F

[〈ij|v̂ |ij〉 − 〈ij|v̂ |ji〉] .

102 / 234


Our Hartree-Fock matrix is thus

ĥHFαβ = 〈α|ĥ0|β〉+N∑

j=1

∑γδ

C∗jγCjδ〈αγ|V |βδ〉AS .

The Hartree-Fock are solved in an iterative waym starting with a guess for thecoefficients Cjγ = δj,γ and solving the equations by diagonalization till the newsingle-particle energies �HFi do not change anymore by a prefixed quantity.Normally we assume that the single-particle basis |β〉 forms an eigenbasis for theoperator ĥ0, meaning that the Hartree-Fock matrix becomes

ĥHFαβ = �αδα,β +N∑

j=1

∑γδ


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Hartree-Fock algorithmOur Hartree-Fock matrix is thus

ĥHFαβ = 〈α|ĥ0|β〉+A∑

j=1

∑γδ


Normally we assume that the single-particle basis |β〉 forms an eigenbasis for theoperator ĥ0, meaning that the Hartree-Fock matrix becomes

ĥHFαβ = �αδα,β +A∑

j=1

∑γδ


The Hartree-Fock eigenvalue problem∑β

ĥHFαβCiβ = �HFi Ciα,

can be written out in a more compact form as

ĥHF Ĉ = �HFĈ.

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Hartree-Fock algorithm

The Hartree-Fock equations are, in their simplest form, solved in an iterative way,starting with a guess for the coefficients Ciα. We label the coefficients as C

(n)iα , where

the subscript n stands for iteration n. To set up the algorithm we can proceed asfollows:

1. We start with a guess C(0)iα = δi,α. Alternatively, we could have used randomstarting values as long as the vectors are normalized. Another possibility is togive states below the Fermi level a larger weight.

2. The Hartree-Fock matrix simplifies then to

ĥHFαβ = �αδα,β +N∑

j=1

∑γδ

C∗jγ(0)C(0)jδ 〈αγ|V |βδ〉AS .

Solving the Hartree-Fock eigenvalue problem yields then new eigenvectors C(1)iαand eigenvalues �HFi

(1).

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Hartree-Fock algorithm

1. With the new eigenvalues we can set up a new Hartree-Fock potential

N∑j=1

∑γδ

C∗jγ(1)C(1)jδ 〈αγ|V |βδ〉AS .

The diagonalization with the new Hartree-Fock potential yields new eigenvectorsand eigenvalues. This process is continued till for example∑

p |�HFi(n) − �HFi

(n−1)|m

≤ λ,

where λ is a user prefixed quantity (λ ∼ 10−8 or smaller) and p runs over allcalculated single-particle energies and m is the number of single-particle states.

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Hartree-Fock code

void hart reeFock : : run ( ) {double i n t e r a c t i o n ;/ / −−−−−−−−−−−−−−− Se t t i ng up the HF−

h a m i l t i n i a n using C = 1 −−−−−−−−−−−−−−−mat H;vec E = zeros ( nStates , 1) ;vec ePrev = zeros ( nStates , 1) ;mat C = eye ( nStates , nStates ) ;vec d i f f ;

/ / Hartree−Fock loopi n t h f I t = 0 ;while ( h f I t < HFI te ra t i ons ) {

cout

Hartree-Fock codefor ( i n t alpha = 0; alpha < nStates ; alpha

++) {for ( i n t gamma = 0; gamma < nStates ;

gamma++) {i n t e r a c t i o n = 0;for ( i n t p = 0; p < n P a r t i c l e s ; p

++) {for ( i n t beta = 0; beta <

nStates ; beta ++) {for ( i n t de l t a = 0 ; de l t a <

nStates ; de l t a ++) {i n t e r a c t i o n += C. a t ( p ,

beta ) ∗ C. a t ( p ,de l t a ) ∗matr ixElement ( alpha, beta , gamma,de l t a ) ;

}}

}108 / 234

Hartree-Fock codeH( alpha , gamma) = H(gamma, alpha ) =

h0 ( alpha , gamma) + i n t e r a c t i o n;

