Lecture notes Condensed matter Physics

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Condensed Matter Physics II Robert van Leeuwen University of Jyv¨askyl¨a, Autumn 2009 (Dated: December 1, 2009) Contents I. Condensed Matter: A problem of many particles 2 II. Interacting electrons and nuclei 3 A. The Schr¨odinger equation for electrons and nuclei 3 B. The time-independent Schr¨odinger equation 4 III. The Born-Oppenheimer separation 5 A. General derivation 5 B. Example: diatomic molecules 8 IV. The many-electron problem 11 A. Wave functions and symmetry 11 B. The Hartree-Fock method 16 1. The ground state energy and the Raleigh-Ritz principle 16 2. The Hartree-Fock one-particle equations 18 C. The density functional method 22 1. The Hohenberg-Kohn mappings 22 2. The Hohenberg-Kohn variational principle 24 3. The Kohn-Sham construction 24 4. The Hellman-Feynman theorem 27 5. The coupling constant integration 27 6. The exchange-correlation hole 29 7. Splitting into exchange and correlation 31 V. The electron gas 33 A. One-particle equations 33 B. Kohn-Sham equations for the electron gas 33 C. Pair-correlationfunction and exchange energy 35 D. Hartree-Fock equations for the electron gas 38 E. Thomas-Fermi theory of charge screening 39 F. Static linear response from the Schr¨odinger equation 40 G. Lindhard theory of charge screening 44 H. Plasmons 47 1. Time-dependent many-particle systems 47 2. The plasma oscillation 50 3. Time-dependent linear response 52 I. Time-dependent density functional theory 55 1. The frequency dependent Lindhard function and the plasma dispersion 57 VI. Symmetry 61 A. Lattice symmetry and Bloch states 61 1. One-particle equations in periodic systems 61 2. Periodicity in one dimension 612 3. Periodicity in more dimensions 63 4. Wigner-Seitz and other primitive cells 66 5. Bloch’s theorem 68 VII. Optical properties 71 A. Excitons 71 VIII. Phonons 76 IX. The electron-phonon interaction and superconductivity 80 A. Functional derivatives and variational equations 80 References 84 I. CONDENSED MATTER: A PROBLEM OF MANY PARTICLES All matter (atoms, molecules, solids) is made out of electrons and atomic nuclei. The simplest ”condensed matter” system is simply an electron circling a proton, also known as the hydrogenatom. When we increase the number of electrons and nuclei we are describing molecules. These systems have so many diverse properties that there is a whole field of science,calledchemistry,devotedtoit. Whenthesesystemsbecomemacroscopicandvisible to the human eye, we call them solids. We thus see that the fields of atomic, molecular and solid state physics are strongly related, they deal with systems made out of the same constituents. Thebeauty ofthis factisthatessentiallyallthepropertiesofallthese systems can be deduced from one basic physical equation, known as the Schr¨odinger equation. This a quantum mechanical equation since the internal properties of molecules and solids are determined by electronic interactions at an atomic scale at which classical physics is not applicable anymore. The time-dependent Schr¨odinger equation, in short, has the following form ˆ i∂ Ψ(t)=H(t)Ψ(t) Ψ(t )= Ψ (1) t 0 0 ˆ In this equation the Hamiltonian H(t) incorporates all information on the kinetic and po- tential energies of all particles and the wavefunction Ψ(t) depends on all coordinates (space and spin) of all the particle and encodes the complete time-evolution of the system, when aninitial state Ψ at timet is specified. Fromthe wavefunction we cancalculateallvalues 0 0 of physical observables of interest. In quantum mechanics these observables are described ˆ by Hermitian operatorsO and their values are evaluated as expectation values ˆ ˆ hOi(t) =hΨ(t)OΨ(t)i (2) The main task of a condensed matter physicist is therefore well-defined: Solve the many- particle Schr¨odinger equation and calculate all properties of interest. However, due to the large number of variables involved, this is not possible in practice, except for the smallest systems. The condensed matter physicists have therefore developed a large number of techniques to attack this difficult mathematical problem. One the one hand there are methodsthattrytoapproximatethewavefunctionsuchasisdoneintheHartree-Fockmean- fieldapproach. Onthe otherhand, there areapproachesthatconstructmodel Hamiltonians that focus on a particular aspect, such as the BCS model for superconductivity. And then there are more systematic approaches based on many-body perturbation theory. In this course we try to give an systematic overview of all these approaches and try to find a unified approach to describe systems ranging from atoms and molecules to solids.3 II. INTERACTING ELECTRONS AND NUCLEI A. The Schr¨odinger equation for electrons and nuclei ˆ The HamiltonianH(t) of a system of N electrons and N atomic nuclei has the form e n ˆ ˆ ˆ ˆ ˆ ˆ ˆ H(t) =T +T +V +V +V +V (t) (3) e n ee en nn ext ˆ ˆ In this expression T and T are the kinetic energy of electrons and nuclei which have the e n explicit form N e X 2 2 ˆ T = − ∇ (4) e i 2m i=1 N n 2 X 2 ˆ T = − ∇ (5) n R α 2M α α=1 where m is the electron mass and M is the mass of nucleus α. We denote the electronic α coordinatesbyr andthenuclearonesbyR . TheCoulombrepulsionsbetweentheelectrons i α and those between the nuclei is given by N e 2 X 1 e ˆ V = (6) ee 2 r −r i j i6=j N n 2 X 1 Z Z e α β ˆ V = (7) nn 2 R −R α β α6=β where−eisthenegativechargeoftheelectronandZ ethepositivechargeofatomicnucleus α α. The attraction between the lectrons and the nuclei is given by N N e n XX 2 Z e α ˆ V =− (8) en r −R i α i=1 α=1 ˆ Then we finally allow for the presence of an external field V (t) acting on the electrons ext and the nuclei. This can, for instance, be a laser field or a homogeneous electric field. The general form of this external field is N N e n X X ˆ V (t) = v(r ,t)+ V(R ,t) (9) ext i α i=1 α=1 where v and V are the external potentials acting on the