Notes on Density Functional Theory
|
|
- Emmeline Skinner
- 5 years ago
- Views:
Transcription
1 Notes on Density Functional Theory Rocco Martinazzo Contents 1 Introduction 1 Density Functional Theory 7 3 The Kohn-Sham approach 11 1 Introduction We consider here a system with N electrons subjected to some external potential. The Hamiltonian of this system takes the following general form: where is the kinetic energy operator, H = T + V ee + V ext T = N i=1 p i m e V ee = 1 4πɛ 0 e r ij i,j i (r ij = r i r j ) is the potential term arising from the electron-electron (repulsive) interaction and V ext = N v(r i ) i=1 is the external eld acting (in the same way) on each electron of the system. In the molecular Hamiltonian the external potential takes the form v(r) = 1 4πɛ 0 1 M α=1 Z α e R α r
2 where the sum runs over nuclear labels and R α specify the position of the nuclei. The electronic system Hamiltonian is completely specied once N and v are given. We may emphasize this fact by writing H H N v Electrons, like any other sets of identical particles, are indistinguishable from each other and this implies that for any state Ψ Ψ(x 1, x,..., x N ) = Ψ(x, x 1,..., x N ) =... holds. Here, x i = r i σ i is a 4-dimensional vector specifying the position and the spin of the i-th electron, i.e. the normalization condition Ψ(x 1, x,..., x N ) d N Γ = 1 means d N Γ = d 3 r 1 d 3 r...d 3 r N σ 1,σ,...,σ N (σ i takes only two values specifying, e.g., the projection of the electron spin along a given axis). Since electrons are fermions, Ψ(x 1, x..x i..x j..x N ) = Ψ(x 1, x..x j..x i..x N ) The wave function Ψ describes a particular state of the N-electron system. In coordinate representation it is a very complicated, complex-valued function depending on 4N variables which allows us to determine any system property (for the given state). Indeed, for any observable A we have: A = Ψ A Ψ = Ψ (x 1...x N )AΨ(x 1...x N )d N Γ where A is the quantum-mechanical operator corresponding to the observable A. Here, since electrons are equal to each other in any respect, for A to be experimentally relevant it must be a symmetric function of the electron coordinates. Usually we are interested in one-, two- or few electrons properties. Oneelectron operators are of the form A = N i=1 where a i acts on the i-th electron only. Two-electron operators are of the form A = N i,j i Thus, for example, in our general Hamiltonian T and V ext are one-electron operators and V ee is a two electron operator. m-electron operators are dened in a analogous way. We now show that in order to compute 1-, - or few electron properties Ψ is much more than we need. Let's rst consider a one-electron operator and its expectation value a i a ij
3 N A = i=1 Ψ (x 1 x...x i...x N )a i Ψ(x 1 x...x i...x N )d N Γ = N Ψ (x 1 x...x i...x N )a 1 Ψ(x 1 x...x i...x N )d N Γ where the second line follows from the symmetry properties of the wave function. Now Ψ (x 1 x...x i...x N )a 1 Ψ(x 1 x...x i...x N )d N Γ = a1 [Ψ(x 1 x...x N )Ψ (x 1x...x N )] x 1 =x1 d N Γ where x 1 = x 1 has to be replaced after a acted on the quantity between brackets. Then N a 1 [Ψ(x 1 x...x N )Ψ (x 1x...x N )] x 1 =x 1 dx 1 d N 1 Γ = a 1 γ(x 1 x 1) x 1 =x 1 dx 1 where γ(x 1 x 1) = N d N 1 Γ Ψ(x 1 x...x N )Ψ (x 1x...x N ) γ(x x) dened in this way is known as rst-order reduced density matrix. Note that n(r) = γ(rσ rσ) = N d N 1 Γ Ψ(xx...x N ) σ is the electron density, correctly normalized to the number of particles: d 3 rn(r) = N γ is an extended density wich allows us to compute any one-electron property. With the same token if A is a two-electron operator we have A = N(N 1) Ψ a ij Ψ = i,j i a 1 γ () (x 1 x x 1x ) dx 1 dx x 1 =x 1;x =x where γ () (x 1 x x 1x ) = N(N 1) d N Γ Ψ(x 1, x,..., x N )Ψ (x 1, x,..., x N ) 3
4 is the second-order reduced density matrix. In general, for an m-electron operator A (m), A (m) = a 1..m γ (m) (x 1 x..x m x 1x..x m) x i =x i dx 1..dx m where γ (m) (x 1..., x m x 1..., x m) = ( N m ) d N m Γ Ψ(x 1...x m, x m+1..., x N )Ψ (x 1..., x m, x m+1..., x N ) is the m th -order reduced density matrix. Note, that once an m-th order reduced density matrix is known, any density matrix of order n < m can easily be obtained by properly integrating the known density matrix, m N m+1 γ (m 1) (x 1..x m 1 x 1..x m 1) = dxm γ (m) (x 1..x m 1 x m x 1..x m 1x m ) Since we are only interested in 1-, -, or few electron properties, the above arguments suggest that we might replace the complicated function Ψ with some sympler function. In particular, our general hamiltonian contains only 1- and - electron operators: if we were able to compute γ () than we could solve any electronic problem, with as many electron as we like! This is the essence of the density matrix theory of the electronic problem. Unfortunately, such a theory has enormous diculties and cannot be used at present for all but a few model problems. The reason is that is not yet clear how to select a general function ρ(x 1 x x 1x ) in such a way that it represents a second order density matrix of some electron system. However, a powerful theory based on the simpler electronic density n(r) turns out to be possible. As we shall see in the following it has the only problem that, in practice, it cannot be considered as truly ab initio. This Density Functional Theory (DFT) is possible because of a number of facts. 1. We are usually interested in stationary states, that is we replace the Time- Dependent Schrodinger Equation with the equation for stationary states HΨ = EΨ This is because we are only interested in situations in which all electrons are bound to a given charged (stationary) core. Note that for such states d Ψ A Ψ dt = 0 for any (stationary, i.e. not explicitly time dependent) observable A.. We are usually interested in the ground-state Ψ 0, i.e. the state of minimum energy. This is so because in molecules electronic energy levels are wellseparated in energy, and at normal temperature only the ground-state is populated. Actually, this condition cannot be satised in metallic systems where excited electronic states with vanishing small excitation energy can be accessed even at very low temperatures (electron-hole excitations). 4