}}/ / Computing the HF one−body energieseig sym (E, C, H) ;C = t rans (C) ;h f I t ++;/ / Convergence t e s td i f f = E − ePrev ;i f ( abs ( d i f f .max ( ) ) < t h resho ld )

break ;ePrev = E;

}double E0 = calcEnergy (C) ;cout

Koopman’s theorem and interpretation of theHartree-Fock energies

We have defined

E [ΦHF(N)] =N∑

i=1

〈i|ĥ0|i〉+12

N∑ij=1

〈ij|V |ij〉AS ,

where ΦHF(A) is the new Slater determinant defined by the new basis of Eq. (23) for Nelectrons (same Z ). If we assume that the single-particle wave functions in the newbasis do not change when we remove one electron or add one electron, we can thendefine the corresponding energy for the N − 1 systems as

E [ΦHF(N − 1)] =N∑

i=1;i 6=k〈i|ĥ0|i〉+

12

N∑ij=1;i,j 6=k

〈ij|V |ij〉AS ,

where we have removed a single-particle state k ≤ F , that is a state below the Fermilevel.

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Calculating the difference

E [ΦHF(N)]− E [ΦHF(N − 1)] = 〈k |ĥ0|k〉+12

N∑i=1;i 6=k

〈ik |V |ik〉AS12

N∑j=1;j 6=k

〈kj|V |kj〉AS ,

which becomes

E [ΦHF(N)]− E [ΦHF(N − 1)] = 〈k |ĥ0|k〉+12

N∑j=1

〈kj|V |kj〉AS

which is just our definition of the Hartree-Fock single-particle energy

E [ΦHF(N)]− E [ΦHF(N − 1)] = �HFk

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Similarly, we can now compute the difference (we label the single-particle states abovethe Fermi level as abcd > F )

E [ΦHF(N + 1)]− E [ΦHF(N)] = �HFa .

This equation and the one on the previous slide, are linked to Koopman’s theorem. In

atomic physics it is used to define the electron ionization or affinity energies.

Koopman’s theorem states that the ionization energy of closed-shell systems is given

by the energy of the highest occupied single-particle state.

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Discussion of the project

We will now set up the Hartree-Fock program by using hydrogen-like single-particle

functions for the 1s, 2s and 3s orbitals. We will apply these basis functions only to the

helium and beryllium atoms. For all other systems, we will use so-called Gaussian type

of orbitals (GTO). These will be introduced after we have set up our first Hartree-Fock

programs. For Hydrogen like-orbitals, the matrix elements involving the Coulomb

interaction, can all be evaluated in a closed form, see the table below for all integrals

involving these orbitals. Your first step is to write a function which reads in these

integrals and sets up the antisymmetrized matrix elements. (Hint: the table lists only

the integrals, there is no spin).

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Discussion of the projectThe table contains the matrix elements for the radial integrals to be used for the directpart and the exchange part. You will need these integrals in solving the Hartree-Fockequations for the helium and beryllium atoms when using hydrogen-like single-particlestates. The table lists only the integrals, no spin is included!∫

ψ∗µ(ri )ψ∗ν(rj )V (rij )ψµ(ri )ψν(rj )dri drj ,

Table: Closed form expressions for the Coulomb matrix elements(see project for all elements). The nomenclature is 1 = 1s, 2 = 2sand 3 = 3s, with no spin degrees of freedom.

〈11|V |11〉 = (5Z )/8 〈11|V |12〉 = (4096√

2Z )/64827〈11|V |13〉 = (1269

√3Z )/50000 〈11|V |21〉 = (4096

√2Z )/64827

〈11|V |22〉 = (16Z )/729 〈11|V |23〉 = (110592√

6Z )/24137569〈11|V |31〉 = (1269

√3Z )/50000 〈11|V |32〉 = (110592

√6Z )/24137569

〈11|V |33〉 = (189Z )/32768 〈12|V |11〉 = (4096√

2Z )/64827. . . . . . . . . . . .