electrons and the nuclei. For instance, a time-dependent homogeneous electric field E(t) would give potentials v(r,t) = −er · E(t) and V(R ,t) = Z eR · E(t). Another example would be the α α α presence of a magnetic field, that couples to the spins of the electrons and nuclei. Such external fields will play a role in the discussion of the optical and magnetic properties ofmoleculesandsolids,butwewillnotconsidertheminthebeginningoftheselecturenotes. After having defined the Hamiltonian we consider the wave function of all the electrons and nuclei. In general for a system of many particles the wave function is a function of time and of all the variables that characterize the particles, i.e. both position in space and all the internal degrees of freedom. For electrons the only internal degree of freedom is the4 spin. Nuclei have more internal degrees of freedom, but the only internal degree of freedom that has any effect on the electronic structure is its spin. The general many-particle wave function is thus of the form Ψ= Ψ(r σ ,...,r σ ,R s ,...,R s ,t). (10) 1 1 N N 1 1 N N e e n n In this equationσ denotes the spin coordinate of electroni. Since the electron is a particle i 1 withspin thespincoordinatecanonlyattaintwovalues,usuallydenotedwithσ = +1,−1 i 2 orσ =↑,↓. Thespinvariablesofnucleusαaredenotedwiths . The nucleicanhaveawide i α range of spins depending on their content of protons and neutrons. However, in practice the nuclear spin plays a minor role in the electronic structure apart from some special circumstances and therefore in this lecture we will often ignore the nuclear spins. We will further often treat the spatial and spin variables together and therefore we introduce the notation x = (r ,σ ) (11) i i i X = (R ,s ) (12) α α α In this notation the wave function is thus written more compactly as Ψ= Ψ(x ,...,x ,X ,...,X ,t). (13) 1 N 1 N e n We can compactify even more by defining the collective variables x = (x ,...,x ) (14) 1 N e X = (X ,...,X ) (15) 1 N n and write the wave function as Ψ= Ψ(x,X,t) (16) The time-dependent Schr¨odinger equation has then the explicit form ˆ i∂ Ψ(x,X,t)=H(t)Ψ(x,X,t) Ψ(x,X,t )= Ψ (x,X) (17) t 0 0 ˆ where H(t) is the Hamiltonian of Eq.(3). If we neglect external magnetic fields, then the spin variables do not occur in the Hamiltonian. Nevertheless, the spin variables have a very large influence on the energy spectrum. The main reason for this is that the electrons are fermionic particles and the wave function for fermions is required to be anti-symmetric in interchange of both the space and spin variables of two electrons (or more generally all the variables that define the fermionic particle), i.e. it has to satisfy Ψ(...x ...x ...)=−Ψ(...x ...x ...) (18) i j j i This means that not all solutions to the Schr¨odinger equation are allowed, but only those that are anti-symmetric. This considerably reduces the number of eigenstates and has a profound impact on the electronic structure of atoms, molecules and solids. For instance, the complete structure of the periodic table of elements can be explained on the basis of this principle. We will discuss this in more detail later. B. The time-independent Schr¨odinger equation The time-dependent Schr¨odinger equation of Eq.(17) is very hard to solve in general. To simplifythe situationsomewhatwewillfirstconsidertheequationintheabsenceofexternal fields such that the Hamiltonian is time-independent and given by ˆ ˆ ˆ ˆ ˆ ˆ H =T +T +V +V +V (19) e n ee en nn5 Equation (17) then attains the form ˆ i∂ Ψ(x,X,t)=HΨ(x,X,t) Ψ(x,X,t )= Ψ (x,X) (20) t 0 0 in which only the wave function is time-dependent. We now formally solve this initial value problem (and in fact our derivation will not depend on the actual form of the Hamiltonian as long as it is time-independent). Let now Φ denote the eigenstates of the Hamiltonian i ˆ H, i.e. ˆ HΦ (x,X)=E Φ (x,X) (21) i i i This equation is usually referred to as the time-independent Schr¨odinger equation. Since the Hamiltonian is a Hermitian operator its eigenstates form a complete orthonormal set of functions. This means that hΦΦ i=δ (22) i j ij whereδ =1 wheni =j and zero otherwise. We further defined the inner product between ij two functions as Z ∗ hΦΨi = dxdXΦ (x,X)Ψ(x,X) (23) and we used the short notation Z Z X dxdX= dr ...dr dR ...R (24) 1 N 1 N e n σ ...σ ,s ...s 1 N 1 N e n i.e. sum of all spin variables and integration over all spatial variables. Since the set of functionsΦ formacompletesetanyfunctioncanbeexpandedinthem. Thisisinparticular i true for the initial state Ψ of Eq.(20) and we can therefore write 0 X Ψ (x,X)= c Φ (x,X) (25) 0 i i i where c = hΦΨ i. It is then easy to check that the general solution of the initial value i i 0 problem of Eq.(20) is given by X −iE (t−t )/ i 0 Ψ(x,X,t)= hΦΨ ie Φ (x,X) (26) i 0 i i Therefore for time-independent Hamiltonians, knowledge of the eigenfunctions and eigen- values of the Hamiltonian determines the solution of all possible initial value problems for the time-dependent Schr¨odinger equation. We therefore first consider the task of finding the eigenfunctions of the coupled electron-nuclear problem. III. THE BORN-OPPENHEIMER SEPARATION A. General derivation For a system of electrons and nuclei we consider the eigenvalue problem: ˆ HΨ (x,X)=E Ψ (x,X) (27) i i i ˆ where the Hamiltonian H is given in Eq.