Density Functional Theory PDF

Density Functional Theory Frank De Proft, Paul Geerlings Research Group of General Chemistry (ALGC), Faculty of Sciences and Bio-engineering Sciences, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium http:...

Density Functional Theory Frank De Proft, Paul Geerlings Research Group of General Chemistry (ALGC), Faculty of Sciences and Bio-engineering Sciences, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium http://algc.research.vub.be [email protected] [email protected] 1 Outline of this Lecture Part I. The Basics of DFT 1. Introduction: the electron density and the need for a Density Functional Theory 2. The basic theorems: the Hohenberg-Kohn Theorems and the Variational Equation 3. The implementation: Kohn-Sham Theory Practical implementation 3.1. Introduction 3.2. The Levy Constrained Search Approach 3.3. Kohn-Sham DFT Part II. Computational DFT 1. The Kohn-Sham Equations 1.1. Derivation of the Canonical Kohn-Sham Equations 1.2. An Alternative View 1.3. Practical Solution of the Kohn-Sham Equations 2  3. The implementation: Kohn-Sham Theory (Previous lecture) 3.1. Introduction The first Hohenberg-Kohn permits us to write the electronic energy as a functional of the density Nucleus electron attraction energy (function of rho) F HK = Hohenberg-Kohn Z function, consist of kinetic energy and electron- E[ρ] = Vne[ρ] + T [ρ] + Vee[ρ] = ρ(r)v(r)dr + FHK[ρ] electron repulsion energy integral (electron density * external potential) part = nucleus attraction part where T [ρ] is the kinetic energy, Vee[ρ] is the electron–electron interaction energy, (electron-electron repulsion) and FHK[ρ] is a universal functional of ρ FHK[ρ] = T [ρ] + Vee[ρ] = hΨ|T̂ + V̂ee|Ψi = expectation value for the ground state wavefunction of the kinetic energy and the electron-electron repulsion The second Hohenberg–Kohn theorem allowed us to introduce a variational prin- ciple -> Ground state electron density can be obtained by minimizing the energy respecting to changes of the electron density. We therefore need to minimise the energy with respect to density variations, R subject to the constraint ρ(r)dr = N. == density constraint Minimizing is done with a constrain: the integration of the density must always give the total amount of electrons 3  R So for our DFT minimisation with constraint ρ(r)dr = N , we have  Z  δ E[ρ] − µ ρ(r)dr − N =0 δρ(r) Constraint Quantity you want to minimize δE[ρ] −µ=0 δρ(r) Finally, given that Z E[ρ] = ρ(r)v(r)dr + FHK[ρ] we have µ = chemical potential of the system = extrernal pot. + der. of Hohenberg-Kohn function --> You δFHK[ρ] can solve for the exact density. When you insertt the ground µ = v(r) + state density in this equation --> This quantity is constant δρ(r) throughout the molecule and equal to the electronic chemical potential of the system This is the Euler–Lagrange equation, which can be solved for the exact density 4  We now consider this variational principle in detail In the Hohenberg-Kohn analysis, a minimization over all electron densities asso- ciated with an external potential is performed Density positive everywhere in space R For any trial density ρ̃(r) (with ρ̃(r) ≥ 0 and ρ̃(r)dr = N ), associated with some external potential ṽ(r), the energy obtained from the functional Always lager than the true ground state E[ρ̃] = Vne[ρ̃] + T [ρ̃] + Vee[ρ̃] ≥ E0 energyof the system Only when it is the same, it will be equal where E0 is the true ground state energy The condition and which are associated