〈33|V |33〉 = (17Z )/256

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Discussion of the project

Set up the Hartree-Fock equations for the ground states of helium and beryllium withthe electrons occupying the respective ’hydrogen-like’ orbitals 1s, 2s and 3s. There isno spin-orbit part in the two-body Hamiltonian. Find also the total binding ground stateenergy and compare this to the energy obtaining without performing the Hartree-Fockcalculations. The experimental values are −2.904 atomic units (a.u.) and −14.67 a.u.for the helium and beryllium atoms, respectively.In setting up the equations make sure your Hartree-Fock matrix is block diagonal in thespin quantum numbers. Since there is no spin-dependent part in the Hamiltonian,states with the same l and n will be degenerate in energy with respect to the spinprojections.

The code you develop here will serve as a starting point for the inclusion of GTOs later

in this course. We will discuss object orientation next week.

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Topics for Week 8, February 17-21

Classes and object orientation in C++

I Simple classes in C++I TemplatesI How to object orient the Hartree-Fock programI Next week we will discuss new orbital functions, the

so-called Gaussian-type orbitals

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Object orientation

Why object orientation?

I Three main topics: objects, class hierarchies and polymorphism

I The aim here is to be to be able to write a more general code which can easilybe tailored to new situations.

I Polymorphism is a term used in software development to describe a variety oftechniques employed by programmers to create flexible and reusable softwarecomponents. The term is Greek and it loosely translates to ”many forms”.

Strategy: try to single out the variables needed to describe a given system and those

needed to describe a given solver.

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Object orientation

In programming languages, a polymorphic object is an entity, such as a variable or aprocedure, that can hold or operate on values of differing types during the program’sexecution. Because a polymorphic object can operate on a variety of values and types,it can also be used in a variety of programs, sometimes with little or no change by theprogrammer. The idea of write once, run many, also known as code reusability, is animportant characteristic to the programming paradigm known as Object-OrientedProgramming (OOP).OOP describes an approach to programming where a program is viewed as acollection of interacting, but mostly independent software components. These softwarecomponents are known as objects in OOP and they are typically implemented in aprogramming language as an entity that encapsulates both data and procedures.

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Programming classes

In Fortran a vector or matrix start with 1, but it is easy to change a vector so that it

starts with zero or even a negative number. If we have a double precision Fortran

vector which starts at −10 and ends at 10, we could declare it asREAL(KIND=8):: vector(−10:10). Similarly, if we want to start at zero and endat 10 we could write REAL(KIND=8):: vector(0:10). We have also seen thatFortran allows us to write a matrix addition A = B + C as A = B + C. This meansthat we have overloaded the addition operator so that it translates this operation into

two loops and an addition of two matrix elements aij = bij + cij .

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Programming classes

The way the matrix addition is written is very close to the way we express this relationmathematically. The benefit for the programmer is that our code is easier to read.Furthermore, such a way of coding makes it more likely to spot eventual errors as well.In Ansi C and C++ arrays start by default from i = 0. Moreover, if we wish to add twomatrices we need to explicitely write out the two loops as

for ( i =0 ; i < n ; i ++) {for ( j =0 ; j < n ; j ++) {

a [ i ] [ j ]=b [ i ] [ j ]+ c [ i ] [ j ]}

}

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Programming classes

However, the strength of C++ is the possibility to define new data types, tailored tosome particular problem. Via new data types and overloading of operations such asaddition and subtraction, we can easily define sets of operations and data types whichallow us to write a matrix addition in exactly the same way as we would do in Fortran.We could also change the way we declare a C++ matrix elements aij , from a[i][j] to saya(i, j), as we would do in Fortran. Similarly, we could also change the default rangefrom 0 : n − 1 to 1 : n.To achieve this we need to introduce two important entities in C++ programming,

classes and templates.

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Programming classes

The function and class declarations are fundamental concepts within C++. Functions

are abstractions which encapsulate an algorithm or parts of it and perform specific

tasks in a program. We have already met several examples on how to use functions.

Classes can be defined as abstractions which encapsulate data and operations on

these data. The data can be very complex data structures and the class can contain

particular functions which operate on these data. Classes allow therefore for a higher

level of abstraction in computing. The elements (or components) of the data type are

the class data members, and the procedures are the class member functions.

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Programming classes

Classes are user-defined tools used to create multi-purposesoftware which can be reused by other classes or functions.These user-defined data types contain data (variables) andfunctions operating on the data.A simple example is that of a point in two dimensions. The datacould be the x and y coordinates of a given point. Thefunctions we define could be simple read and write functions orthe possibility to compute the distance between two points.

