payne2密度泛函理論_圖文

1、 Born-Oppenheimer approximation Hohenberg-Kohn theorems Kohn-Sham implementation The E xc functional The local density approximation (LDA Limits of current implementations of DFT nucleuselectronit is free of adjustable parametersit treats the electrons explicitlycost of the calculation limits system

2、 size and simulation timeee e N N N e N e N V V V T T H += , where(iJ e N e N i J e N e N E H r R r R , , , , , , =Born-Oppenheimer approximation decouples the electronic problemfrom the ionic problem. The electronic problem is:(ii E H r r =Electron mass much smaller than nuclear mass:Timescale asso

3、ciated with nuclear motion much slower than that associated with electronic motionElectrons follow instantaneously the motion of the nuclei,remaining always in the same stationary state of the Hamiltonian(i J N i J e N r R r R , , =(iJ e N e N i J e N e N E H r R r R , , , , , , =depends only parame

4、trically on R J Theorem 1.The external potential is uniquely determined by the electronic charge density -n (r -so the total energy is a unique functional of the density-E n !Theorem 2.The density which minimises the energy is the ground state density and the minimum energy is the ground state energ

5、y:0E n E Min n=Hohenberg & Kohn, PRB 136, 864 (1964there is a universal functional E n which could be minimisedto obtain the exact ground state density and energy. Thomas-Fermi-Dirac Model (1929: First model where the energy is expressed in terms of the electron density Kohn-Hohenberg-Sham (1964

6、:Exact unique relationship between groundstate energy and electron densityE. Fermi, Nobel laureate 1938W. Kohn, Nobel laureate (in chemistry! 1998 Does not depend on V ext (thespecific system: it is a universal functional. The exact dependence on n (r is unknownE xc must be approximated in applicati

7、ons(n E n E n T n T n E H e e s xc += nxcFor the homogeneous electron gas the exact dependence of xc (n can be computed by Quantum Monte Carlo.The idea: build Exc from the knowledge of the exchange-correlation energy per particle, xc , of the homogeneous electron gasKohn & Sham, PRA 140, 1133 (1

8、965Ceperley & Alder, PRL 45, 566 (1980nn (=r r r d n n Exc LDA xc ( (Picture courtesy of Andreas SavinKohn & Sham, PRA 140, 1133 (1965 The exchange correlation hole, P xc (r 1, r 2, is the probability of finding an electron at r 2 given that there is an electron at r 1. It is the hole the el

9、ectron at r 1digs for itself in the surrounding electronic density.There are a number of properties which will be satisfied by the exact exchange correlation hole. For instance it should normalise to exactly one electron:1, (221=r r r d Pxc LDA satisfies this rule.Jones & Gunnarsson, Rev. Mod. P

10、hys. 61, 689(1979 How can Ve-e be reasonable if Pxc is wrong ? Spherical average of Pxc Ve-e depends only on the spherical average of Pxc LDA works in part because it generates a reasonable estimate of the spherical average Jones et. al. 1979 The LDA energy densities The difference between the exact

11、 (V-QMC and LDA energy density in bulk silicon (au Exchange Correlation Hood et al PRB 57 8972 (1998 So Why does the LDA work ? Exact properties of the xc-hole maintained The electron-electron interaction depends only on the spherical average of the xc-hole this is reasonably well reproduced The err

12、ors in the exchange and correlation energy densities tend to cancel Some things that do not work in this approach Van der Waals interactions: due to mutual dynamical charge polarisation of the atoms not properly included in any existing approximations to Exc Excited states: DFT is a ground state the

13、ory (ways forward: time-dependent DFT, GW, Non Born-Oppenheimer processes (i.e, non-radiative transitions between electronic states Self-interaction problem: each electron lives in the field created by all electrons including itself ( ways forward: SIC, hybrid DFT Disadvantages of DFT It only applie

14、s to the electronic groundstate (or an electronic system in thermal equilibrium. Have to use approximations to the true density functional. Not possible to predict error in the value of any particular property. Not possible to systematically improve accuracy of calculation. Advantages of DFT Kohn wa

15、s awarded the Nobel prize for DFT - this has endowed DFT with a prestige that makes it hard to criticise! It offers very good scaling of computational cost with system size. It allows calculations to be performed on large and complex systems. Given the very large number of DFT calculations the likel

16、y accuracy of property prediction for many properties/systems is known. Some myths and half-truths about DFT DFT tells you nothing about the excited states. Although the bandgap is wrong, the wavefunctions of excited states are meaningful. Density functional theory can fail. The true density functio

17、nal gets all groundstate energies and densities correct, but it only manages this by psychopathic behaviour. Infinitely small changes in the electron density produce large changes in the XC potential infinitely far away. Useful References Hohenberg & Kohn, PRB 136, 864 (1964 Kohn & Sham, PRA 140, 1133 (1965 R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989 M. C. Payne et al., Rev.