(19). This is equation presents a rather difficult many-particle problem. To simplify the problem we are going to make use of the fact that6 the mass of the atomic nuclei is much larger than those of the electrons. This will allow us to a very good approximations to separate the electronic from nuclear degrees of freedom. ˆ To derive this scheme we first split the Hamiltonian H as ˆ ˆ ˆ ˆ H =H (r)+H (R)+V (r,R) (28) e n en where we defined the collective coordinates r = (r ...r ) and R = (R ...R ) and 1 N 1 N e n defined several parts as N N e e 2 2 X X 1 e 2 H (r) = T (r)+V (r)=− ∇ + (29) e e ee i 2m 2 r −r i j i=1 i6=j N N n n X 2 X 2 1 Z Z e α β 2 H (R) = T (R)+V (R) =− ∇ + (30) n n nn R α 2M 2 R −R α α β α=1 α6=β N N e n 2 XX Z e α V (r,R) = − (31) en r −R i α i=1 α=1 ˆ We therefore split the Hamiltonian in a purely electronic part H , a purely nuclear part e ˆ ˆ ˆ H and a coupling V . We now note that the nuclear kinetic term T is proportional to n en n 1/M and since M ≫ m one could therefore as a first guess neglect the nuclear kinetic α α ˆ ˆ energyT (R) with respect to the electronic kinetic energy T (r). If we do that the nuclear n e coordinates only appear as parameters in the Hamiltonian. Since the nuclear repulsion ˆ term V (R) is then simply a constant, it then is natural to look at the eigenstates of nn ˆ H (r)+V (r,R), i.e. e en ˆ (H (r)+V (r,R))Φ (x,R)=ǫ (R)Φ (x,R) (32) e en m m m The functions Φ are functions of all the space and spin variables of the electrons, but m dependparametricallyonthenuclearpositionsR. Alsotheeigenenergiesareparametrically dependent on the nuclear positions. Since the functions Φ are, for each configuration R, m eigenfunctions of a Hermitian operator they form a complete orthonormal set with respect to an inner product in the electronic Hilbert space, i.e. we have Z ∗ δ =hΦΦ i = dxΦ (x,R)Φ (x,R) (33) lm l m e m l where we used the short notation Z Z Z X dx= dr ... dr (34) 1 N e σ ...σ 1 N e Let us now consider a true eigenstate of Eq.(27). The eigenfunctions Ψ (x,X) in this i equation can, for a fixed set of arguments X, be regarded as a function of the electronic coordinates only. It can therefore be expanded in terms of the functions Φ of Eq.(32) as m follows: X Ψ (x,X) = χ (X)Φ (x,R) (35) i li l l were the χ (X) are some coefficients to be determined. By multiplying this equation from li ∗ the left with Φ and integrating over the electronic coordinates we find m Z ∗ χ (X) = dxΦ (x,R)Ψ (x,X) (36) mi i m7 We now insert the expansion of Eq.(567) back into the eigenvalue Eq.(27) X X ˆ ˆ ˆ ˆ (T (R)+V (R)+H (r)+V (r,R))χ (X)Φ (x,R)=E χ (X)Φ (x,R) (37) n nn e en li l i li l l l ∗ We multiply this equation from the left by Φ and integrate over the electronic coordinates m With help of Eq.(32) and the orthonormality relation (33) this can then be rewritten as X (ǫ (R)+V (R))χ (X)+ hΦ T (χ Φ )i =E χ (X) (38) m nn mi m n li l e i mi l We can work out further the term under summation sign ˆ ˆ hΦ T (χ Φ )i = δ T (R)χ (X)+χ (X)hΦ T (R)Φi m n li l e lm n li li m n l e Z N n 2 X ∗ − dxΦ (x,R)∇ Φ (x,R)·∇ χ (X) (39) R l R li m α α M α α With this expression Eq.(38) can be written as X (T (R)+ǫ (R)+V (R))χ (X)+ A (R)χ (X) =E χ (X) (40) n m nn mi ml li i mi l where we defined the terms Z N n X 2 ∗ ˆ A (R)=hΦ T (R)Φi − dxΦ (x,R)∇ Φ (x,R)·∇ (41) ml m n l e R l R m α α M α α Wehavethereforefoundthattheeigenvalueequationforthecoupledelectron-nuclearsystem is equivalent to the set of equations (32), (567) and (40). So far no approximations were made. However, in practice we find that the coefficients A are often very small. The lm Born-Oppenheimer (BO) approximation consists in the neglect of all these terms (we will justify this later). In that case Eq.(40) simplifies to to (T (R)+ǫ (R)+V (R))χ (X) =E χ (X) (42) n m nn mi i mi This approximationamounts to saying that restricting the sum in Eq.(567)to a single term already gives a good approximate eigenfunction. This can be seen as follows. According to Eq.(39) neglect of the coefficients A amounts to saying that lm ˆ ˆ T (R)(χ (X)Φ (x,R))≈ Φ (x,R)T (R)χ (X) (43) n mi m m n mi i.e. it amount to a neglect of the nuclear derivatives of the function Φ . If we make this m approximation then it follows from Eq.(32) and Eq.(42) that ˆ ˆ ˆ Hχ (X)Φ (x,R)= (H (r)+H (R)+V (r,R))χ (X)Φ (x,R) mi m e n en mi m ˆ ˆ ≈ χ (X)(H (r)+V (r,R))Φ (x,R)+Φ (x,R)(H (R)+V (R))χ (X) mi e en m m n nn mi = ǫ (R)χ (X)Φ (x,R)+(E −ǫ (R))χ (X)Φ (x,R) m mi m i m mi m = E χ (X)Φ (x,R) (44) i mi m We thus see that with approximation (43) the function Ψ = χ Φ is an eigenfunction i mi m ˆ if the complete electron-nuclear Hamiltonian H. Therefore the Born-Oppenheimer (BO) approximation can also be viewed as making an product Ansatz for the eigenfunctions of the coupled electron-nuclear problem.8 From Eq.(42) we see that the within the Born-Oppenheimer approximation the index i labels eigenstates of nuclear wavefunctions χ in potential ǫ (R) +V (R) for various mi m nn values of m. It is therefore useful to use the labeling i = (m,k) where k refers to the k-th excited state of Eq.(42). For convenience we also write χ = χ = χ and mi m(m,k) mk E = E = E . Then the Born-Oppenheimer approximation can be summarized in i (m,k) mk terms of the following equations Ψ (x,X)=χ (X)Φ (x,R) (45) (m,k) mk m ˆ (H (r)+V (r,R))Φ (x,R)=ǫ (R)Φ (x,R) (46) e en m m m (T (R)+ǫ (R)+V (R))χ (X)=E χ (X) (47) n m nn mk mk mk The BO approximation therefore allows us to divide the problem of interacting electrons and nuclei into two separate problems. We have to solve the purely electronic problem of Eq. (46) for fixed nuclei and subsequently, with the obtained energy surfaces ǫ (R), the m purely nuclear problemofEq.(47). We will see that many properties ofmolecules and solids can be understood from the electronic equation Eq.(46) alone and we will devote several chapters to that problem. However, before we do that, we will illustrate the BO equations with some more specific examples. B. Example: diatomic molecules As a first illustration of the BO oppenheimer Eqs.(46) and (47) we consider the case of a diatomic molecule. For the diatomic molecule has two atomic nuclei with charges Z e and 1 Z e, massesM andM and nuclear coordinatesR andR . For the moment we leave the 2 1 2 1 2 numberofelectronstobearbitary. Forexample,thehydrogenmoleculeH hastwonucleiof 2 positive charge +e and two negatively charged electrons, and the nitrogen molecule N has 2 two nuclei with positive charge of +7e and 14 negatively charged electrons. The electronic ˆ part H still has the general form of Eq.