with some external potential ṽ(r) has been explicitly mentioned as a restriction for densities to be eligible for use in the variational procedure The trial densitye must be associated with some external potential, otherwise you fall outside of the action of the Hohenberg-Kohn theory (because there is the link between rho an v ) This restriction: v-representability problem for electron densities A density is defined to be v-representable if it is the density associated with an antisymmetric wave function and a Hamilton operator with some kind of external potential, which should not necessarily be a Coulomb potential PN 1 2 This Hamiltonian should thus be of the form of the form H = − i 2 ∇i + 5  PN PN 1 i v(ri) + i local potential that works on electron i As a consequence, the density of a noninteracting system can be expressed exactly as For HF, the density is equal to the sum of the square of the orbital present in the Slater determinant X ρ= |ψi|2 i Real system & use non-interacting reference system, that have the same density as the true system 13  This is the expression for the electron density of a single Slater determinant (no proof) Now, it will be presumed that this density expression spans all possible N -electron densities, interacting or not This means we are invoking a noninteracting reference system for which the density is exactly the exact ground state density of the interacting system For an N -electron system, this determinant has the form χi(x1) χj (x1)... χk (x1) χi(x2) χj (x2)... χk (x2) Ψ(x1, x2,... , xN ) = (N !)−1/2...... χi(xN ) χj (xN )... χk (xN ) Spin orbitals (spin coordinates) −1/2 where (N !) is a normalization factor. In this expression, the χ indicate spin orbitals, written as a product of a spatial function ψ and a spin function, α or β One can prove that the kinetic energy of a single Slater determinant, describing 14  the non-interacting system, can be written as Instead to direct approximate kin. T --> introduce non-interacting reference N system, exactly described by Slater X 1 Determinant, that have the same density Ts[ρ] = Ts[{ψm}] = hψm| − ∇2|ψmi as the true system. And then look over all these non-interacting reference system m 2 for the true ground-state energy. sum over all molecular orbitals Next, Vee[ρ] needs to be approximated as an explicit functional of the density In practice, Vee[ρ] is decomposed as Hartree energy functional and remaining Q Vee[ρ] = EH[ρ] + Q[ρ] Hartree functional = classical electrostatic repulsion Here, EH[ρ] is the Hartree energy functional: energy for the electron density rho(r) ρ(r)ρ(r0) Z Z 1 0 Classical self-repulsion EH[ρ] ≡ J[ρ] = drdr 2 |r − r0| Exactly known This represents the classical electrostatic repulsion energy for the charge distri- bution ρ(r); this quantity is calculated exactly Q[ρ], the non-classical part of the electron-electron repulsion remains to be ap- proximated Since Ts[ρ] is not equal to T [ρ], their difference is combined with Q[ρ] to define 15  the exchange-correlation energy EXC[ρ] Ts can be computed exactly, J can also be computed, only Exc is unknown FHK[ρ] = T [ρ] + Vee[ρ] = Ts[ρ] + J[ρ] + EXC[ρ] with Difference of true T with Ts Omvormen naar Exc = EXC[ρ] = (T [ρ] − Ts[ρ]) + Q[ρ] = (T [ρ] − Ts[ρ]) + (Vee[ρ] − J[ρ]) Exchange correlation functional As can be seen, EXC contains both a kinetic energy and a potential energy part The Kohn-Sham total energy functional is thus given as Before: E[p] = T[p] + Vee[p] + { p(r)v(r) dr Z Exc containst the corrections