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Programming classesC++ has a class complex in its standard template library (STL).The standard usage in a given function could then look like

/ / Program to c a l c u l a t e a d d i t i o n and m u l t i p l i c a t i o no f two complex numbers

using namespace s td ;#include #include #include i n t main ( ){

complex x ( 6 . 1 , 8 . 2 ) , y ( 0 . 5 , 1 . 3 ) ;/ / w r i t e out x+ycout

Programming classesWe proceed by splitting our task in three files.We define first a header file complex.h which contains the declarations of the class.The header file contains the class declaration (data and functions), declaration ofstand-alone functions, and all inlined functions, starting as follows

# i fndef Complex H#define Complex H/ / va r ious inc lude statements and d e f i n i t i o n s#include / / Standard ANSI−C++

inc lude f i l e s#include #include . . . .

class Complex{ . . .d e f i n i t i o n o f v a r i a b l e s and t h e i r character} ;/ / dec l a r a t i ons o f var ious f u n c t i o n s used by the

c lass. . .#endif

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Programming classes

Next we provide a file complex.cpp where the code and algorithms of differentfunctions (except inlined functions) declared within the class are written. The filescomplex.h and complex.cpp are normally placed in a directory with other classes andlibraries we have defined.

Finally,we discuss here an example of a main program which uses this particular class.

An example of a program which uses our complex class is given below. In particular we

would like our class to perform tasks like declaring complex variables, writing out the

real and imaginary part and performing algebraic operations such as adding or

multiplying two complex numbers.

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Programming classes

#include "Complex.h". . . o ther include and dec l a ra t i o nsi n t main ( ){

Complex a ( 0 . 1 , 1 . 3 ) ; / / we dec lare a complexv a r i a b l e a

Complex b ( 3 . 0 ) , c (5 .0 ,−2.3) ; / / we dec larecomplex v a r i a b l e s b and c

Complex d = b ; / / we dec lare a newcomplex v a r i a b l e d

cout

Programming classes

We include the header file complex.h and define four differentcomplex variables. These are a = 0.1 + ı1.3, b = 3.0 + ı0 (notethat if you don’t define a value for the imaginary part this is setto zero), c = 5.0− ı2.3 and d = b. Thereafter we have definedstandard algebraic operations and the member functions of theclass which allows us to print out the real and imaginary part ofa given variable.

128 / 234

Programming classes

class Complex{private :

double re , im ; / / r e a l and imaginary pa r tpublic :

Complex ( ) ; / /Complex c ;

Complex ( double re , double im = 0 .0 ) ; / /D e f i n i t i o n o f a complex v a r i a b l e ;

Complex ( const Complex& c ) ; / /Usage : Complex c ( a ) ; / / equate two complex

v a r i a b l e sComplex& operator= ( const Complex& c ) ; / / c = a ;

/ / equate two complex var iab les , same asprev ious

. . . .

129 / 234

Programming classes

˜Complex ( ) {} / /d e s t r u c t o r

double Re ( ) const ; / / double r e a l p a r t= a .Re ( ) ;

double Im ( ) const ; / / double imag par t= a . Im ( ) ;

double abs ( ) const ; / / double m = a . abs( ) ; / / modulus

f r iend Complex operator+ ( const Complex& a ,const Complex& b ) ;

f r iend Complex operator− ( const Complex& a ,const Complex& b ) ;

f r iend Complex operator∗ ( const Complex& a ,const Complex& b ) ;

f r iend Complex operator / ( const Complex& a ,const Complex& b ) ;

} ;

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Programming classes

The class is defined via the statement class Complex. We mustfirst use the key word class, which in turn is followed by theuser-defined variable name Complex. The body of the class,data and functions, is encapsulated within the parentheses{...};.

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Programming classes

Data and specific functions can be private, which means thatthey cannot be accessed from outside the class. This meansalso that access cannot be inherited by other functions outsidethe class. If we use protected instead of private, then data andfunctions can be inherited outside the class.

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Programming classes

The key word public means that data and functions can beaccessed from outside the class. Here we have defined severalfunctions which can be accessed by functions outside the class.The declaration friend means that stand-alone functions canwork on privately declared variables of the type (re, im). Datamembers of a class should be declared as private variables.