(29) but the nuclear part and the electron-nuclear e interaction of Eqs.(30) and Eq.(31) simplify to 2 2 2 Z Z e 1 2 2 2 H (R) = T (R)+V (R) =− ∇ − ∇ + (48) n n nn R R 1 2 2M 2M R −R 1 2 1 2   N e 2 2 X Z e Z e 1 2 V (r,R) = − + (49) en r −R r −R i 1 i 2 i=1 Within the BO approximation we first solve the electronic problem of Eq.(46) for fixed nuclear positions. For the diatomic molecule this amounts to solving the equation "   N e 2 X 2 Z e Z e 1 2 ˆ H (r)− + Φ (x,R ,R )=ǫ (R ,R )Φ (x,R ,R ) (50) e m 1 2 m 1 2 m 1 2 r −R r −R i 1 i 2 i=1 We will not solvethis equationhere, but focus for the moment onthe nuclear problem. It is clearthatbysimplytranslatingthewholemoleculeinspace,wewillnotalteritseigenvalues, and thereforeǫ (R ,R )=ǫ (R +a,R +a) for an arbitrary vectora. In particular we m 1 2 m 1 2 can choosea =−R from which we see immediately thatǫ only depends on R −R i.e. 2 m 1 2 ǫ (R ,R ) = ǫ (R −R ). It is also clear that the eigenenergies cannot depends on the m 1 2 m 1 2 orientation of the molecule, and can therefore only depend on the lenght R −R of the 1 2 vector connecting the nuclear positions. This then means that the nuclear equation (47) attains the form   2 2 2 Z Z e 1 2 2 2 − ∇ − ∇ +ǫ (R −R )+ χ (X)=E χ (X) (51) m 1 2 km km km R R 1 2 2M 2M R −R 1 2 1 29 This is now a simple two-particle equation in which the potential only depends on the relative distance between the particles. For this reason it is useful to introduce relative and center-of-mass coordinates R = R −R (52) 1 2 M R +M R 1 1 2 2 R = (53) CM M +M 1 2 For simplicity we neglect the spin-dependence of the nuclear wavefunctions (or assume that the nuclei have spin zero) in which case we can write χ (X) = χ (R). In terms of the km km new coordinates we have a new wave function ϕ (R,R ) = χ (R ,R ). This wave km CM km 1 2 function satisfies the equation   2 2 2 Z Z e 1 2 2 2 − ∇ − ∇ +ǫ (R)+ ϕ (R,R )=E ϕ (R,R ) (54) m km CM km km CM R R CM 2M 2μ R tot whereM =M +M is the total nuclear mass andμ=M M /(M +M ) is the reduced tot 1 2 1 2 1 2 mass. Exercise. Derive this equation. The hamiltonian of Eq. (54) is a sum of terms depending only on R and terms CM depending only on R, and therefore separable. It is the eigen solution is then readily seen tobeoftheformofaplanewavedescribingthecenter-of-massandawavefunctiondepending only on the relative coordinateR 1 ip·R CM ϕ(R,R )= √ e φ (R) (55) CM rm V with V the volume of space (we assume that the molecule is enclosed in a large volme V) and where the relative wave function φ satisfies the equation mr   2 2 Z Z e 1 2 2 − ∇ +ǫ (R)+ φ (R)=E φ (R) (56) m rm rm rm R 2μ R and 2 p E = +E (57) km rm 2M tot The label k = (p,r) is therefore a multilabel describing the center-of-mass momentum p andinternalstater. The problemis nowreducedto the simple one-particleequation(56)in a centralpotential. The solution ofthis problemis well-knownfrom the coursesin quantum mechanics (see e.g. the hydrogen atom) If we write the kinetic energy operator in spherical coordinates then we find that J φ (R)=F (R)Y (Ω) (58) rm vJm M J where R = R and where Y (Ω) is a spherical harmonic function of the angles Ω, corre- M sponding to angular momentum quantum number J with M =−J,...,+J. The spherical harmonic functions describe an overall rotational state of the molecule. We further intro- duced the notationr =(vJM). The radial function F (R) satisfies the equation vJm     2 2 2 2 d 2 d J(J +1) Z Z e 1 2 − + + +ǫ (R)+ F (R)=E F (R) (59) m vJm vJm vJm 2 2 2μ dR RdR 2μR R10 The functions F describe the vibrational states of the molecule. Due to the large value vJM of the reduced mass μ the solutions F described narrow functions peaked around the vJM minimum of the potential (we will justify this a posteriori). We can there make a harmonic 2 2 approximationandexpandthepotentialarounditsminimum. Theterm J(J+1)/(2μR ), again due to the large value of μ, has a negligible contribution to the position of this minimum, and we determine the equilibrium distance by the requirementdV /dR(R )= m 0,m 0 where we define the potential V (R) by m 2 Z Z e 1 2 V (R) =ǫ (R)+ (60) m m R The harmonic approximation for the potential V (R) is then m 1 2 V (R)=V (R )+ K (R−R ) (61) m m 0,m m 0,m 2 2 2 where K =d V (R )/dR . Then Eq.(59) in the harmonic approximation becomes m m 0,m "   2 2 2 d 2 d J(J +1) 1 2 − + + +V (R )+ K (R−R ) F (R)=E F (R) m 0,m m 0,m vJm vJm vJm 2 2 2μ dR RdR 2μR 2 0,m (62) 2 2 2 2 Since we replaced the term J(J +1)/(2μR ) by the constant J(J +1)/(2μR ) the 0,m functions F do not depend on J anymore and we can denote them with F . We can vJm vm then write Eq.(62) as     2 2 d 2 d 1 2 ˜ − + + K (R−R ) F (R) =E F (R) (63) m 0,m vm vm vm 2 2μ dR RdR 2 ˜ where we definedE by the equation vm 2 J(J +1) ˜ E =E +V (R )+ (64) vJm vm m 0,m 2 2μR 0,m Now Eq.(A3) looks very much like the equation for a harmonic oscillator apart from the term (1/R)(d/dR). However, we will show that the contribution of this term is small and that hence the equation can indeed be reduced to that of the harmonic oscillator   2 2 d 1 2 ˜ − + K (R−R ) F (R) =E F (R) (65) m 0,m vm vm vm 2 2μdR 2 The solution of this equation are well-known from the lectures on quantum mechanics and given by Hermite polynomials H times a Gaussian function v 1 1 2 − Q 2 F (R)= p H (Q)e (66) vm v 1 v 2 π 2 v 1/2 1/2 where Q = (μω /) (R−R ) and ω = (K /μ) and v is an integer. The corre- m 0,m m m sponding eigenenergies are 1 ˜ E =(v+ )ω (67) vm m 2 IfwenowcollectourresultsthenwefindthatthecompleteBorn-Oppenheimerwavefunction and energy for the diatomic molecule is given by 2 1 1 1 ip·R − Q J CM 2 Ψ (x,X) = Φ (x,R)√ e p H (Q)e Y (Ω) (68) pvJMm m v M 1 V v 2 π 2 v 2 2 p 1 J(J +1) E = +V (R )+(v+ )ω + (69) pvJm m 0,m m 2 2M 2 2μR tot 0,m11 Now that we have derived these equations, let us see how we can interpret them. The energy expression of Eq.