EKS[ρ] = Ts[ρ] + J[ρ] + EXC[ρ] + ρ(r)v(r)dr This is a brilliant decomposition: Ts[ρ] and J[ρ] are given by exact expressions and the unknown functional EXC[ρ] is a relatively small part of the energy This is an exact theory ! --> replaced two big quantities that are unknown by two big quantities that are known and one small quantity that is unknown (a correction) Next: minimization of the Kohn-Sham energy EKS with respect to the orbitals ψi 16 we've seen the philosophy, now we need to implement the Kohn-Sham energy 17 1. The Kohn-Sham Equations (Only need to know the final results!) 1.1. Derivation of the Canonical Kohn-Sham Equations Minimize EKS with respect to the orbitals {ψi} These orbitals are present in the Slater Determinant (exact wave function for a non-interacting system) δEKS =0 δψi subject to hψi|ψj i = δij. --> Minimization with constratin, the orbitals should remain orthonormal --> Undetermineted Lagrange multipliers We define the functional Ω[{ψi}] of the N orbitals X N X N Z  Ω[{ψi}] = EKS[{ψi}] − εij ψi∗(r)ψj (r)dr − δij i j The constraint (orthonormality) For EKS[{ψi}] to be a minimum, it is necessary that Lagrange multiplier δΩ[{ψi}] = 0 --> This condition must be fullfilled to obtain lowestenergy We first explicitly write EKS[{ψi}] as a function of the N orbitals {ψi} 18  (Lecture, have a look at it) N Z   N Z X 1 X EKS[{ψi}] = ψi∗(r) − ∇2 ψi(r)dr + v(r)ψi∗(r)ψi(r)dr i 2 i N N ψi∗(r)ψi(r)ψj∗(r0)ψj (r0) Z Z 1 XX 0 + drdr 2 i j |r − r0| + EXC[{ψi}] We now perform the variations for each of these terms Kinetic energy Ts[{ψi}] N Z   X 1 δTs[{ψi}] = δ ψi∗(r) − ∇2 ψi(r)dr i 2 N Z   Z    X 1 1 = δψi∗(r) − ∇2 ψi(r)dr + ψi∗(r) − ∇2 δψi(r)dr i 2 2 N Z   Z   ∗  X 1 1 = δψi∗(r) − ∇2 ψi(r)dr + δψi∗(r) − ∇2 ψi(r)dr i 2 2 N Z   X 1 = δψi∗(r) − ∇2 ψi(r)dr + c.c. i 2 19  Nucleus-electron attraction energy Vne[{ψi}] XN Z δVne[{ψi}] = δ v(r)ψi∗(r)ψi(r)dr i N Z X = v(r) [ψi∗(r)δψi(r) + ψi(r)δψi∗(r)] dr i N Z X = δψi∗(r)v(r)ψi(r)dr + c.c. i 20  Classical electron-electron repulsion energy Vee[{ψi}] N N Z Z 1 XX δψi∗(r)ψi(r)ψj∗(r0)ψj (r0) 0 δVee[{ψi}] = drdr 2 i j |r − r0| N X N Z Z 1 X ψi∗(r)δψi(r)ψj∗(r0)ψj (r0) 0 + drdr 2 i j |r − r0| N X N Z Z 1 X ψi∗(r)ψi(r)δψj∗(r0)ψj (r0) 0 + drdr 2 i j |r − r0| N N Z Z 1 XX ψi∗(r)ψi(r)ψj∗(r0)δψj (r0) 0 + drdr 2 i j |r − r0| N X N Z Z X δψi∗(r)ψi(r)ψj∗(r0)ψj (r0) 0 = 0 drdr + c.c. i j |r − r | N Z Z 0  X ρ(r ) = δψi∗(r) 0 dr0 ψi(r)dr + c.c. i |r − r | 21  Finally, we work out the variation for the exchange-correlation energy EXC[ρ] = EXC[{ψi}] Don't need to know the derivation N Z N Z X δEXC X δEXC ∗ δEXC[{ψi}] = δψi(r)dr + ∗ δψi (r)dr i δψi(r) i δψi (r) N Z    X δEXC δρ(r) = δψi(r)dr i δρ(r) δψi(r) N Z    X δEXC δρ(r) ∗ + ∗ δψ i (r)dr i δρ(r) δψi (r) N Z   X δEXC = δψi∗(r) ψi(r)dr + c.c. i δρ(r) The derivative occurring in this equation is the exchange-correlation potential δEXC vXC ≡ δρ(r) 22  Also N X Z N X Z N X Z εij δ ψi∗(r)ψj (r)dr = εij δψi∗(r)ψj (r)dr + εij ψi∗(r)δψj (r)dr ij ij ij N X Z N X Z = εij δψi∗(r)ψj (r)dr + εji ψj∗(r)δψi(r)dr ij ji N X Z N X Z ∗ = εij δψi∗(r)ψj (r)dr + ε∗ij δψi∗(r)ψj (r)dr ij ij N X Z = εij δψi∗(r)ψj (r)dr + c.c. ij As a result, the first variation Ω[{ψi}] becomes   XN Z   1  N X  δΩ = δψi∗(r) − ∇2 + v(r) + vJ(r) + vXC(r) ψi(r) − εij ψj (r) dr  2  i j + c.c. = 0 Since the change in δψi∗(r) is arbitrary, it must be that the quantity between {} 23  is zero for all i; therefore,   N 1 2 X − ∇ + v(r) + vJ(r) + vXC(r) ψi(r) = εij ψj (r) 2 j Since (εij ) is a Hermitian matrix, it can