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Programming classes

The first public function we encounter is a so-called constructor,which tells how we declare a variable of type Complex and howthis variable is initialized. We have chose three possibilities inthe example above:

I A declaration like Complex c; calls the member functionComplex() which can have the following implementation

Complex : : Complex ( ) { re = im = 0 . 0 ; }

meaning that it sets the real and imaginary parts to zero.Note the way a member function is defined. Theconstructor is the first function that is called when an objectis instantiated.

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Programming classes

I Another possibility is

Complex : : Complex ( ) {}

which means that there is no initialization of the real andimaginary parts. The drawback is that a given compiler canthen assign random values to a given variable.

I A call like Complex a(0.1,1.3); means that we could call themember function Complex(double, double)as

Complex : : Complex ( double re a , double im a ){ re = re a ; im = im a ; }

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Programming classes

The simplest member function are those we defined to extractthe real and imaginary part of a variable. Here you have torecall that these are private data, that is they invisible for usersof the class. We obtain a copy of these variables by defining thefunctions

double Complex : : Re ( ) const { return re ; }} / /g e t t i n g the r e a l pa r t

double Complex : : Im ( ) const { return im ; } / /and the imaginary pa r t

Note that we have introduced the declaration const. What doesit mean? This declaration means that a variabale cannot bechanged within a called function.

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Programming classes

If we define a variable as const double p = 3; and then try tochange its value, we will get an error when we compile ourprogram. This means that constant arguments in functionscannot be changed.

/ / const arguments ( i n f u n c t i o n s ) cannot be changed:

void myfunc ( const Complex& c ){ c . re = 0 . 2 ; /∗ ILLEGAL ! ! compi ler e r r o r . . . ∗ / }

If we declare the function and try to change the value to 0.2, thecompiler will complain by sending an error message.

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Programming classes

If we define a function to compute the absolute value ofcomplex variable like

double Complex : : abs ( ) { return s q r t ( re ∗ re + im∗im ) ;}

without the constant declaration and define thereafter afunction myabs as

double myabs ( const Complex& c ){ return c . abs ( ) ; } / / Not ok because c . abs ( ) i s

not a const func .

the compiler would not allow the c.abs() call in myabs sinceComplex::abs is not a constant member function.

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Programming classes

Constant functions cannot change the object’s state. To avoidthis we declare the function abs as

double Complex : : abs ( ) const { return s q r t ( re ∗ re +im∗ im ) ; }

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Programming classes

C++ (and Fortran) allow for overloading of operators. Thatmeans we can define algebraic operations on for examplevectors or any arbitrary object. As an example, a vectoraddition of the type c = a + b means that we need to write asmall part of code with a for-loop over the dimension of thearray. We would rather like to write this statement as c = a+b; asthis makes the code much more readable and close to eventualequations we want to code. To achieve this we need to extendthe definition of operators.

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Programming classes

Let us study the declarations in our complex class. In our mainfunction we have a statement like d = b;, which means that wecall d.operator= (b) and we have defined a so-calledassignment operator as a part of the class defined as

Complex& Complex : : operator= ( const Complex& c ){

re = c . re ;im = c . im ;return ∗ th is ;

}

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Programming classes

With this function, statements like Complex d = b; orComplex d(b); make a new object d , which becomes a copy ofb. We can make simple implementations in terms of theassignment

Complex : : Complex ( const Complex& c ){ ∗ th is = c ; }

which is a pointer to ”this object”, ∗this is the present object, so∗this = c; means setting the present object equal to c, that isthis−>operator= (c);.

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Programming classes

The meaning of the addition operator + for Complex objects isdefined in the function Complex operator+ (const Complex& a,const Complex& b); //a+b The compiler translates c = a + b; intoc = operator+ (a, b);. Since this implies the call to function, itbrings in an additional overhead. If speed is crucial and thisfunction call is performed inside a loop, then it is more difficultfor a given compiler to perform optimizations of a loop.