(69) has a clear physical interpretation. The first term on the righthand side is simply the kinetic energy of the center-of-mass of the molecule. The remaining terms are more interesting and describe the internal state of the molecule. The first term V (R ) essentially gives the position of the electronic levels of the molecule. m 0,m Typical energy differences between energy surfaces ǫ (R) is a of the order of eV’s in most m molecules. The third term describes the internuclear vibrations of the molecule with typical values ofω between the vibrationallevels of the order of tenths of an eV. Finally the last m term in Eq.(69) describes the rotational levels of the rigidly rotating molecule. The typical differences in rotation energies are of the order of hundredths of an eV. IV. THE MANY-ELECTRON PROBLEM A. Wave functions and symmetry We now consider the purely electronic problem of Eq.(46). ˆ ˆ ˆ (T (r)+V (r)+V (r,R))Φ (x,R) =ǫ (R)Φ (x,R) (70) e ee en m m m Since we in following chapters are not interested in the dependence on the nuclear positions R we will stopwriting them asarguments inthe wavefunction. It is also notnecessaryto to indicate use the subindexe for electrons,as we will only discuss electrons. We will therefore in the following consider general Hamiltonians of the form ˆ ˆ ˆ ˆ H =T +V +W (71) where N X 2 2 ˆ T = − ∇ (72) i 2m i=1 N X ˆ V = v(r ) (73) i i=1 N X 1 ˆ W = w(r ,r ) (74) i j 2 i= 6 j 2 For the case that w(r ,r )=e /r −r and i j i j Nn X Z e α v(r) =− (75) r−R α α=1 we recover the Hamiltonian of Eq.(70). We now study eigenstates of Hamiltonian (71) ˆ ˆ ˆ (T +V +W)Ψ (x) =E Ψ (x) (76) n n n How do the eigenfunctions of this equation look like? A first observation is that the Hamil- tonian is invariant under permutation of all electronic coordinatesr , i.e. i ˆ ˆ H(r ,...,r ) =H(r ,...,r ) (77) 1 N P(1) P(N) where P is a permutation of the labels (1,...,N). This implies that if Ψ(x ,...,x ) is an 1 N ˆ eigenstate of Hamiltonian H then also ′ Ψ (x ,...,x )= Ψ(x ,...,x ) (78) 1 N P(1) P(N)12 for an arbitrary permutation P is also an eigenstate with the same eigenenergy. This is readily derived as follows ′ H(r ,...,r )Ψ (x ,...,x ) = H(r ,...,r )Ψ(x ,...,x ) 1 N 1 N 1 N P(1) P(N) ˆ = H(r ,...,r )Ψ(x ,...,x ) P(1) P(N) P(1) P(N) = EΨ(x ,...,x ) P(1) P(N) ′ = EΨ (x ,...,x ) (79) 1 N It is not difficult to find some more concrete examples. Let us, for simplicity, take the ˆ interaction to be zero (i.e. W = 0) and let ϕ (x) be the eigenfunctions of i 2 2 h(r) =− ∇ +v(r). (80) 2m So we have h(r)ϕ (r) =ǫ ϕ (r). (81) i i i Let us further consider the case that we deal with only three particles, such that the Hamil- tonian is given by ˆ H =h(r )+h(r )+h(r ) (82) 1 2 3 Then, for instance, the 3-particle wavefunction Ψ(x ,x ,x )=ϕ (x )ϕ (x )ϕ (x ) (83) 1 2 3 1 1 2 2 3 3 ˆ is and eigenfunction ofH: ˆ HΨ(x ,x ,x )= (ǫ +ǫ +ǫ )Ψ(x ,x ,x ). (84) 1 2 3 1 2 3 1 2 3 (Check this yourself). So the eigenvalue is simplyE =ǫ +ǫ +ǫ . It is then easy to check 1 2 3 that, for instance ′ Ψ (x ,x ,x ) =ϕ (x )ϕ (x )ϕ (x ) (85) 1 2 3 1 3 2 1 3 2 (i.e we applied permutation P(1),P(2),P(3) = 3,1,2) is an eigenfunction with the same eigenenergy. For the case that the one-particle energies ǫ ,ǫ and ǫ are all different there 1 2 3 are in fact 3 = 6 different eigenfunctions with eigenenergy equal to E = ǫ +ǫ +ǫ . 1 2 3 However, it turns out that not all solutions are realized in nature, but only those that are either completely symmetric or completely anti-symmetric in the interchange of variables of particles of the same type, i.e. only those wave functions that satisfy Ψ(...x ,...,x ...)=±Ψ(...x ,...,x ...) (86) i j j i when the coordinates of any pair of particles is interchanged. When interchange of any two particles yields a minus sign we say that the wavefunction is anti-symmetric, and when it yields a plus sign we say that the wave-function is symmetric. Those particles that are describedbyanti-symmetricwavefunctionsarecalledfermionsandthosethataredescribed by symmetric ones are called bosons. Paulidemonstrated in his famous spin-statistics theo- remthatfermionsmustbe particleswithhalf-integerspinandthatbosonsmustbe particles with integer spin. The proof of this theorem relies on the theory of relativity and invokes local causility and the transformation property of spinors under Lorentz transformations. These matters would, however, be the topic of a completely different lecture. In this course13 we mainly dealwith electronswhich arespinhalf particlesand accordingto Pauli’s theorem therefore described by anti-symmetric wave functions that satisfy Ψ(...x ,...,x ...)=−Ψ(...x ,...,x ...) (87) i j j i 2 For wavefunctions with this property the probability distribution Ψ is completely sym- metric such that for any permutationP we have that 2 2 Ψ(x ,...,x ) =Ψ(x ,...,x ) (88) 1 N P(1) P(N) This a function with has the following interpretation. The quantity 2 3 3 Ψ(x ,...,x ) d r ...d r (89) 1 N 1 N represents the probability to find an electron with space-spin coordinate x = (r σ ) in 1 1 1 3 a volume d r around point r , another electron with space-spin coordinate x = (r σ ) 1 1 2 2 2 3 in a volume d r around point r , etc. Eq.(88) now implies that this probability dis- 2 2 tribution is completely symmetric and does not distinguish configurations that differ in the permutations of arbitrary particle coordinates. This property is not shared by the 3-particle wave functions of Eqs.(83) and (85) as they assign specific particle coordinates to specific one-particle states ϕ . These wave functions therefore describe distinguishable i particles. As mentioned before, the systems that we find in nature are not described by by such wave functions. Only probability distributions that satisfy the symmetry requirement Eq.(88) and therefore lead to completely indistinguishable particles are realized in nature. If we take Eq.