be diagonalized by a unitary transforma- tion, to yield the Kohn-Sham orbital equations in their canonical form   1 − ∇2 + veff (r) ψi(r) = εiψi(r) This can be done for all orbitals i 2 Kohn-Sham orbitals equation --> Used to optimize the Kohn-Sham orbitals --> Gives the energy of the system This expression you need to know effective potential only depending on the position r 24  You obtain orbitals, and the energy associated Kinetic energy operator   1 2 with these orbitals, the orbital energies − ∇ + veff (r) ψi(r) = εiψi(r) 2 Local effective potential, which is the function derivate of nucleu-electron with attraction energy, repulsion and exchange correlation Classic Coulumb repulsion potential veff (r) = v(r) + vJ(r) + vXC(r) exchange correlation potential due to the nuclei (external potential) As can be seen, there are three local potential terms in this expression, all func- tional derivatives of Kohn-Sham energy contributions 1. v(r), the functional derivative of the nuclear-electron attraction energy: Z  δVne[ρ] δ = ρ(r)v(r)dr = v(r) δρ(r) δρ(r) 2. vJ(r), the functional derivative of the classical electrostatic repulsion energy for the charge distribution ρ(r): 0 0  Z Z  Z δJ[ρ] δ 1 ρ(r)ρ(r ) 0 ρ(r ) 0 = drdr = dr = vJ(r) δρ(r) δρ(r) 2 |r − r0| |r − r0| 3. vXC(r), the functional derivative of the exchange-correlation energy: δEXC[ρ] = vXC(r) δρ(r) 25  EXC[ρ] can be further decomposed as EXC[ρ] = EX[ρ] + EC[ρ] Exchange contribution and correlation contribution so that It is done seperately vXC(r) = vX(r) + vC(r) 26  One can also determine the energy of a single Slater determinant and minimize this energy with respect variations in the orbitals that compose this determinant These HF equations can be compared to the KS equation. KS are exact, the HF equations are approximated. This results in the famous Hartree-Fock equations Because the assumption that the system is described by one Slater determinant. In KS, we also do that, but we introduced a correction for this We can now compare the Kohn-Sham equations (exact) to the Hartree-Fock equations (approximate); the Hartree-Fock equations have the following form:   v(r) and vj are also present in KS 1 2 − ∇ + v(r) + vJ(r) − K̂ ψi(r) = εiψi(r) equation. But instead of vxc, we have K operator, the non-local exchange 2 operator. This operator is only determined on orbital i. In this equation, K̂ is the non-local exchange operator 2/3 terms are identical, operator is the difference " N Z # X ψ ∗(r0)ψi(r0) It exchange i and b --> non-local operator, b 0 K̂ψi(r) = dr ψ b (r) because it is non-local because r and r' are |r − r0| exchanged. " Nb Z # Exchange operator. X ψ ∗(r0)P12ψb(r0) b 0 = dr ψi(r) |r − r0| In KS: local potential. b In Hartree-Fock, there is only a non-local exchange contribution In Kohn-Sham, there is a local exchange-correlation potential vXC(r) which in- cludes kinetic and electrostatic contributions 27  We now only need to find suitable approximations for vXC(r) KS is an exact method: exact form of the exchange correlation function --> exact solution of the schrodinger equation. Now: find suitable approximations for the exchange correlations 'Only', but it's highly complex to find approximations 28 1.2. An Alternative View Instead minimizing the KS energy expression with respect to the orbitals present in the Slater determinant. We could also minimize the KS energy with respect to the density of the system (like did before) We will take an alternative look at the minimization of EKS