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Programming classes

The solution to this is to inline functions. We discussed inliningin chapter 2 of the lecture notes. Inlining means that thefunction body is copied directly into the calling code, thusavoiding calling the function. Inlining is enabled by the inlinekeyword

i n l i n e Complex operator+ ( const Complex& a , constComplex& b )

{ return Complex ( a . re + b . re , a . im + b . im ) ; }

Inline functions, with complete bodies must be written in theheader file complex.h.

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Programming classes

Consider the case c = a + b; that is,c.operator= (operator+ (a,b)); If operator+, operator= and theconstructor Complex(r,i) all are inline functions, this transformsto

c . re = a . re + b . re ;c . im = a . im + b . im ;

by the compiler, i.e., no function calls

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Programming classes

The stand-alone function operator+ is a friend of the Complexclass

class Complex{

. . .f r iend Complex operator+ ( const Complex& a ,

const Complex& b ) ;. . .

} ;

so it can read (and manipulate) the private data parts re and imvia


{ return Complex ( a . re + b . re , a . im + b . im ) ; }

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Programming classes

Since we do not need to alter the re and im variables, we canget the values by Re() and Im(), and there is no need to be afriend function


{ return Complex ( a .Re ( ) + b .Re ( ) , a . Im ( ) + b . Im ( ) ); }

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Programming classes

The multiplication functionality can now be extended toimaginary numbers by the following code

i n l i n e Complex operator∗ ( const Complex& a , constComplex& b )

{return Complex ( a . re ∗b . re − a . im∗b . im , a . im∗b . re +

a . re ∗b . im ) ;}

It will be convenient to inline all functions used by this operator.

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Programming classes

To inline the complete expression a∗b;, the constructors andoperator= must also be inlined. This can be achieved via thefollowing piece of code

i n l i n e Complex : : Complex ( ) { re = im = 0 . 0 ; }i n l i n e Complex : : Complex ( double re , double im ){ . . . }i n l i n e Complex : : Complex ( const Complex& c ){ . . . }i n l i n e Complex : : operator= ( const Complex& c ){ . . . }

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Programming classes

/ / e , c , d are complexe = c∗d ;/ / f i r s t compi ler t r a n s l a t i o n :e . operator= ( operator∗ ( c , d ) ) ;/ / r e s u l t o f nested i n l i n e f u n c t i o n s/ / opera tor = , opera tor ∗ , Complex ( double , double =0) :e . re = c . re ∗d . re − c . im∗d . im ;e . im = c . im∗d . re + c . re ∗d . im ;

The definitions operator− and operator/ follow the same set up.

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Programming classes

Finally, if we wish to write to file or another device a complexnumber using the simple syntax cout

Programming classes, templates

What if we wanted to make a class which takes integers orfloating point numbers with single precision? A simple way toachieve this is copy and paste our class and replace doublewith for example int.C++ allows us to do this automatically via the usage oftemplates, which are the C++ constructs for parameterizingparts of classes. Class templates is a template for producingclasses. The declaration consists of the keyword templatefollowed by a list of template arguments enclosed in brackets.

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Programming classesWe can therefore make a more general class by rewriting ouroriginal example as

templateclass Complex{private :

T re , im ; / / r e a l and imaginary pa r tpublic :

Complex ( ) ; / /Complex c ;

Complex (T re , T im = 0) ; / / D e f i n i t i o n o f acomplex v a r i a b l e ;

Complex ( const Complex& c ) ; / /Usage : Complex c ( a ) ; / / equate two complex

v a r i a b l e sComplex& operator= ( const Complex& c ) ; / / c = a ;

/ / equate two complex var iab les , same asprev ious

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Programming classesWe can therefore make a more general class by rewriting ouroriginal example as

˜Complex ( ) {} / /d e s t r u c t o r

T Re ( ) const ; / / T r e a l p a r t = a .Re ( ) ;T Im ( ) const ; / / T imag par t = a . Im ( ) ;T abs ( ) const ; / / T m = a . abs ( ) ; / /

modulusf r iend Complex operator+ ( const Complex& a ,

const Complex& b ) ;f r iend Complex operator− ( const Complex& a ,

const Complex& b ) ;f r iend Complex operator∗ ( const Complex& a ,

const Complex& b ) ;f r iend Complex operator / ( const Complex& a ,

const Complex& b ) ;} ;

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Programming classes

What it says is that Complex is a parameterized type with T asa parameter and T has to be a type such as double or float.The class complex is now a class template and we woulddefine variables in a code as

Complex a ( 1 0 . 0 , 5 . 1 ) ;Complex b (1 ,0 ) ;

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Programming classes

Member functions of our class are defined by preceding thename of the function with the template keyword. Consider thefunction we defined asComplex:: Complex (double re a, double im a). We would rewritethis function as

templateComplex : : Complex (T re a , T im a ){ re = re a ; im = im a ; }

The member functions are otherwise defined following ordinarymember function definitions.