(88) for the probability distribution as an indistinguishability postulate in many-particle quantum mechanics, then is not difficult to derive that the corresponding wave functions must be either symmetric or anti-symmetric. The indistinguishability postulate leads to some rather counterintuitive consequences. For example, if in classical mechanics we consider the collision of two point particles of the same mass but different momenta would always say that the particle coming from the left had, for instance, a higher momentum than the particle coming from the right. In quantum mechanics this is no longer possible. If two electrons with different momenta collide then we cannot say any longer which particle has which momentum, the only thing we can say that they ”have” different momenta. Letusnowturnbacktoourexampleofthreeparticles. Sowhatshouldnowbethe proper anti-symmetic 3-particle eigenfunctions of Eq.(82)? It is not difficult to check that 1 Ψ(x ,x ,x ) = √ ϕ (x )ϕ (x )ϕ (x )+ϕ (x )ϕ (x )ϕ (x )+ϕ (x )ϕ (x )ϕ (x ) 1 2 3 1 1 2 2 3 3 1 2 2 3 3 1 1 3 2 1 3 2 6  − ϕ (x )ϕ (x )ϕ (x )−ϕ (x )ϕ (x )ϕ (x )−ϕ (x )ϕ (x )ϕ (x ) (90) 1 2 2 1 3 3 1 3 2 2 3 1 1 1 2 3 3 2 is such an eigenfunction. First of all, since every term in Eq.(90) is an eigenfunction with eigenenergy ǫ +ǫ +ǫ also the linear combination is an eigenfunction with that energy. 1 2 3 Further using the fact that all the terms in this expression are orthogonal to each other it is immediately seen that the function is correctly normalized. Such wave functions are obtained by applying the an anti-symmetrization operatorA to a given wave function. The action of this operator on a given wave-function is defined as X 1 P AΨ(x ,...,x )= √ (−1) Ψ(x ,...,x ) (91) 1 N P(1) P(N) N P P wherethesumrunsoverallpermutationsP andwhere(−1) isthesignofthepermutation. The number P is the number of two-particle interchanges or transpositions in which the14 permutation can be decomposed. For example, the wave function of Eq.(90) is obtained by applying the operatorA to the 3-particle wave function of Eq.(83). In this case the 3 = 6 permutations were P = 1,(13)(12),(12)(13),(12),(13),(23) with P = 0,2,2,1,1,1. The expression Eq.(90) can be more elegantly rewritten as the determinant ϕ (x ) ϕ (x ) ϕ (x ) 1 1 2 1 3 1 1 √ Ψ(x ,x ,x ) = ϕ (x ) ϕ (x ) ϕ (x ) (92) 1 2 3 1 2 2 2 3 2 3 ϕ (x ) ϕ (x ) ϕ (x ) 1 3 2 3 3 3 ˆ Itis noweasyto guesswhatallthe eigenstatesofHamiltonianH ofEq.(82)wouldbe. They are given by the wave functions ϕ (x ) ϕ (x ) ϕ (x ) i 1 j 1 k 1 1 √ Ψ (x ,x ,x ) = ϕ (x ) ϕ (x ) ϕ (x ) (93) ijk 1 2 3 i 2 j 2 k 2 3 ϕ (x ) ϕ (x ) ϕ (x ) i 3 j 3 k 3 and these functions have eigenenergies E =ǫ +ǫ +ǫ . We also see that if two indices ijk i j k within the triplet(ijk)arethe same thentwocolumns ofthe determinantinEq.(93)arethe same and the determinant is identically zero. Therefore all the indices i,j and k must be different. Much less accurately we can say that no two fermions can occupy the same state. This statement is often referred to as the Pauli principle. Furthermore changing the order ijk of the indices only changes the wave function up to a sign. Therefore to give a unique representation of all the eigenstates we have to fix the order. This is most easily done by requiring that ij k. It is now easy to imagine what the eigenstates of a system of N noninteracting fermions would look like. The Hamiltonian is given by ˆ H =h(r )+...+h(r ) (94) 1 N and the eigenfunctions are given by ϕ (x ) ... ϕ (x ) i 1 i 1 1 N 1 . . . . Ψ (x ,...,x ) = √ (95) i ...i 1 N 1 N . . N ϕ (x ) ... ϕ (x ) i N i N 1 N where i i ... i and the eigenvalues are given by E = ǫ +...+ǫ . The 1 2 N i ...i i i 1 N 1 N wave functions of the form of Eq.(95) are often referred to as Slater determinants. From the definition of the determinant one can see that an equivalent way of writing Eq.(95) is the anti-symmetrized product X 1 P Ψ (x ,...,x )= √ (−1) ϕ (x )...ϕ (x ) (96) i ...i 1 N i P(1) i P(N) 1 N 1 N N P Since the value of the determinant does not change when we interchange rows and columns this can be equivalently written as X 1 P Ψ (x ,...,x ) = √ (−1) ϕ (x )...ϕ (x ) (97) i ...i 1 N i 1 i N 1 N P(1) P(N) N P where the permutation now acts on the labelsi of the one-particle states instead of acting k on the coordinatesx . This equationis useful for calculating expectationvalues and matrix k elements of Slater determinants. Using it we can then prove that whenever the one-particle states ϕ are orthonormal in the one-particle Hilbert space, i.e. i Z ∗ hϕϕ i= dxϕ (x)ϕ (x) =δ (98) i j j ij i15 (they need not necessarily be eigenstates of a one-particle Hamiltonian ) then the Slater determinants Ψ and Ψ are orthonormal as well in the many-particle Hilbert i ...i j ...j 1 N 1 N space, i.e. Z ∗ hΨ Ψ i = dx ...dx Ψ (x ,...,x )Ψ (x ,...,x ) i1...iN j1...jN 1 N i ...i 1 N j1...jN 1 N 1 N = δ ...δ (99) i j i j 1 1 N N This can be shown directly from Eq.(503). We have Z ∗ hΨ Ψ i= dx ...dx Ψ (x ,...,x )Ψ (x ,...,x ) i ...i j ...j 1 N 1 N j ...j 1 N 1 N 1 N i ...i 1 N 1 N Z X ′ 1 P+P ∗ ∗ = (−1) dx ...dx ϕ (x )...ϕ (x )ϕ (x )...ϕ (x ) 1 N 1 N j ′ 1 j ′ N i i P(1) P(N) P (1) P (N) N ′ P,P Z Z X ′ 1 P+P ∗ ∗ = (−1) dx ϕ (x )ϕ (x )... dx ϕ (x )ϕ (x ) 1 1 j ′ 1 N N j ′ N i i P(1) P (1) P(N) P (N) N ′ P,P X 1 ′ P+P = (−1) δ ...δ (100) i j i j P(1) ′ P(N) ′ P (1) P (N) N ′ P,P Our conclusions follow not immediately from the last line of this equation. If the sets of indices (i ...i ) and (j ...j ) differ in at least one label then this is also true for the 1 N 1 N ′ sets (i ...i ) and (j ′ ...j ′ ) for any permutation P and P . So in that case P(1) P(N) P (1) P (N) every term in sum in the last line of Eq.(100) is zero. The only remaining case is when (i ...i ) =(j ...j ). But in that case the sum in the last line of Eq.(100) gives 1 N 1 N X X 1 ′ 1 P+P (−1) δ ...δ = 1 =1 (101) i i i i ′ ′ P(1) P (1) P(N) P (N) N N ′ P,P P where we used that the terms of the sum on the righthand side are only nonvanishing ′ when P = P . This proves Eq.