E(KS) [p] = Ts(p) + { p(r)v(r) dr + J[p] + Ex[p] Minimising the Kohn–Sham energy expression Z E = ρ(r)v(r)dr + Ts[ρ] + J[ρ] + EXC[ρ] with respect to the density (subject to fixed N ) gives the Euler equation Minimize the energy with respect δTs[ρ] to constraint µ = vs(r) + δρ(r) Compare the equation with the equation before: it looks like there is no electron- electron repulsion. But it is there (in vs). And T/p is changed by Ts/p. True system where is changed by Ts and last term is dropped. And v is replaced by vs(r). KS -> electrons are not interacting with each other, δJ[ρ] δEXC[ρ] vs = v(r) + + δρ(r) δρ(r) This equation,which yields the exact density of the system, is just the conventional DFT Euler equation δT [ρ] δVee[ρ] µ = v(r) + + δρ(r) δρ(r) for a system of non-interacting electrons (T = Ts; Vee = 0) moving in an external potential vs(r) (the Kohn-Sham potential). Other interpreation veff = vs 29  The interacting system (i.e. electrons moving in an external potential v(r) and repelling each other) is thus modelled by a non-interacting systems where the electrons move in an effective potential vs(r) without considering the electron- electron repulsing directly Both systems have exactly the same density The Figure below shows a plot of the external potential v(r) = −2 r (blue) and the effective potential vs(r) (pink) for the Helium atom; also given is the exact electron density, plotted as 4πr2ρ(r) (red) He-atom: it shows. The blue potential (external potential) -2/r. Effective potential runs parallel, but is not the same Electron density for the different angles, radial distrubution Vs(r) V(r) Reprinted from K. Burke, J. Chem. Phys. 136, 150901 (2012). 30  Two non-interacting electrons, which doubly occupy the 1s of vs(r), thus produce the exact density of the Helium atom, i.e. r (See lecture at 23:30) ρ(r) ψ1s(r) = 2 As we have seen previously, the Hamiltonian for a system of non-interacting elec- trons moving in an external potential vs(r) has the following form N N Exact wave function of this is a Slater determinant X 1 X Ĥ = − ∇2i + vs(ri) i 2 i We know that the exact wavefunction for this Hamiltonian is just a single deter- minant constructed from orbitals that are the solutions to   1 2 These orbitals in this Slater determinant are − ∇ + vs(r) ψi(r) = iψi(r) solutions of the KS equations 2 The density and kinetic energy of this non-interacting system are just N X Density ρ(r) = ψi2(r) i 31  and N 1 X Kinetic energy hψi| − ∇2|ψii Ts[ρ] = i 2 R The total energy can then be evaluated as E = ρ(r)v(r)dr + Ts[ρ] + J[ρ] + EXC[ρ]. Total KS energy If you would have the exact exchange correlation functional, then you would have the exact energy and density An exact theory! The exact EXC yields the exact E and ρ(r) Nobel Prize, 1998 32  The largest part of T [ρ] is now described exactly; we now just need to approximate the smaller EXC[ρ]. In KS, by incoving a non-interacting reference system. The largest part is exact, but EXC we need to approximate. 