學(xué)習(xí)資源量子化學(xué)課件2012

1、Density Functional TheoryA ground state effective one-particle theory:accounting for both exchange and correlationeffects in a one-particle pictureWave function vs. densityA wave function (3N coordinates) contains too much information more than needed, and not observable. The complexity of a wave fu

2、nction increases sharply with N.The electron density only depends on 3 coordinates, independently of the number of electrons N.Is it possible to express E and other properties solely in terms of density ?Functional vs. functionA function is a prescription for producing a number from a set of variabl

3、es (coordinates).A functional is a prescription for producing a number from a function. A wave function and the electron density are thus functions, while an energy depending on them is a functional.1927 Thomas-Fermi model: 1930 Thomas-Fermi-Driac model: (TF&TFD: no shell structure, zero binding ene

4、rgies for molecules)1951 Slaters method：(exact T, )1964 Hohenberg-Kohn theorems (1980s Levy-Lieb)1965 Kohn-Sham equation, LDA ( )Applications: atoms, molecules, solids, ab initio MD, etcDensity Functional Theory7.1 The Hohenberg-Kohn Theorem In 1964, Hohenberg and Kohn proved that (Phys. Rev. 136, 1

5、3864 (1964) )all properties are functionals of v-representable for -non-degenerate ground states -local (multiplicative) external potential one-to-one mapping:Proof:The electronic Hamiltonian isit is produced by charges external to the system of electrons.In DFT, is called the external potential act

6、ing on electron i, sinceOnce the external potential the electronic wave functions and allowed energies of the molecule are and the number of electrons n are specified, determined as the solutions of the electronic Schrdinger equation. Now we need to prove that the ground-state electron probability d

7、ensity a) Sincedetermines the number of electrons.b) To see thatdetermines the external potential, we supposethat this is false and that there are two external potentialsand(differingby more than a constant) that each give rise to the same ground-state electrondensity.the number of electrons. the ex

8、ternal potential (except for an arbitrary additive constant) determinesthe exact ground-state wave function and energy of the exact ground-state wave function and energy of LetSinceanddiffer by more than a constant,andmust be different functions.Proof:Assume thusthuswhich contradicts the assumptionf

9、unction, the exact ground-state wave function state energy for a given Hamiltonian If the ground state is nondegenerate, then there is only one normalizedthat gives the exact groundAccording to the variation principle, suppose that If thenis any normalizedwell-behaved trial variation function. Now u

10、se as a trial function with the HamiltonianthenSubstituting givesLetbe a function of the spatial coordinatesof electron i,thenUsing the above result, we getSimilarly, if we go through the same reasoning with a and b interchanged, we getBy hypothesis, the two different wave functions give the same el

11、ectron. Putting and adding the above two inequalitiesdensity: yieldpotentials could produce the same ground-state electron density must be false. This result is false, so our initial assumption that two different externalTherefore, the ground-state electron density determines both theexternal potent

12、ial and the number of electrons (and therefore the Hamiltonian). The undetermined additive constants in the potential merely affects the zero point of the bability densityand other properties”emphasizes the dependence of the external potential differs for different molecules.“For systems w

13、ith a nondegenerate ground state, the ground-state electrondetermines the ground-state wave function and energy, whichHowever, the functionalsare unknown.is also written asThe functionalindependent of the externalonisPotential (universal).7.2 The Hohenberg-kohn variational theorem“For every trial de

14、nsity functionthat satisfiesandfor all, the following inequality holds:, is the true groundstate energy.”Proof:Letsatisfy thatandHohenberg-Kohn theorem, determines the external potential and this in turn determines the wave functiondensity . By the,that corresponds to the .where( has to be v-represe

15、ntable !)with Hamiltonian. According to the variation theoremLet us use the wave functionas a trial variation function for the moleculeSince the left hand side of this inequality can be rewritten asOne gets states. Subsequently, Levy proved the theorems for degenerate ground states. Hohenberg and Ko

16、hn proved their theorems only for nondegenerate groundHohenberg-Kohn Theorem(Levys constrained-search)Non-degeneracy not requiredVR not required (a density is VR if it is a ground state density of some Vext)local is N-representable only ! (known condition, from an electronic wavefunction)7.3 The Koh

17、n-Sham method If we knew the ground-state electron density molecular properties from, the Hohenberg-Kohntheorem tells us that it is possible in principle to calculate all the ground-statewithout recourse to the molecular wave function. 1965, Kohn and Sham devised a practical method for finding andfr

18、om. Phys. Rev., 140, A 1133 (1965). and for finding DFT Kohn-Sham Scheme Quasi-Particle Schemeelectrons that each experiences the same local external potential the ground-state electron probability density equal to the exact of the molecule we are interested in:. Kohn and Sham considered a fictitiou

