BO approximation波近似

1、Fundamentals of DFTR. WentzcovitchU of MinnesotaVLab Tutorial Hohemberg-Kohn and Kohn-Sham theorems Self-consistency cycle Extensions of DFT BO approximation IrrintVeT - Basic equations for interacting electrons and nuclei Ions (RI ) + electrons (ri )2222222,1222IJIioniIiiji IIi IeiIIIJijZ Z eZ eeHm

2、rRMRRrr inttotextion ionion ionHTVVEHE IRR ionTextVion ionE22( )2ionItotIIHHRM |eleltotion ionelelHEREThis is the quantity calculatedby total energy codes.PseudopotentialsNucleusCore electronsValence electronsV(r)1.00.50.0-0.50Radial distance (a.u.)rRl (r)123453s orbital of SiReal atomPseudoatomrIon

3、 potentialPseudopotential1/2 Bond lengthBO approximation Born-Oppenheimer approximation (1927) Ions (RI ) + electrons (ri ) 2222( )2totIIIERRRMR 2( )2IJtotIJIJZ ZeERE RRR( )totIIERFR ( )totlmlmER 22( )1det0totIJIJERR RM M IRR Molecular dynamicsLattice dynamicsforcesstressesphononsElectronic Density

4、Functional Theory (DFT) (T = 0 K) Hohemberg and Kohn (1964). Exact theory of many-body systems. 3int( )( ) ( )|eleltotion ionextion ionelelHERETVd rVr n rEDFT1Theorem I: For any system of interacting particles in an external potential Vext(r), the potential Vext(r) is determined uniquely, except for

5、 a constant, by the ground state electronic density n0(r).Theorem II: A universal functional for the energy En in terms of the density n(r) can be defined, valid for any external potential Vext(r). For any particular Vext(r), the exact ground state energy is the global minimum value of this function

6、al, and the density n(r), that minimizes the functional is the ground state density n0(r). Proof of theorem I Assume Vext(1)(r) and Vext(2)(r) differ by more than a constant and produce the same n(r). Vext(1)(r) and Vext(2)(r) produce H(1) and H(2) , which have different ground state wavefunctions,

7、(1) and (2) which are hypothesized to have the same charge density n(r). It follows that Then andAdding both which is an absurd! (1)(1)(1)(1)(2)(1)(2)EHH (2)(1)(2)(2)(2)(2)(2)(1)(2)(2)HHHH (2)3(1)(2)0( )( )( )extextEd r VrVrn r(1)(2)3(1)(2)0( )( )( )extextEEd r VrVrnr(2)(1)3(2)(1)0( )( )( )extextEEd

8、 r VrVrnr(2)(1)(1)(2)EEEEHohemberg and Kohn, Phys. Rev. 136, B864 (1964)Proof of theorem II Each Vext(r) has its (R) and n(r). Therefore the energy Eel(r) can be viewed as a functional of the density. Consider and a different n(2)(r) corresponding to a different It follows that (1)(1)(1)(1)(2)(1)(2)

9、EHH (1)(1)(1)(1)(1)HKEEnH int ( ) ( )HKextion ionEnT nEndrVr n rE ( ) ( )HKextion ionFndrVr n rE(1)( )extVr(2)( )extVrHohemberg and Kohn, Phys. Rev. 136, B864 (1964)The Kohn-Sham Ansatz 3int ( ) ( )extE nT nEnd rVr n r ( ) ( ) HartreeextxcE nT nEndrVr n rEnReplacing one problem with another(auxiliar

10、y and tractable non-interacting system) Kohn and Sham(1965) Hohemberg-Kohn functional:How to find n?iiipmnT221)()()(rrrniii ( ) ( )( )Hartreedr n r n rErrrKohn and Sham, Phys. Rev. 140, A1133 (1965) Kohn-Sham equations: (one electron equation) )()()()()(222rrrVrVrVmiiixcHartreeext)()()(rrrndrrnnErVH

11、artreeHartree)()(rnnErVxcxcWith is as Lagrange multipliers associated with the orthonormalization constraint and anddft2Minimizing En expressed in terms of the non-interacting system w.r.t. s, while constraining s to be orthogonal:,|iji j Exchange correlation energy and potential: By separating out

12、the independent particle kinetic energy and the long range Hartree term, the remaining exchange correlation functional Excn can reasonably be approximated as a local or nearly local functional of the density. ( , )( ),( )( )( )xcxcxcxcEnn rVrnrn rn rn rwithand Local density approximation (LDA) uses

13、xcn calculagted exactly for the homogeneous electron system ( )( , )xcxcEndrn rn rQuantum Monte Carlo by Ceperley and Alder, 1980 Generalized gradient approximation (GGA) includes density gradients in xcn,n Meaning of the eigenvalues and eigenfunctions:Eigenvalues and eigenfunctions have only mathem

14、atical meaning in the KS approach. However, they are useful quantities and often have good correspondence to experimental excitation energies and real charge densities. There is, however, one important formal identityThese eigenvalues and eigenfunctions are used for more accurate calculations of tot

15、al energies and excitation energy.The Hohemberg-Kohn-Sham functional concerns only ground state properties.The Kohn-Sham equations must be solved self-consistentlyiidEdnSelf consistency cycle0( )innr0inV n( )( )( )( )22( )( )( )( )( )2iiiiininoutoutextHartreexciiiVrVrVrrrm ( )ioutnrioutV n1iiiininou

16、tV nV nV n1ii ( )( )iioutinnrnruntilExtensions of the HKS functional Spin density functional theory The HK theorem can be generalized to several types of particles. The most important example is given by spin polarized systems. ( )( )( )n rn rn r( )( )( )s rn rn r , HKEEn s22( , )( , )( )( )( )2extH

17、artreexciiiVr sVr sVrrrm Finite T and ensemble density functional theory The HK theorem has been generalized to finite temperatures. This is the Mermin functional. This is an even stronger generalization of density functional. , , HKelF n TEn TT Sln (1)lnBiiiiSkfff 11 expiiBelfk T( )( )( )iiiin rfrr

18、D. Mermim, Phys. Rev. 137, A1441 (1965)Wentzcovitch, Martins, Allen, PRB 1991Use of the Mermin functional is recommended in the study of metals. Even at 300 K, statesabove the Fermi level are partially occupied.It helps tremendously one to achieve self-consistency. (It stops electrons from “jumping” from occupied to empty states in one step of the cycle to the next.)This was a s