5 Stationarity allows us to use the variational principle ( ) Ψ H Ψ δ = 0 Ψ Ψ or simply δψ H Ψ = 0 if the additional norm-conserving condition δψ Ψ = 0 is enforced. In addition, the ground state has the minimum energy property E(Φ) = Φ H Φ E 0 = Ψ 0 H Ψ 0 which holds for any trial, normalized wavefunction Φ. Note: Functionals and functional derivatives. The map H R Φ E(Φ) = Φ H Φ Φ Φ is called energy functional and the results E(Φ) E 0 is called wave function variational principle. In general, given a space of functions E a map E R( or also C) Φ F [Φ] is called a functional. Thus, for example, E H [n] = d 3 r 1 d 3 n(r 1 )n(r ) r r 1 r T T F [n] = A d 3 rn 5 3 (r) are functionals of the electronic density. The functional derivative δf dened by the equation δf = d 3 r δf (r)(r) is δf where δf = F [n + ] F [n]. In other words, is a function of r (and, in general, also a functional of n for any r) which, once multiplied by and integrated over r, gives the variation of F. More formally, let us consider the following ordinary function of λ F [n + λ] 5
6 where n and are two functions of the domain 1 E. The variation of F is dened with the help of the standard derivative with respect to λ, according to δf [n, ] = d F [n + λ] dλ λ=0 This variation, or also dierential is, in general, a functional of both n and. If it is linear in, that is δf [n, ] = [n] where [n] is a linear operator from E to R(C), then we usually write [n]φ = d 3 r δf (r)φ(r) and δf, the previously dened functional derivative, is the integral representation of the operator [n]. Thus, for example, T T F [n] = A d 3 rn 5 3 (r) δt T F [n, ] = A d 3 r 5 3 n 3 (r)(r) δt i.e.: T F (r) = A 5 3 n 3 (r). With the second (formally more correct) denition δt T F [n, ] = d dλ A dr(n + λ) 5 3 = λ=0 = A d 3 r 5 3 (n + λ) 3 (r) = A d 3 r 5 λ=0 3 n 3 (r)(r) For E H we have δe H [n, ] = d 3 r 1 d 3 r (r 1 )n(r ) + n(r 1 )(r ) r 1 r = = + ( ) d 3 r 1 d 3 n(r ) r (r 1 ) r 1 r ( ) d 3 r d 3 n(r 1 ) r 1 (r ) = r 1 r = d 3 r δe H (r)(r) with δe H (r) = d 3 r n(r ) r r. 1 If E is not linear one has to consider the curves n(λ) such that n(0) = n and n (0) =. 6
7 Since we are only interested in the ground-state Ψ of our N-electron system we may write v, N Ψ N v X = X [v, N] where the Ψ N v is a functional of v (and a function of N) and X is any groundstate property of the system. The above relationship means that once N is xed any ground-state property is a functional of v, v X = X [v] The above relation holds thanks to the chain of relationships v Ψ [v] Ψ X [Ψ] The only assumption we need is that the ground-state is non degenerate, in such a way that the potential uniquely determines the ground-state function (apart from an irrelevant phase factor). Density Functional Theory The beauty and power of the so-called Density Functional Theory relies on two very simple theorems, known as Hohenberg-Kohn theorems, whose results can be summarized as follow. In studying ground-state electronic properties one can replace the ground-state electronic wave function Ψ(x 1,..., x N ) with the simple electron density n(r). In other words one can use the electron density as fundamental variable: the ground-state wave function (and therefore any groundstate property) is a functional of the electron density, Ψ = Ψ [n]. These properties follow from two very simple theorems: Theorem I The density n uniquely determines v, that is v = v [n] or, in other words, for each electron density n there exists one and only one potential v (mathematically, n! v) such that n is its ground-state density. Proof. We use the reductio ad absurdum procedure rst used by Hohenberg and Kohn. Suppose there exist two potentials v 1, v whose ground-state density is n, i.e v 1 H v1 Ψ 1 n Let's consider where H v1 H v V ext 1 = v H v Ψ n = T + V ee + V ext 1 = U + V ext 1 = T + V ee + V ext = U + V ext N i=1 v (i) 1 V ext = N i=1 v (i) 7
8 Here U is the internal energy operator, U = T + V ee, a universal operator (in the sense that it is the same for any N-electron system). We have (E(Φ) E 0 ) Ψ H v1 Ψ = Ψ U Ψ + drv 1 (r)n(r) > > Ψ 1 H v1 Ψ 1 Ψ 1 U Ψ 1 + drv 1 (r)n(r) i.e. Ψ U Ψ > Ψ 1 U Ψ 1 With the same token, from Ψ 1 H v Ψ 1 > Ψ H v Ψ it follows Ψ 1 U Ψ 1 > Ψ U Ψ Note that we have implicitly assumed that Ψ is uniquely determined by v, which is only possible if the ground-state is not degenerate (as it almost always happens). Furthermore, il v 1 and v give accidentaly the same Ψ (e.g. because v 1 = v + c) in any case Ψ is still uniquely given by n. Note. To be precise, n is a function with the following properties, n(r) 0 d 3 rn(r) = N n Ψ v v Here the last condition means that n must be an electronic density derivable from a ground state wavefunction Ψ v for some external potential v. This kind of densities are called v-representable. The above theorem makes use of (i) the non-degenerate nature of the groundstate, and (ii) the possibility to identify the class of v-representable densities. We will relax these two assumptions with a second, more general formulation. Before doing this, however, let us state the second Hohenberg-Kohn theorem, which proves to be extremely useful for the application of the theory. Theorem II For any given external potential v a density variational principle can be formulated, i.e. there exist a functional, called energy functional E v [n] such that E [n] E v 0 where E 0 is the ground-state energy. The equality sign holds if and only if n is the ground-state density. Proof. Let us consider v xed. Then for any Φ giving the density n we may write Φ H v Φ = Φ U Φ + d 3 rv(r)n(r) 8