33  KS philosophy we've seen: non-interacting reference system -> evaluate the largest part fo 1.3. Practical solution of the Kohn-Sham Equations the kinetic energy, also classical e-e repulsion. Only the exchange correlation is unknown Essentially all practical calculations involve the expansion of KS orbitals in an atomic basis set: Just like solve HF, we use a basis set. All practical Basis functions are centered on the calculations that solve the KS equations, they K different atoms in the molecule. involve the expansion of KS orbitals in an atomic X basis set. --> Linear combinations of atomic centered basis functions with expansion ψi(r) = Cαiφi(r) i = 1, 2,... , K Optimal KS orbitals = obtain the optimal coefficients. coefficient C. α The atom-centered basis functions used are normally the same as the ones used in wave mechanics for expanding the Hartree-Fock orbitals The problem of calculating the Kohn-Sham orbitals reduces to the problem of calculating the set of expansion coefficients Cαi Not looking at orbitals, but to the coefficients We can obtain a matrix equation for the Cαi by substituting the linear expansion for the KS orbitals into the Kohn-Sham equation Replace the orbitals by C in the KS equation.  X 1 2 X − ∇ + vs(r) Cνiφν (r) = εi Cνiφν (r) 2 ν ν By multiplying by φ∗µ(r) on the left and integrating, we turn the integro-differential equation into a matrix equation Multiple the equation on the left with mnu* & integrate;, the obtained matrix equation will be solved Z   Z X 1 X Cνi φ∗µ(r) − ∇2 + v(r) + vJ (r) + vXC(r) φν (r)dr = εi φ∗µ(r)φν (r)dr ν 2 ν 34  or More shorthand: instead vs write the kohn-sham operator X Z XZ Cνi φ∗µ(r)ĥKSφν (r)dr = εi φ∗µ(r)φν (r)dr ν ν We now define the following matrices (notes 32:40) – The overlap matrix Sµν You can evaluate the integrals in the overlap matrix Z analytically Sµν = φ∗µ(r)φν (r)dr – The matrix Tµν containing the kinetic energy integrals Z   This can also be evalualated analytically 1 Tµν = φ∗µ(r) − ∇2r φν (r)dr 2 nucl – The matrix Vµν containing the nuclear attraction integrals This can also be evalualated analytically Z Z " # X ZA nucl ∗ ∗ Vµν = φµ(r)v(r)φν (r)dr = φµ(r) − φν (r)dr |r − RA| A To introduce the matrix related to the classical electron-electron repulsion term, 35  we first turn to the expression of the electron density with this basis set X XXX ∗ ∗ ρ(r) = ψi (r)ψi(r) = Cµi Cνiφ∗µ(r)φν (r) i i µ ν (see 39:20) We now define the density matrix Pµν as X ∗ Pµν = Cµi Cνi i so that the electron density becomes XX ρ(r) = Pµν φ∗µ(r)φν (r) µ ν We can now define the matrix Jµν as the matrix containing the contribution from 36  the classical electron-electron repulsion Pink is equal to vJ -> Density appearing 0 Z Z  ρ(r ) Jµν = φ∗µ(r) 0 dr 0 φν (r)dr |r − r |  ∗ 0 0 Z  XX ∗ φλ(r )φσ (r ) 0 = Pλσ φµ(r) 0| dr φν (r)dr σ |r − r λ Z ∗ XX φµ(r)φ∗λ(r0)φν (r)φσ (r0) 0 electron-repulsion integral or = Pλσ 0| drdr two-electron integral σ |r − r λ X X = Pλσ hµλ|νσi This can also be evalualated analytically λ σ where we have introduced the two-electron integrals hµλ|νσi written in the physi- cists’notation XC Finally, we have the contribution from exchange-correlation Vµν vXC put in Z This integral can not be evaluated analytically, but you need to obtain it by numerical integration XC Vµν = φ∗µ(r)vXC(r)φν (r)dr Contrary to the integrals occurring in the other matrix elements, these integrals can NOT be evaluated analytically and need to be obtained by numerical integra- tion In particular, they cannot be evaluated analytically due to fractional powers of the 37  density (see later) For their evaluation, we must use numerical quadrature Consider the general molecular integral Decompose the integral that is over allspace into components localised on the nuclei --> Z e.g. F = ρ5/3 define atomic weights, which are near unity F (r)dr nucleus, and zero when you are far away from that nucleus First, the integrand is decomposed into components localised on the nuclei To do this, we define atomic weights