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Programming classes

Here follows a very simple first class

/ / Class to compute the square o f a numberclass Squared{

public :/ / De fau l t cons t ruc to r , not used hereSquared ( ) {}

/ / Overload the f u n c t i o n opera tor ( )double operator ( ) ( double x ) {

return x∗x ;}

} ;

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Programming classes

and we would use it as

#include using namespace s td ;

i n t main ( ) {Squared s ;cout

The Hartree-Fock code

It can be useful to split the code in a system and a solver part,where the system part contains information about variousquantum numbers, the interaction elements such as theexpectation value of the one-body and two-body interactionsand other properties pertinent to a given system. The solverpart needs only to deal with the explicit solution algorithm andtakes as input for example only indices relevant to a givensingle-particle system. Here’s an example which contains

I A basis classI An interaction class, the Coulomb interaction hereI And a solver class, which here contains only the

Hartree-Fock method

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The Hartree-Fock code: the main program

#include #include #include #include #include #include "coulomb.h"#include "basis.h"#include "hartreeFock.h"#include

using namespace arma ;using namespace std ;

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The Hartree-Fock code: the main program/ / The main programi n t main ( i n t argc , char∗∗ argv ) {

i n t n P a r t i c l e s = 4;i n t dim = 3;i n t n = 20; / / number o f s ing le−p a r t i c l e s ta tesdouble w = 1;double t h resho ld = 1e−4;i n t HFI te ra t i ons = 30;

/ / Se t t i ng up the basisbasis harm = basis ( dim , n , w) ;harm . createBas is ( ) ;harm . genera te In te rac t ionE lements ( ) ;vector s ta tes = harm . getSPS ( ) ;map i n tE = harm . g e t I n t ( ) ;

/ / Running Hartree−Fock c a l c u l a t i o n shart reeFock HF = hart reeFock ( s ta tes , nPa r t i c l es

, i n tE ) ;HF. run ( ) ;. . . / / more statements , p r i n t e tcreturn 0;

}

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The Hartree-Fock code: the basis class

# i f n d e f BASIS H#define BASIS H#include #include #include #include #include #include

using namespace std ;using namespace arma ;

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class basis {public :

bas is ( ) ;bas is ( i n t dim , i n t n , double w) ;bas is ( const basis& o r i g ) ;v i r t u a l ˜ bas is ( ) ;void createBas is ( ) ;void genera te In te rac t ionE lements ( ) ;vector getSPS ( ) {

return s ta tes ;} ;vector get IE ( ) {

return s ta tes ;} ;

map g e t I n t ( ) {return i e ;

} ;

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private :s t r i n g basisName ;i n t dim ;i n t n ;double w;i n t maxRange ;vector s ta tes ;vector i n te rac t i onE lemen ts ;map i e ;

} ;#endif /∗ BASIS H ∗ /

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The Hartree-Fock code: the basis class and specificfunctions

. . . / / more statementsbasis : : bas is ( i n t dim , i n t n , double w) : dim ( dim ) ,

n ( n ) , w(w) {}

void basis : : c reateBas is ( ) {

/ / Generat ing a l l s i n g l e p a r t i c l e s ta tes andenergies

vec s ta te ( 4 ) ;i n t counter = 0 ;/ / s p e c i f i c casefor ( i n t k = 0; k

The Hartree-Fock code: the basis class and specificfunctions

. . . . .void basis : : genera te In te rac t ionE lements ( ) {

double E;i n t n1 , l1 , s1 , n2 , l2 , s2 , n3 , l3 , s3 , n4 , l4 ,

fys4411/fys9411 computational physics ii …...fys4411/fys9411 computational physics ii university...

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