9 Thanks to theorem I we can now focus on Φ = Ψ [n] where Ψ is the ground-state wavefunction for some unspecied potential v uniquely determined by n. Then the quantity E v [n] = Ψ[n] H v Ψ[n] is a functional of the electron density and we know that E v [n] E v 0 the equality being satised by the ground-state density n of our problem only (since in that case v = v is our xed external potential). Please note that we have used again the v-representability property. Note also that E v [n] = U [n] + d 3 rv(r)n(r) and this functional can be minimized subjected to the constraint N = d 3 rn(r) This means ( ( δ E v [n] µ )) d 3 rn (r) N = 0 that is δu [n] + d 3 rv(r)(r) µ d 3 r(r) = 0 (where µ is the Langrange multiplier related to the above normalization contraint) or also ( δu = δu (r)(r)d3 r ) δu (r) + v(r) = µ In principle, knowing U we should solve this equation for n(n = n µ ) and µ has to be adjusted in order to satisfy the normalization constraint, i.e. n µ d 3 r = N. In practice, U is an unknown functional and this means that it must be approximated in some way. This explains why the theory is not ab-initio, though it comes from rst-principles. Note: Lagrange multipliers and constrained minimization. Suppose we have a function f = f(x 1, x ) to be minimized. In the absence of constraints we use df = f x 1 dx 1 + f x dx = 0 f x 1 = f x = 0 9
10 since dx 1, dx are arbitrary (i.e. one can take for example dx = 0). If the function f has to be minimized with the constraint g(x 1, x ) = 0 we can no longer consider dx 1 and dx arbitrary. Rather, we have dg = g x 1 dx 1 + g x dx = 0 This means that the vector dx = (dx 1 dx ) t must be orthogonal to the vector g = ( g g x 1 x ) t, i.e., g dx = 0. The stationary condition then reads df = f dx = 0 dx g Here g is the orthogonal complement of g, that is the (linear) space of vectors orthogonal to g, and f = ( f f x 1 x ) t is the gradient vector. It follows f dx = g = V {g} where V {g} is the space spanned by g, V = {u such that u = λg, with λ R(C)}, or also f λg = 0 This equation has to be solved in conjuction with the constraint g(x) = 0, which determines the value of the parameter λ, the so-called Lagrange multiplier. Note that the above equation is equivalent to the constraintfree stationarity condition d(f λg) = 0 of the Lagrange function L λ = f λg. Analogous relations hold for functions of more than two variables, in the presence of a number of constraints. Before considering the problem of how determining U (at least approximately) let us consider an alternative derivation of Theorems I and II. This is due to Levy and has the merit of removing some restrictions. From the wave function variational principle we have E0 v = min Ψ H v Ψ Ψ where minimization is over all possible normalized states. We can separate the minimization in two steps by grouping wavefunctions which give the same density min Ψ min n min Ψ n where Ψ n means a Ψ which gives the density n. Now, let us consider min Ψ H v Ψ = min Ψ U Ψ + d 3 rn(r)v(r) Ψ n Ψ n This is a well-dened functional of n, 10
11 E v [n] = min Ψ U Ψ + Ψ n d 3 rn(r)v(r) We also have E0 v = min E v [n] n which means that E v [n] is minimum when n is the ground-state density. It follows the following stationary condition δu (r) + v(r) = µ where µ is the Lagrange multiplier arising from the constraint dxn(x) = N. This equation tells us that v is uniquely determined by n, or by one of the ground-state densities if the ground-state is degenerate. We therefore have removed the constraint of a non degenerate ground-state. We also note that minimization of E v is performed over all n which come from a wavefunction: it is no longer necessary that Ψ is a ground-state wavefunction for some potential. Densities satisfying n(r) 0 d 3 rn(r) = n n Ψ are called N representable, a condition weaker than the v representability. N-representability is satised by any reasonable density. Specicaly, it can be shown that n is N-representable if and only if n(r) 0 n(r)d 3 n 1 r = n d 3 r < 3 The Kohn-Sham approach The energy density functional, E v [n] = min Ψ n Ψ U Ψ + d 3 rn(r)v(r) = = U [n] + d 3 rn(r)v(r) is quite hard to approximate because of the presence of the term U [n] = min Ψ n Ψ T + V ee Ψ which includes both kinetic and electronic repulsion terms. Let us then consider a dierent form of this term. See Parr&Yang, Density Functional Theory of Atoms and Molecules, p