(99). We therefore conclude that the set of Slater determi- nants form of a complete orthonormal set of wave functions that span the Hilbert space of anti-symmetric wave functions. This implies that any anti-symmetric wavefunction can be expanded in these states. More precisely, if we are given an arbitrary many-electron wave function Φ then we can always write X Φ(x ,...,x ) = c Ψ (x ,...,x ) (102) 1 N i ...i i ...i 1 N 1 N 1 N i ...i 1 N This expression provides us with a way to solve the correlated many-electron problem of Eq.(76). We write expand the eigenfunctions of this equation, which we will denote by Φ , n in Slater determinants X (n) Φ (x) = c Ψ (x) (103) n I I I where we introduced the short notation for a multi-indexI = (i ...i ) withi ...i . 1 N 1 N If we insert Eq.(103) into the Schr¨odinger equation of Eq.(76) we obtain X X (n) (n) ˆ c HΨ (x)=E c Ψ (x) I n I I I I I ∗ IfwemultiplybothsidesoftheequationwithΨ andintegrateoverallelectroniccoordinates J we then obtain, with help of the orthonormality condition Eq.(99), the following eigenvalue16 (n) equation for the coefficients c I X (n) (n) H c =E c (104) JI n I J I where we introduced the matrix ˆ H =hΨ HΨ i (105) JI J I We have therefore found a well-defined recipe to solve the many-particle Schr¨odinger equa- tion. Wefirstselectasetoforthonormalone-particlefunctionsϕ (x). Thenwesubsequently i construct all possible Slater determinants Ψ in terms of these one-particle functions. We I subsequently calculate all matrix elements H of Eq.(319) and diagonalize this matrix. JI However, in practice this can only be done for the smallest systems. First of all, in practice we canonly select a finite numberi = 1,...,M of one-particlestates out of theseM states k we can construct   M M = N N(N−M) multi-index pairs I = (i ...i ). This leads to very large dimensions of the matrix H 1 N JI (n) and the vectorsc . The problem essentially grows factorially. If we consider, for instance I a system with N = 10 electrons, such as the neon atom or the water molecule, then if (n) we choose a a set of M = 20 one-particle states the lenght of the vectors c is already I 184756. If we decide to do it more accurately and include twice as many states M = 40 (n) the dimension of the vectors c already increases to 847660528. To find a practical way I to solve this many-body problem we therefore have to find other methods. The first such method that the we will consider the is Hartree-Fock method. This is discussed in the next section. B. The Hartree-Fock method 1. The ground state energy and the Raleigh-Ritz principle We start by considering again the Schr¨odinger equation of Eq.(76). We define a ground state Ψ to be a state with the lowest possible energy, i.e. 0 E ≤E ∀n (106) 0 n There could be several linearly independent wave functions with the lowest energy. If this is the case the ground state is called degenerate. This happens for instance in open shell iα atoms. When the groundstate is unique ( up to a trivialphase factore ) the groundstate is called nondegenerate. Since any state Ψ in the Hilbert space can be expressed as a linear combination of the eigenstates we can write X Ψ(x) = ) (107) c Ψ (x n n n where, sincehΨ Ψ i =δ , it is easy to see that when Ψ is normalized to one that n m nm X 2 c = 1 (108) n n17 Then we also see that X X X ∗ 2 2 ˆ ˆ hΨHΨi= c c hΨ HΨ i= E c ≥E c =E (109) m n m n n 0 n 0 n n,m n,m n where we used that E ≥E . So we obtain that the expectation value of the Hamiltonian n 0 of any normalized state,hΨΨi= 1, has a lower bound given by the ground state energy of the Hamiltonian, i.e. ˆ hΨHΨi≥E (110) 0 This is the Rayleigh-Ritzprinciple. This principle is veryuseful to find approximateground state wave functions and both the Hartree-Fock and density-functional method that we will discuss below are based on this principle. In order to proceed we need a more explicit expression of the expectation value on the right hand side of Eq.(110). We do this separately for the one- and two-particle terms in Eq.(76). For this purpose let us write the Hamiltonian in this equation as ˆ ˆ ˆ H =H +W 0 ˆ ˆ where the term H only contains the one-body parts of the Hamiltonian and W only the 0 two-body parts. Then we have N X ˆ ˆ ˆ H =T +V = h(r ) (111) 0 i i=1 ˆ where h(r) is given by Eq.(80). The expectation value ofH is then given by 0 Z Z N X ∗ ∗ ˆ hΨH Ψi= dxΨ (x)h(r )Ψ(x)=N dxΨ (x)h(r )Ψ(x) 0 i 1 i=1 whereinthesecondstepweusedtheanti-symmetryofthewave-functionwhichmakesevery term in the sum on the right hand side give the same contribution. Now the last term can be written as Z  Z  ∗ ′ N dx h(r ) dx ...dx Ψ (x ,x ...x )Ψ(x ,x ...x ) 1 1 2 N 2 N 1 2 N 1 ′ x =x 1 1 ′ In this expression the operator h(r ) first acts on the integrand before the limit x =x is 1 1 1 ˆ taken. With help of this expression the expectation value of H is then finally given by 0 Z ′ ˆ hΨH Ψi= dx h(r )γ(x ,x ) (112) 0 1 1 1 ′ 1 x =x 1 1 where we defined the so-called one-particle density matrix γ as Z ′ ∗ ′ γ(x ,x )=N dx ...dx Ψ (x ,x ...x )Ψ(x ,x ...x ) (113) 1 2 N 2 N 1 2 N 1 1 Itthenremainstofindanexpressionfortheexpectaionvalueofthetwo-particleinteractions. For this quantity we can calculate that Z Z N X 1 1 ∗ ∗ ˆ hΨWΨi= dxΨ (x)w(r ,r )Ψ(x)= N(N −1) dxΨ (x)w(r ,r )Ψ(x) i j 1 2 2 2 i= 6 j18 where again due to the anti-symmetry of the wave function every of theN(N−1) terms in the sum gives the same contribution. If we now define the two-particle density matrix by Z ′ ′ ∗ ′ ′ Γ(x ,x ,x ,x )=N(N−1) dx ...dx Ψ (x ,x ,x ...x )Ψ(x ,x ,x ...x ) (114) 1 2 3 N 3 N 1 2 3 N 1 2 1 2 then the expectation value of the two-particle interactions can be written as Z 1 ˆ hΨWΨi= dx dx w(r ,r )Γ(x ,x ,x ,x ) (115) 1 2 1 2 1 2 1 2 2 Collecting our results we therefore obtain the following expression for the total energy ex- pectation value of the wave function Ψ Z Z 1 ′ ˆ hΨHΨi= dx h(r )γ(x ,x ) + dx dx w(r ,r )Γ(x ,x ,x ,x ) (116) 1 1 1 ′ 1 2 1 2 1 2 1 2 1 x =x 1 1 2 We will use this general result to calculate the total energy of a Slater determinant wave function in the next section. Before we do that we will derive a few useful relations that we will use later. First of all 2. The Hartree-Fock one-particle equations The Hartree-Fock method is based on the Rayleigh-Ritz variational principle together with the expression for the energy expectation value for a single Slater determinant wave function. SinceweareinterestedinthegroundstateenergyweconsideraSlaterdeterminant in which the lowestN states ϕ (x) are occupied: i ϕ (x ) ... ϕ (x ) 1 1 N 1 1 . . . . Ψ (x ,...,x ) = √ (117) 1...N 1 N . . N ϕ (x ) ... ϕ (x ) 1 N N N Let us now calculate the total energy expectation value for this wave function. According to Eq.