wA(r), which are near unity near nucleus A, P but zero near all other nuclei, and which satisfy A wA(r) = 1 --> Weight the space by doing this Sum of all weights = 1 Inserting this into the first equation gives Insert the sum, move the summation in front of the integral --> weighted integral that Z Z X ! you can integrated around nucleus A XZ XZ F (r)dr = wA(r) F (r)dr = wA(r)F (r)dr = FA(r)dr A A A where FA(r) is a function localised around nucleus A. Each localised integral is then evaluated using standard r, θ, φ numerical integration, centered on that nucleus Z X FA(r)dr ≈ FA(ri)wi i 38  or Z∞ Zπ Z2π IA = FA(r, θ, φ)r2 sin θdrdθdφ Volume elements in polar coords 0 0 0 Function around nucleus A P X Q X ≈ Wprad Wqang FA(rp, θq, φq) p q In typical applications, one places 1000-10000 grid points (i.e. p × q) per atom You must have high density grids These points will be the most dense where vXC(r) varies the most in regions where the exchange correlation function varies the most When all the matrix elements have been evaluated, we can solve the KS matrix C = coefficient matrix equation --> To be solved for the coefficients, but these coefficients are also present in h. This means that you will have these equations self- consistency --> Guess of the coefficients, you construct the KS matrix, you solve the equation which give a new set of coefficients hKSC = SCe and solve again in the KS matrix, a new versionof KS matrix... In this equation, C is a K × K square matrix of the expansion coefficients Cµi   Repeat this until you obtain the coefficients that you obtain from the equation, when you reused them in C11 C12... C1K the new version in the KS matrix, you must again   obtain the same coefficients --> self-consistent field  C21 C22... C2K  C= ......     CK1 CK2... CKK 39  e is a diagonal matrix of the orbital energies εi   ε1    ε2 0  = ...   0   Open-shell: electrons not coupled two by two in the εK orbitals --> We will not look at this here The SCF equations can now be solved just like in Hartree-Fock; for open-shell systems, one must use the unrestricted version The only difference is that the exchange contribution contribution to the Fock matrix is replaced by an exchange-correlation contribution Remark : These Hartree-Fock exchange integral matrix elements are given by Z ∗ XX φµ(r)φ∗σ (r0)φλ(r)φν (r0) 0 Kµν = Pλσ 0| drdr σ |r − r λ X X = Pλσ hµσ|λνi Computation scale is determined by the step that has the most computational cost: the λ σ evaluation of the two-electron integrals. These integrals depend on the product of 4 basis functions --> evaluating scales to the Just as Hartree-Fock, DFT thus formally scales as M 4 number of basis set ^4 40 References Section on v representability and Levy’s constrained search adapted from – W. Koch and M. C. Holthausen, A Chemist’s Guide to Density Functional Theory, Second Edition, Wiley-VCH, Weinhein, 2002. – D. J. Tozer (Durham University, UK): Lecture notes for the European Summer School of Quantum Chemistry (ESQC), used with permission. Section 3.3: adapted from A. Szabo and N. S. Ostlund, Modern Quantum Chem- istry: Introduction to Advanced Electronic Structure Theory, Dover Publications Inc., New York, 1996 Section 3.4.: A. D. Becke, Perspective: Fifty years of density-functional theory in chemical physics, J. Chem. Phys. 140, 18A301 (2014). Section 1.2. and 1.3.: adapted from D. J. Tozer (Durham University, UK): Lecture notes for the European Summer School of Quantum Chemistry (ESQC), used with permission. 41