12 Consider an alternative world, which we call the Kohn-Sham world, where electrons do not interact with each other. In this world U KS [n] T KS [n] = min Ψ T Ψ Ψ n where the minimum has to be searched among those independent N-electron wavefunctions that gives the density n, i.e. Slater determinants Ψ KS = 1 φ KS 1 φ KS 1...φ KS N N n(r) = i,σ φ KS i,σ (r) For any density n there exist a potential v KS (r) in which n is the ground-state density δt KS (r) + v KS(r) = µ and such potential determines also the orbitals ( ) p + v KS (r) φ KS i,σ = ɛ i φ KS i,σ m e If the above stationary condition in the Kohn-Sham world has to be equivalent to δu (r) + v(r) = µ (we absorb any possible deerence between µ and µ in v(r)) we have to put v KS (r) = δu (r) δt KS (r) + v(r) In this expression, U comes as the sum of a kinetic functional, the trivial part of the electron-electron interaction functional (the Hartree functional) and an exchange-correlation functional E xc U[n] = T [n] + v H [n] + E ex[n] and thus, absorbing the (reasonably small) dierence T [n] T KS [n] into the working exchange and correlation functional we arrive at E xc [n] := U[n] T KS [n] v H [n] v KS (r) = v(r) + δv H (r) + δe xc (r) Therefore, one can solve the Kohn-Sham equation for the Kohn-Sham orbitals {φ KS i,σ } { } p m e + v(r) + δv H (r) + δe xc (r) 1 φ KS i,σ (r) = ɛ i φ KS i,σ (r)
13 and the ground-state density n(r) = i,σ φ KS i (r) until self-consistency is reached, i.e. until the Kohn-Sham density n above matches the density n used to build up the Kohn-Sham Hamiltonian Once done, the total, ground-state energy is given by E v = T KS [ n] + v H [ n] + E xc [ n] + d 3 r n(r)v(r) Θ(µ ɛ i ) ψ i T ψ i + v H [n] + E xc [n] + d 3 rn(r)v(r) i where n is the self-consistent solution of the Kohn-Sham equation and { 1 x 0 Θ(x) = 0 x < 0 This is the popular Kohn-Sham method: the only problem left is now to approximate E xc [n]. Note that from a computational point of view the method is almost equivalent to the Hartree-Fock method. This, however, does not means that the ground-state wavefunction is a Slater determinant! The latter is the groundstate wave function of a ctitious independent electron problem with potential v KS. Orbitals appear in the theory only as a tool to represent the ground-state density, and to write down an exact expression for the kinetic energy functional. Example We show here how to devise approximate functionals, focusing on the exchange part of the exchange-correlation functional. If we consider a uniform electron gas in the Hartree-Fock approximation the exchange energy (per unit volume) can be easily computed for any value of the density n, ɛ HF ex could be written as = ɛ HF ex (n), and thus an approximate exchange functional E ex = d 3 rɛ HF ex (n(r)) This is the Local Density Approximation to the exchange functional, also known as Slater exchange (J.C. Slater, Phys. Rev. B, 81 (1951) 385). The calculation is straightforward if we remember that in Hartree-Fock theory the exchange operator takes the form J(x x ) = γ(x x ) r r where the rst-order density matrix reads as γ(x x ) = v φ v (x)φ v(x ) For a uniform electron gas (writing x = (r, σ)) φ v (x) φ k,σ (x) = 1 V e ikr δ σσ 13
14 where the set of k vectors span the Fermi sphere N = 1 = d 3 k = 8 k k k k F 3 k πk F k = (π) 3 /V being the volume of k-space occupied by each k-vector. That is 3 3π n = k 3 F is the relation linking the Fermi momentum to the electron gas density n. It follows γ αα (r r ) = φ α k(r)φ α k(r ) = 1 d 3 ke ik(r r ) kv k k k F i.e., after a straightforward integration, γ αα (r r ) f(ξ F ) = 3n sin(ξ F ) ξ F cos(ξ F ) ξ 3 F where ξ F = k F r with r = r r. Notice that f n/ as r 0, as it should be since f in such limit represents the density of the α electrons (similarly for the β species). Thus, J αα (r r ) = J ββ (r r ) = f(k F r) r Now, the exchange energy is given by E ex = 1 ν J ν k J αα k ν k where for each k k J αα k = 1 V 1 V d 3 r 1 d 3 r J αα (r 1 r )e ik(r r1) d 3 r 1 d 3 r e ik(r r1) f(k F r r 1 ) r r 1 The integral can be simplied by introducing the relative and the center of mass coordinates (a transformation with unit Jacobian), r = r r 1 R = r 1 + r 3 With the same token one can obtain the energy (per unit volume) of the free-electron gas, i.e. the quantity which denes the Thomas-Fermi functional. Thus E = X k k m e = k m e Zk kf d 3 kk V 10π m e k 5 F ɛ T F [n] = E V = 3 3 9π 4/3 10m e n 5/3 14
15 namely k J αα k = Summing over k we obtain E ex = k J αα k = 1 k k d 3 re ikr f(k F r) r d 3 r f(k F r) r d 3 ke ikr = V k k F d 3 r f(k F r) r and thus, introducing ξ = k F r, E ex = V 9n kf 4 d 3 [sin(ξ) ξcos(ξ)] ξ ξ 7 i.e. ɛ = E ex V = C k 4 F = Cn 4/3 where C is a (positive) numerical coecient 4. 4 Note that the integral above is well dened, since for ξ 0 we have sinξ ξcosξ ξ 3 /3. 15
Introduction to density-functional theory. Emmanuel Fromager
Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Institut
More information1 Density functional theory (DFT)
1 Density functional theory (DFT) 1.1 Introduction Density functional theory is an alternative to ab initio methods for solving the nonrelativistic, time-independent Schrödinger equation H Φ = E Φ. The
More informationIntroduction to Density Functional Theory
Introduction to Density Functional Theory S. Sharma Institut für Physik Karl-Franzens-Universität Graz, Austria 19th October 2005 Synopsis Motivation 1 Motivation : where can one use DFT 2 : 1 Elementary
More informationDept of Mechanical Engineering MIT Nanoengineering group
1 Dept of Mechanical Engineering MIT Nanoengineering group » To calculate all the properties of a molecule or crystalline system knowing its atomic information: Atomic species Their coordinates The Symmetry
More informationSolid State Theory: Band Structure Methods
Solid State Theory: Band Structure Methods Lilia Boeri Wed., 11:15-12:45 HS P3 (PH02112) http://itp.tugraz.at/lv/boeri/ele/ Plan of the Lecture: DFT1+2: Hohenberg-Kohn Theorem and Kohn and Sham equations.