(116) we have to calculate first the one-particle and two-particle density matrices of Eq.(113) and (114) for our wave function. The calculation is not difficult, but to not interrupt the story too much the derivation is given in the Appendix. The result is that for a Slater determinant wave function N X ′ ∗ ′ γ(x ,x ) = ϕ (x )ϕ (x ) (118) 1 i 1 1 i 1 i=1 and ′ ′ ′ ′ ′ ′ Γ(x ,x ,x ,x )=γ(x ,x )γ(x ,x )−γ(x ,x )γ(x ,x ) (119) 1 2 1 2 2 1 1 2 1 2 1 2 With these expressions inserted in Eq.(116) the energy expectation value becomes Z N X ∗ ˆ Eϕ = hΨHΨi= dxϕ (x)h(r)ϕ (x) i i i i=1 Z 1 + dx dx w(r ,r )(γ(x ,x )γ(x ,x )−γ(x ,x )γ(x ,x )) (120) 1 2 1 2 1 1 2 2 1 2 2 1 2 where the one-particledensity matrixγ in the interactionterm is givenby expression(118). We have written the left hand side of the equation as Eϕ. This is to indicate that i19 this energy expression depends on what form of one-particle wave functions ϕ we insert. i This quantity is therefore an object that assigns a number to a function. Such an object is also called a functional. It is common notation in physics to enclose the arguments of a functional by square brackets. Due to the Rayleigh-Ritz principle we know that this functional is bounded from below by the true ground state energy of the many-particle system. This implies Eϕ≥E (121) i 0 So far we havenot specified the formof the functionsϕ . The only property that we used in i deriving expression (120) for the total energy is that they are orthonormal. Equation (121) nowprovidesus with anoptimalwayofchoosingthe one-particlestates: We choosethem of such a form that they minimize the value ofEϕ. Since the minimum of a function has i the property that its derivative at that point vanishes, this means that we have to require that δEϕ i =0 (122) δϕ (x) i Note that this is not a usual derivative, but a so-called functional derivative. Functional derivatives are, however evaluated in very much the same way as usual derivatives, by con- sidering small changes in their arguments. We will show this soon below. For completeness a smallappendix onthe calculationof functional derivatives is added to these lecture notes. Let us continue with Eq.(122). This equation can, however, not be applied straightfor- wardly as in the construction of Eϕ we assumed that the one-particle states ϕ were i orthonormal, i.e., they satisfy the relations Z ∗ dxϕ (x)ϕ (x) =δ (123) j ij i Therefore the wave function variations in Eq.(122) must be such that the perturbed wave functions satisfy the orthonormality relations. The problem of finding minima with a con- straintisacommononeinphysics. Theproblemcanbeconvenientlydealtwithbymeansof the method of Lagrange multipliers. For our case this means that we consider the modified functional Z N X ∗ Lϕ=Eϕ− λ dxϕ (x)ϕ (x) (124) i i ij j i i,j=1 ∗ with Lagrange multipliers λ =λ and consider the equations ij ji δLϕ i =0 (125) δϕ (x) i where we now are allowed to make free variations in the wave functions ϕ . One can show i thatthesolutionoftheseequationsareequivalenttothesolutionofEq.(122)withconstraints Eq.(123). In making the variations in the wave functions ϕ one should take into account i that they are in general complex and we can therefore make separate variations in the real and imaginary part. As shown in the Appendix, this is equivalent to making independent ∗ variations in the ϕ andϕ in Lϕ. This means that Eq.(125) is equivalent to i i i ∗ ∗ δLϕ ,ϕ δLϕ,ϕ i i i i = 0 , = 0 (126) ∗ δϕ (x) δϕ (x) i i20 ∗ where in the first term we keep ϕ fixed and make variation with respect to ϕ and in the i i ∗ second equation we keep ϕ fixed and make variation with respect to with respect to ϕ . i i ∗ However, the functionalLϕ ,ϕ has the property that i i ∗ ∗ ∗ Lϕ ,ϕ =Lϕ ,ϕ (127) i i i i and in that case the second equation in Eq.(126) is simply the complex conjugate of the first equation and does not give new information. We therefore only need to consider the ∗ first equation in Eq.(126). Let us therefore make a variationϕ (x) in Eq.(124). We have i Z N X ∗ δL = δE− λ dxδϕ (x)ϕ (x) ij i i ij Z Z N X ∗ ∗ = dxδϕ (x)h(r )ϕ (x)+δW − λ dxδϕ (x)ϕ (x) (128) i i ij i i i ij where we defined the variation in the interaction term δW by Z  1 δW = dx dx w(r ,r ) δγ(x ,x )γ(x ,x )+γ(x ,x )δγ(x ,x ) 1 2 1 2 1 1 2 2 1 1 2 2 2  −δγ(x ,x )γ(x ,x )−γ(x ,x )δγ(x ,x ) 1 2 2 1 1 2 2 1 Z   = dx dx w(r ,r ) δγ(x ,x )γ(x ,x )−δγ(x ,x )γ(x ,x ) (129) 1 2 1 2 1 1 2 2 1 2 2 1 where in the last step of this equation used the symmetry of the integrand. Now we further ∗ have to calculate δγ for a change δϕ (x) in the orbitals i N X ∗ δγ(x ,x ) = δϕ (x )ϕ (x ) (130) 1 2 1 i 2 i i If we insert this back into Eq.(129) we obtain Z   ∗ δW = dx dx δϕ (x )w(r ,r ) ϕ (x )γ(x ,x )−ϕ (x )γ(x ,x ) 1 2 1 1 2 i 1 2 2 i 2 2 1 i Z ∗ = dx δϕ (x )V (x ) (131) 1 1 i 1 i where we defined Z Z V (x) =ϕ (x ) dx v(r ,r )γ(x ,x )− dx w(r ,r )ϕ (x )γ(x ,x ) (132) i i 1 2 1 2 2 2 2 1 2 i 2 2 1 With Eq.(131) the variational expression for δL of Eq.(128) then becomes   Z N N X X ∗   δL= dxδϕ (x) h(r)ϕ (x)+V (x)− λ ϕ (x) (133) i i ij j i i j and therefore N X δL =h(r)ϕ (x)+V (x)− λ ϕ (x) (134) i i ij j ∗ δϕ (x) i j

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