19、s reference system of n noninteractingthat makesof the reference systemSince the electrons do not interact with each another in the reference system,the Hamiltonian of the reference system iswhereis the one-electron Kohn-Sham Hamiltonian. The exact wave function of a non-interacting H is a Slater de

20、terminant !Thus, the ground-state wave functionof the reference system is: is a spin functionorbital energies.are Kohn-ShamDefineas follows:ground-state electronic kinetic energysystem of noninteracting electrons.(either)is the difference in the averagebetween the molecule and the reference The quan

21、tityrepulsion energy.units) for the electrostatic interelectronic is the classical expression (in atomicRemember thatWith the above definitions, can be written asDefine the exchange-correlation energy functional asNow we haveside are easy to evaluate fromget a good approximation to to the ground-sta

22、te energy. The fourth quantity accurately. The key to accurate KS DFT calculation of molecular properties is to The first three terms on the rightis a relativelysmall term, but is not easy to evaluate and they make the main contributions.Now we need explicit equations to find the ground-state electr

23、on density.same electron density as that in the ground state of the molecule: is readily proved thatSince the fictitious system of noninteracting electrons is defined to have the, itground-state energy by varying to minimize the functional can vary the KS orbitals minimize the above energy expressio

24、n subject to the orthonormality constraint: The Hohenberg-Kohn variational theorem tells us that we can find the so as. Equivalently, instead of varyingweThus, the Kohn-Sham orbitals are those thatwith the exchange-correlation potential defined by(If is known, its functional derivative is also known

25、.)The KS operator exchange operators in the HF operator are replaced by state-independent, and accounts for both exchange and correlation. is the same as the HF operator except that the, which is local,It is a basic feature of the KS solution of an N electron system that all occupied and virtual orb

26、itals are obtained with the same local, state-independent KS potential, so that they “feel” the effective field of N-1 electrons. Because of this, the energies of the KS virtual orbitals represent an “excited” electron interacting with N-1 electrons, rather than an “extra” electron interacting with

27、N electrons. This is a large and important difference with Hartree-Fock, where the LUMO orbital energy represents approximately the energy of an added electron(in the field of all other N electrons), and therefore is physically more like an electron affinity than like an energy of an excited electro

28、n. KS orbital energy differences are therefore much better approximations to vertical excitation energies than HF orbital energy differences are.XC functionals : LDA & GGAGeneralized gradient approximation (GGA): (GGA era 1985-present)at r depends on the density and its gradient (+ higher terms) at

29、rat r depends on the shape of everywhere (nonlocal)General form:at r depends only on the density at r (exact for h.e.g.)Local density approximation (LDA): (LDA era 1965-1985)Courtesy : Claudia Ambrosch (Graz)GGA follows LDA !LDA exchangefunctional of (r)composite function of rLDA correlationGGA (J.

30、P. Perdew 1986)Popular GGAx functionalsLee-Yang-Parr (LYP) (no LDA)Self-interaction errorSelf-interaction error (Zhang & Yang, JCP 109, 2604 (1998)SIE is enhanced for systems with a fractional number of electrons, resulting in too negative energy ! (e.g., too low reaction barriers and charge transfe

31、r states)Ensemble of N+q:If self-interaction freeBut in generalDelocalization error and fractional chargesstatic correlation error and fractional spinsEnergy versus number of electronsW. Yang et al., Science 321, 792 (2008)Delocalization error and fractional chargesstatic correlation error and fract

32、ional spinsComments on the DFT methods:(1) The KS equations are solved in a self-consistent fashion, like the HF equations.(2) The computation time required for a DFT calculation formally scales as thethird power of the number of basis functions. Since only strongly occupied(3) There is no wave func

33、tion. The interacting and non-interacting systems share the same true density rather than the determinantal wave function like HF.(4) The KS orbitals can be used for qualitative MO discussion, like the HF orbitals.(5) Koopmans theorem doesnt hold here, exceptorbitals enter the density, the requireme

34、nt on basis set is much less demanding.(6) All electrons feel the same (N-1)-electron fields. Therefore, is the zeroth order approximation for excitation energy(7) KS-LUMO approximates an excited electron, instead of an added electron (HF)(8) Janak theorem: no occupied orbital should be lower than v

35、irtual orbitals (9) Vxc contains more physics than Exc. LDA/GGA Vxc falls off too quickly.(10) Road map of DFT: Jacobs ladder (J. P. Perdew)LDA:GGA:meta-GGA:hyper-GGA:Generalized RPA: virtual orbitals (vdW)OccupiedOrbitals only(poor vdW)(11) Breakthroughs: (1) 1986 Perdews GGAc, 1988 Beckes GGAx (2)

36、 1988 Beckes quadrature (3) 1993 Beckes hybrid functional (workhorse for organic chemistry)(12) physicists functionals vs. chemists functionals Approximate DFT is the best semi-empirical or industry standard method. It yields better results than HF, and very often even better than MP2. But there is no clear way to improve DFT systematically. Namely, DFT