More informationThe potential of Potential Functional Theory
The potential of Potential Functional Theory IPAM DFT School 2016 Attila Cangi August, 22 2016 Max Planck Institute of Microstructure Physics, Germany P. Elliott, AC, S. Pittalis, E.K.U. Gross, K. Burke,
More informationSolutions to selected problems from Giuliani, Vignale : Quantum Theory of the Electron Liquid
Solutions to selected problems from Giuliani, Vignale : Quantum Theory of the Electron Liquid These problems have been solved as a part of the Independent study class with prof. Neepa Maitra. Compiled
More informationDensity matrix functional theory vis-á-vis density functional theory
Density matrix functional theory vis-á-vis density functional theory 16.4.007 Ryan Requist Oleg Pankratov 1 Introduction Recently, there has been renewed interest in density matrix functional theory (DMFT)
More informationThermal Density Functional Theory in Context
Thermal Density Functional Theory in Context Aurora Pribram-Jones, Stefano Pittalis, E.K.U. Gross, and Kieron Burke Department of Chemistry, University of California, Irvine, CA 92697, USA CNR Istituto
More informationDensity Functional Theory. Martin Lüders Daresbury Laboratory
Density Functional Theory Martin Lüders Daresbury Laboratory Ab initio Calculations Hamiltonian: (without external fields, non-relativistic) impossible to solve exactly!! Electrons Nuclei Electron-Nuclei
More informationQuantum Mechanical Simulations
Quantum Mechanical Simulations Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332, U.S.A. yan.wang@me.gatech.edu Topics Quantum Monte Carlo Hartree-Fock
More informationLecture 8: Introduction to Density Functional Theory
Lecture 8: Introduction to Density Functional Theory Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science
More informationDensity Functional Theory for Electrons in Materials
Density Functional Theory for Electrons in Materials Richard M. Martin Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign 1 Density Functional Theory for
More informationDensity Functional Theory (DFT)
Density Functional Theory (DFT) An Introduction by A.I. Al-Sharif Irbid, Aug, 2 nd, 2009 Density Functional Theory Revolutionized our approach to the electronic structure of atoms, molecules and solid
More information3: Density Functional Theory
The Nuts and Bolts of First-Principles Simulation 3: Density Functional Theory CASTEP Developers Group with support from the ESF ψ k Network Density functional theory Mike Gillan, University College London
More informationCHEM6085: Density Functional Theory
Lecture 5 CHEM6085: Density Functional Theory Orbital-free (or pure ) DFT C.-K. Skylaris 1 Consists of three terms The electronic Hamiltonian operator Electronic kinetic energy operator Electron-Electron
More informationMO Calculation for a Diatomic Molecule. /4 0 ) i=1 j>i (1/r ij )
MO Calculation for a Diatomic Molecule Introduction The properties of any molecular system can in principle be found by looking at the solutions to the corresponding time independent Schrodinger equation
More informationv(r i r j ) = h(r i )+ 1 N
Chapter 1 Hartree-Fock Theory 1.1 Formalism For N electrons in an external potential V ext (r), the many-electron Hamiltonian can be written as follows: N H = [ p i i=1 m +V ext(r i )]+ 1 N N v(r i r j
More informationADIABATIC CONNECTION AND UNIFORM DENSITY SCALING: CORNERSTONES OF A DENSITY FUNCTIONAL THEORY
ADIABATIC CONNECTION AND UNIFORM DENSITY SCALING: CORNERSTONES OF A DENSITY FUNCTIONAL THEORY BY RUDOLPH J. MAGYAR A dissertation submitted to the Graduate School Piscataway Rutgers, The State University
More informationElectron Correlation
Electron Correlation Levels of QM Theory HΨ=EΨ Born-Oppenheimer approximation Nuclear equation: H n Ψ n =E n Ψ n Electronic equation: H e Ψ e =E e Ψ e Single determinant SCF Semi-empirical methods Correlation
More informationAdvanced Solid State Theory SS Roser Valentí and Harald Jeschke Institut für Theoretische Physik, Goethe-Universität Frankfurt
Advanced Solid State Theory SS 2010 Roser Valentí and Harald Jeschke Institut für Theoretische Physik, Goethe-Universität Frankfurt i 0. Literatur R. M. Martin, Electronic Structure: Basic Theory and
More informationTeoría del Funcional de la Densidad (Density Functional Theory)
Teoría del Funcional de la Densidad (Density Functional Theory) Motivation: limitations of the standard approach based on the wave function. The electronic density n(r) as the key variable: Functionals
More informationDensity Functional Theory: from theory to Applications
Density Functional Theory: from theory to Applications Uni Mainz November 29, 2010 The self interaction error and its correction Perdew-Zunger SIC Average-density approximation Weighted density approximation
More informationUNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II
UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II August 23, 208 9:00 a.m. :00 p.m. Do any four problems. Each problem is worth 25 points.
More informationDensity Functional Theory - II part
Density Functional Theory - II part antonino.polimeno@unipd.it Overview From theory to practice Implementation Functionals Local functionals Gradient Others From theory to practice From now on, if not
More informationLecture 9. Hartree Fock Method and Koopman s Theorem
Lecture 9 Hartree Fock Method and Koopman s Theorem Ψ(N) is approximated as a single slater determinant Φ of N orthogonal One electron spin-orbitals. One electron orbital φ i = φ i (r) χ i (σ) χ i (σ)
More informationKey concepts in Density Functional Theory
From the many body problem to the Kohn-Sham scheme ILM (LPMCN) CNRS, Université Lyon 1 - France European Theoretical Spectroscopy Facility (ETSF) December 12, 2012 Lyon Outline 1 The many-body problem
More informationKey concepts in Density Functional Theory (I) Silvana Botti
From the many body problem to the Kohn-Sham scheme European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address: Centre
More informationModule 6 1. Density functional theory
Module 6 1. Density functional theory Updated May 12, 2016 B A DDFT C K A bird s-eye view of density-functional theory Authors: Klaus Capelle G http://arxiv.org/abs/cond-mat/0211443 R https://trac.cc.jyu.fi/projects/toolbox/wiki/dft
More informationStudies of Self-interaction Corrections in Density Functional Theory. Guangde Tu
Studies of Self-interaction Corrections in Density Functional Theory Guangde Tu Theoretical Chemistry School of Biotechnology Royal Institute of Technology Stockholm 2008 c Guangde Tu, 2008 ISBN 978-91-7178-964-8
More informationDensity Functional Theory for Beginners
Density Functional Theory for Beginners Basic Principles and Practical Approaches Fabio Finocchi Institut des NanoSciences de Paris (INSP) CNRS and University Pierre et Marie Curie October 24, 2011 1 To
More informationMean-field concept. (Ref: Isotope Science Facility at Michigan State University, MSUCL-1345, p. 41, Nov. 2006) 1/5/16 Volker Oberacker, Vanderbilt 1
Mean-field concept (Ref: Isotope Science Facility at Michigan State University, MSUCL-1345, p. 41, Nov. 2006) 1/5/16 Volker Oberacker, Vanderbilt 1 Static Hartree-Fock (HF) theory Fundamental puzzle: The
More informationElectrochemistry project, Chemistry Department, November Ab-initio Molecular Dynamics Simulation
Electrochemistry project, Chemistry Department, November 2006 Ab-initio Molecular Dynamics Simulation Outline Introduction Ab-initio concepts Total energy concepts Adsorption energy calculation Project
More informationIndependent electrons in an effective potential
ABC of DFT Adiabatic approximation Independent electrons in an effective potential Hartree Fock Density Functional Theory MBPT - GW Density Functional Theory in a nutshell Every observable quantity of
More information2 Electronic structure theory
Electronic structure theory. Generalities.. Born-Oppenheimer approximation revisited In Sec..3 (lecture 3) the Born-Oppenheimer approximation was introduced (see also, for instance, [Tannor.]). We are
More informationDENSITY FUNCTIONAL THEORY FOR NON-THEORISTS JOHN P. PERDEW DEPARTMENTS OF PHYSICS AND CHEMISTRY TEMPLE UNIVERSITY
DENSITY FUNCTIONAL THEORY FOR NON-THEORISTS JOHN P. PERDEW DEPARTMENTS OF PHYSICS AND CHEMISTRY TEMPLE UNIVERSITY A TUTORIAL FOR PHYSICAL SCIENTISTS WHO MAY OR MAY NOT HATE EQUATIONS AND PROOFS REFERENCES
More informationDensity Functional Theory
Chemistry 380.37 Fall 2015 Dr. Jean M. Standard October 28, 2015 Density Functional Theory What is a Functional? A functional is a general mathematical quantity that represents a rule to convert a function
More informationInstructor background for the discussion points of Section 2
Supplementary Information for: Orbitals Some fiction and some facts Jochen Autschbach Department of Chemistry State University of New York at Buffalo Buffalo, NY 14260 3000, USA Instructor background for
More informationDensity Functional Theory
Density Functional Theory March 26, 2009 ? DENSITY FUNCTIONAL THEORY is a method to successfully describe the behavior of atomic and molecular systems and is used for instance for: structural prediction
More informationComputational Methods. Chem 561
Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationExchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride. Dimer. Philip Straughn
Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride Dimer Philip Straughn Abstract Charge transfer between Na and Cl ions is an important problem in physical chemistry. However,
More informationI. CSFs Are Used to Express the Full N-Electron Wavefunction
Chapter 11 One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N- Electron Configuration Functions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon
More informationWorkshop on: ATOMIC STRUCTURE AND TRANSITIONS: THEORY IN USING SUPERSTRUCTURE PROGRAM
Workshop on: ATOMIC STRUCTURE AND TRANSITIONS: THEORY IN USING SUPERSTRUCTURE PROGRAM PROF. SULTANA N. NAHAR Astronomy, Ohio State U, Columbus, Ohio, USA Email: nahar.1@osu.edu http://www.astronomy.ohio-state.edu/
More informationAll electron optimized effective potential method for solids
All electron optimized effective potential method for solids Institut für Theoretische Physik Freie Universität Berlin, Germany and Fritz Haber Institute of the Max Planck Society, Berlin, Germany. 22
More informationConical Intersections. Spiridoula Matsika
Conical Intersections Spiridoula Matsika The Born-Oppenheimer approximation Energy TS Nuclear coordinate R ν The study of chemical systems is based on the separation of nuclear and electronic motion The
More informationHartree-Fock Theory Variational Principle (Rayleigh-Ritz method)
Hartree-Fock Theory Variational Principle (Rayleigh-Ritz method) (note) (note) Schrodinger equation: Example: find an approximate solution for AHV Trial wave function: (note) b opt Mean-Field Approximation
More informationThe ABC of DFT
1 http://dft.rutgers.edu/kieron/beta The ABC of DFT Kieron Burke and friends Department of Chemistry, Rutgers University, 610 Taylor Rd, Piscataway, NJ 08854 April 7, 2003 2 Contents 1 Overview and elementary
More informationOVERVIEW OF QUANTUM CHEMISTRY METHODS
OVERVIEW OF QUANTUM CHEMISTRY METHODS Outline I Generalities Correlation, basis sets Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density
More informationdf(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation
Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations
More informationA Primer to Electronic Structure Computation
A Primer to Electronic Structure Computation Nick Schafer Fall 2006 Abstract A brief overview of some literature the author read as a part of an independent study on Electronic Structure Computation is
More informationCHEM3023: Spins, Atoms and Molecules
CHEM3023: Spins, Atoms and Molecules Lecture 5 The Hartree-Fock method C.-K. Skylaris Learning outcomes Be able to use the variational principle in quantum calculations Be able to construct Fock operators
More informationLS coupling. 2 2 n + H s o + H h f + H B. (1) 2m
LS coupling 1 The big picture We start from the Hamiltonian of an atomic system: H = [ ] 2 2 n Ze2 1 + 1 e 2 1 + H s o + H h f + H B. (1) 2m n e 4πɛ 0 r n 2 4πɛ 0 r nm n,m Here n runs pver the electrons,
More informationSelf-consistent Field
Chapter 6 Self-consistent Field A way to solve a system of many electrons is to consider each electron under the electrostatic field generated by all other electrons. The many-body problem is thus reduced
More information1. Thomas-Fermi method
1. Thomas-Fermi method We consider a system of N electrons in a stationary state, that would obey the stationary Schrödinger equation: h i m + 1 v(r i,r j ) Ψ(r 1,...,r N ) = E i Ψ(r 1,...,r N ). (1.1)
More informationIntroduction to Electronic Structure Theory
Introduction to Electronic Structure Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2002 Last Revised: June 2003 1 Introduction The purpose of these
More informationMany-Body Perturbation Theory. Lucia Reining, Fabien Bruneval
, Fabien Bruneval Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Belfast, 27.6.2007 Outline 1 Reminder 2 Perturbation Theory
More informationTime-dependent density functional theory
Time-dependent density functional theory E.K.U. Gross Max-Planck Institute for Microstructure Physics OUTLINE LECTURE I Phenomena to be described by TDDFT Some generalities on functional theories LECTURE
More information2.5 Time dependent density functional theory
.5 Time dependent density functional theory The main theorems of Kohn-Sham DFT state that: 1. Every observable is a functional of the density (Hohenger-Kohn theorem).. The density can be obtained within
More informationThe Postulates of Quantum Mechanics Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case: 3-D case
The Postulates of Quantum Mechanics Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about the
More informationThe properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit:
Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k B T µ, βµ 1, which defines the degenerate Fermi gas. In this
More informationELECTRONIC STRUCTURE IN THE ORGANIC-BASED MAGNET LINEAR COMBINATION OF ATOMIC ORBITALS METHOD WITH HUBBARD U. A THESIS IN Physics
ELECTRONIC STRUCTURE IN THE ORGANIC-BASED MAGNET [Fe II (TCNE)(NCMe)2][Fe III Cl4] USING THE ORTHOGONALIZED LINEAR COMBINATION OF ATOMIC ORBITALS METHOD WITH HUBBARD U A THESIS IN Physics Presented to
More informationAn Approximate DFT Method: The Density-Functional Tight-Binding (DFTB) Method
Fakultät für Mathematik und Naturwissenschaften - Lehrstuhl für Physikalische Chemie I / Theoretische Chemie An Approximate DFT Method: The Density-Functional Tight-Binding (DFTB) Method Jan-Ole Joswig
More informationMulti-Scale Modeling from First Principles
m mm Multi-Scale Modeling from First Principles μm nm m mm μm nm space space Predictive modeling and simulations must address all time and Continuum Equations, densityfunctional space scales Rate Equations
More informationAlgorithms and Computational Aspects of DFT Calculations
Algorithms and Computational Aspects of DFT Calculations Part I Juan Meza and Chao Yang High Performance Computing Research Lawrence Berkeley National Laboratory IMA Tutorial Mathematical and Computational
More informationConsequently, the exact eigenfunctions of the Hamiltonian are also eigenfunctions of the two spin operators
VI. SPIN-ADAPTED CONFIGURATIONS A. Preliminary Considerations We have described the spin of a single electron by the two spin functions α(ω) α and β(ω) β. In this Sect. we will discuss spin in more detail
More informationDirect Minimization in Density Functional Theory
Direct Minimization in Density Functional Theory FOCM Hongkong 2008 Partners joint paper with: J. Blauert, T. Rohwedder (TU Berlin), A. Neelov (U Basel) joint EU NEST project BigDFT together with Dr. Thierry
More informationLecture 6. Fermion Pairing. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 6 Fermion Pairing WS2010/11: Introduction to Nuclear and Particle Physics Experimental indications for Cooper-Pairing Solid state physics: Pairing of electrons near the Fermi surface with antiparallel
More information3 Electron-Electron Interaction
3 Electron-Electron Interaction 3.1 Electron-Electron Interaction In the adiabatic approximation the dynamics of the electrons and of the nuclei is decoupled. Still, equations (1.8) and (1.12) describe
More informationSession 1. Introduction to Computational Chemistry. Computational (chemistry education) and/or (Computational chemistry) education
Session 1 Introduction to Computational Chemistry 1 Introduction to Computational Chemistry Computational (chemistry education) and/or (Computational chemistry) education First one: Use computational tools
More informationSchrödinger Equation in Born-Oppenheimer Approximation (1)
Schrödinger Equation in Born-Oppenheimer Approximation (1) Time-independent Schrödinger equation: Solution: Wave-function: (where x=(r,s)) Energy: E 2 Schrödinger Equation in Born-Oppenheimer Approximation
More information3: Many electrons. Orbital symmetries. l =2 1. m l
3: Many electrons Orbital symmetries Atomic orbitals are labelled according to the principal quantum number, n, and the orbital angular momentum quantum number, l. Electrons in a diatomic molecule experience
More information( ( ; R H = 109,677 cm -1
CHAPTER 9 Atomic Structure and Spectra I. The Hydrogenic Atoms (one electron species). H, He +1, Li 2+, A. Clues from Line Spectra. Reminder: fundamental equations of spectroscopy: ε Photon = hν relation
More informationP3317 HW from Lecture 15 and Recitation 8
P3317 HW from Lecture 15 and Recitation 8 Due Oct 23, 218 Problem 1. Variational Energy of Helium Here we will estimate the ground state energy of Helium. Helium has two electrons circling around a nucleus
More informationReikniefnafræði - Verkefni 1 Haustmisseri 2013 Kennari - Hannes Jónsson
Háskóli Íslands, raunvísindasvið Reikniefnafræði - Verkefni 1 Haustmisseri 2013 Kennari - Hannes Jónsson Guðjón Henning 11. september 2013 A. Molecular orbitals and electron density of H 2 Q1: Present
More informationQuantum mechanics can be used to calculate any property of a molecule. The energy E of a wavefunction Ψ evaluated for the Hamiltonian H is,
Chapter : Molecules Quantum mechanics can be used to calculate any property of a molecule The energy E of a wavefunction Ψ evaluated for the Hamiltonian H is, E = Ψ H Ψ Ψ Ψ 1) At first this seems like
More informationThe Hartree-Fock approximation
Contents The Born-Oppenheimer approximation Literature Quantum mechanics 2 - Lecture 7 November 21, 2012 Contents The Born-Oppenheimer approximation Literature 1 The Born-Oppenheimer approximation 2 3
More informationOne-Electron Hamiltonians
One-Electron Hamiltonians Hartree-Fock and Density Func7onal Theory Christopher J. Cramer @ChemProfCramer 2017 MSSC, July 10, 2017 REVIEW A One-Slide Summary of Quantum Mechanics Fundamental Postulate:
More informationToward ab initio density functional theory for nuclei
Toward ab initio density functional theory for nuclei J.E. Drut, R.J. Furnstahl, L. Platter Department of Physics, Ohio State University, Columbus, OH 43210 September 11, 2009 arxiv:0906.1463v2 [nucl-th]
More informationAb initio calculations for potential energy surfaces. D. Talbi GRAAL- Montpellier
Ab initio calculations for potential energy surfaces D. Talbi GRAAL- Montpellier A theoretical study of a reaction is a two step process I-Electronic calculations : techniques of quantum chemistry potential
More informationCHEM6085: Density Functional Theory Lecture 10
CHEM6085: Density Functional Theory Lecture 10 1) Spin-polarised calculations 2) Geometry optimisation C.-K. Skylaris 1 Unpaired electrons So far we have developed Kohn-Sham DFT for the case of paired
More informationSection 10: Many Particle Quantum Mechanics Solutions
Physics 143a: Quantum Mechanics I Section 10: Many Particle Quantum Mechanics Solutions Spring 015, Harvard Here is a summary of the most important points from this week (with a few of my own tidbits),
More informationCLIMBING THE LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS JOHN P. PERDEW DEPARTMENT OF PHYSICS TEMPLE UNIVERSITY PHILADELPHIA, PA 19122
CLIMBING THE LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS JOHN P. PERDEW DEPARTMENT OF PHYSICS TEMPLE UNIVERSITY PHILADELPHIA, PA 191 THANKS TO MANY COLLABORATORS, INCLUDING SY VOSKO DAVID LANGRETH ALEX
More informationKohn Sham density functional theory [1 3] is. Role of the Exchange Correlation Energy: Nature s Glue STEFAN KURTH, JOHN P. PERDEW.
Role of the Exchange Correlation Energy: Nature s Glue STEFAN KURTH, JOHN P. PERDEW Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118 Received 11 March 1999;
More informationSpring College on Computational Nanoscience May Variational Principles, the Hellmann-Feynman Theorem, Density Functional Theor
2145-25 Spring College on Computational Nanoscience 17-28 May 2010 Variational Principles, the Hellmann-Feynman Theorem, Density Functional Theor Stefano BARONI SISSA & CNR-IOM DEMOCRITOS Simulation Center
More informationUNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II
UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II January 22, 2016 9:00 a.m. 1:00 p.m. Do any four problems. Each problem is worth 25 points.
More informationElectronic Structure Calculations and Density Functional Theory
Electronic Structure Calculations and Density Functional Theory Rodolphe Vuilleumier Pôle de chimie théorique Département de chimie de l ENS CNRS Ecole normale supérieure UPMC Formation ModPhyChem Lyon,
More informationLecture-XXVI. Time-Independent Schrodinger Equation
Lecture-XXVI Time-Independent Schrodinger Equation Time Independent Schrodinger Equation: The time-dependent Schrodinger equation: Assume that V is independent of time t. In that case the Schrodinger equation
More informationInteracting Fermi Gases
Interacting Fermi Gases Mike Hermele (Dated: February 11, 010) Notes on Interacting Fermi Gas for Physics 7450, Spring 010 I. FERMI GAS WITH DELTA-FUNCTION INTERACTION Since it is easier to illustrate
More informationPhysics 70007, Fall 2009 Answers to Final Exam
Physics 70007, Fall 009 Answers to Final Exam December 17, 009 1. Quantum mechanical pictures a Demonstrate that if the commutation relation [A, B] ic is valid in any of the three Schrodinger, Heisenberg,
More informationCHAPTER 8 The Quantum Theory of Motion
I. Translational motion. CHAPTER 8 The Quantum Theory of Motion A. Single particle in free space, 1-D. 1. Schrodinger eqn H ψ = Eψ! 2 2m d 2 dx 2 ψ = Eψ ; no boundary conditions 2. General solution: ψ
More informationSection 9 Variational Method. Page 492
Section 9 Variational Method Page 492 Page 493 Lecture 27: The Variational Method Date Given: 2008/12/03 Date Revised: 2008/12/03 Derivation Section 9.1 Variational Method: Derivation Page 494 Motivation
More informationAnswers Quantum Chemistry NWI-MOL406 G. C. Groenenboom and G. A. de Wijs, HG00.307, 8:30-11:30, 21 jan 2014
Answers Quantum Chemistry NWI-MOL406 G. C. Groenenboom and G. A. de Wijs, HG00.307, 8:30-11:30, 21 jan 2014 Question 1: Basis sets Consider the split valence SV3-21G one electron basis set for formaldehyde
More information2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements
1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical
More informationii PREFACE This dissertation is submitted for the degree of Doctor of Philosophy at the University of Cambridge. It contains an account of research ca
QUANTUM MONTE CARLO CALCULATIONS OF ELECTRONIC EXCITATIONS By Andrew James Williamson Robinson College, Cambridge Theory of Condensed Matter Group Cavendish Laboratory Madingley Road Cambridge CB3 0HE
More informationIntermission: Let s review the essentials of the Helium Atom
PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the
More informationIntroduction to DFTB. Marcus Elstner. July 28, 2006
Introduction to DFTB Marcus Elstner July 28, 2006 I. Non-selfconsistent solution of the KS equations DFT can treat up to 100 atoms in routine applications, sometimes even more and about several ps in MD
More informationIntroduction to Density Functional Theory
1 Introduction to Density Functional Theory 21 February 2011; V172 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT 2 3 4 Ab initio Computational
More informationElectronic structure, plane waves and pseudopotentials
Electronic structure, plane waves and pseudopotentials P.J. Hasnip Spectroscopy Workshop 2009 We want to be able to predict what electrons and nuclei will do from first principles, without needing to know
More informationWe can instead solve the problem algebraically by introducing up and down ladder operators b + and b
Physics 17c: Statistical Mechanics Second Quantization Ladder Operators in the SHO It is useful to first review the use of ladder operators in the simple harmonic oscillator. Here I present the bare bones
More information