The Schrödinger equation for many-electron systems
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1 The Schrödinger equation for many-electron systems Ĥ!( x,, x ) = E!( x,, x ) 1 N 1 1 Z 1 Ĥ = " $ # " $ + $ 2 r 2 A j j A, j RAj i, j < i a linear differential equation in 4N variables (atomic units) (3 spatial and 1 spin coordinate for every electron) Time independent Schrödinger equation Born-Oppenheimer approximation Relativistic effects are neglected Neglect of higher order effects (e.g. spin-orbit interaction) No excited states! Ground state (E 0, " 0, # 0 ) ij N " is antisymmetric!
2 The Hartree-Fock approximation The HF method provides the N spin-orbitals! HF i (x) (i=1,2,,n) which define the best single-determinant approximation of " 0 : HF " # $ = 0 1 N! HF ( x )! ( x ) HF!! 1 " # " HF ( x )! ( x ) HF!! N N N N HF HF = # ˆ HF HF E H # = mine! % # " & $ E0 The HF equations (Self-Consistent-Field) 2 ( ) ( ) ( ) 2 ( ) ( ) nonlocal exchange potential # # $ 1!! % &!! ' 2 j r2 j r2 j r2 i r2,( ) + Ven ( r) + 3 dr -! i r (! = "!, ( - * 3 dr ( + j r i i r. 2 j r1 r2 / 0 j r1 r2 1 ( ) ( ) Mean-field approximation $ no electron correlation effects (E c =E 0 -E HF )
3 Beyond Hartree-Fock Many methods/approximations are applicable. Generally expensive and not yet widely applied in periodic systems (CRYSCOR) Eg: MP2, MP3, MP4 CI, CIS, CASSCF, CCSD(T) QMC Is it necessary to solve the Schrödinger equation and determine the 4N dimensional wavefunction in order to compute the ground state energy?
4 The external potential Any energy E accessible to a many-electron system in a stationary state " is given by: 1 * 2 * E =! Ĥ! = "! ( 1,, N )# j! ( 1,, N ) d 1 d N " 2 $% x x x x x x ( x,, x ) ( x,, xn ) " Z dx dx + *! 1 N! 1 A A, j rj " R A $ % j 1 N kinetic energy electron-nuclear attraction + $ % *! 1 N! 1 ( x,, x ) ( x,, x ) r " r i, j< i j i N dx 1 dx N electron-electron repulsion ˆV ( r) ext = " A Z A r! R A The external potential V ext and the number of electrons N completely determine the Hamiltonian. The kinetic energy and e-e repulsion terms are universal.
5 $ The electron density The electron-nuclear attraction and electron-electron repulsion terms are easily rewritten in terms of the one-particle electron density * " ( x 1,, x N )"( x 1,, x N ) 1!( 1) dx1 dxn = r dr1 r1 # R A N r1 # R A $ $!( r ) = N # ( x,, x )#( x,, x ) d" dx dx * 1 1 N 1 N 1 2 N and the pair electron density: * " ( x 1,, x N )"( x 1,, x N ) 1!( 1, 2) dx1 dxn = r r dr1 dr2 r1 # r2 N( N # 1) r1 # r2 $ $ %!( r, r ) = N( N # 1) $ ( x,, x ) $ ( x,, x ) d" d" dx dx * N 1 N N Because electrons are indistinguishable, E results from a sum of integrals which depend at most on six independent spatial coordinates.
6 Hohenberg-Kohn Theorem To find the exact total energy knowledge of the electron charge density "(r) is enough! Hohenberg-Kohn (1964) established the rules of the DFT (Density Functional Theory) to compute the ground state energy as a functional of the electron density The external potential V ext (r) is a unique functional of "(r) to within a constant and, since V ext (r) determines H, the full many-particle ground-state is a unique functional of "(r) References: W. Koch, M. C. Kolthausen, A Chemist s Guide to Density Functional Theory, Wiley-VCH, Weinheim, 2000 R. G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989
7 Hohenberg-Kohn Theorem: proof The proof of the Hohenberg-Kohn theorem is surprisingly simple. Let us suppose there exist two different external potentials (by more than a constant) associated with the same electron density: Vˆ # Hˆ # $ #! % $ " % Hˆ " % Vˆ " ext As a consequence of the variational theorem, the ground state energies corresponding to the trial functions " and "! are E < " Hˆ " = " Hˆ! " + " H-H ˆ ˆ! " = E! + " H-H ˆ ˆ! " 0 0 E 0! < " H ˆ! " = " Ĥ " + " H ˆ!-Ĥ " = E0 # " Ĥ- H ˆ! " but, summing these two expressions, we obtain the following false inequality: E + E! < E! + E We conclude that two V ext that yield the same electron density cannot exist. E = E [!] The energy is a functional of the electron density ext
8 Hohenberg-Kohn Theorem 2 The density which minimises the energy is the ground state density and the minimum energy is the ground state energy. E 0 =!( r )" N [! r ] min E ( ) under the constraint: "!( r)dr = N But... The kinetic energy and the e-e repulion energy are difficult to compute in terms of the electron density: #! ( x,, x )"! ( x,, x ) dx dx * 2 * 1 N j 1 N 1 N V e e 1 2 = #!( r1, r2 ) r " r 1 2 dr dr 1 2 Warning! Unfortunately, not any electron density is acceptable, but only those "(r) which are generated by antisymmetric wavefunctions
9 The Kohn-Sham method Kohn and Sham (1965) proposed to express the electron density in terms of a set of orthonormal single-particle functions: The total energy is then given by: ( r) ( r) 2 i = 1 The ground state density " 0 (r) is obtained by solving a coupled set of one-electron pseudo-schrödinger equations, the Kohn-Sham equations, self-consistently: ( r) = ( r) h! "! ˆKS i i i ˆ KS 1 2!( r ') h = " # + Vext ( r ) + $ dr ' + Vxc( r ) 2 r " r ' N! = # " i ( r) = ( r) + ( r) + ( r) + ( r) KS E " $! #% T " $! #% " $! #% " $! #% " $! # S Eext EC Exc % V xc! E =!" xc
10 The non-interacting system There exists an effective mean field potential which, when applied to a system of non-interacting fermions, will generate the exact ground state energy and charge density. 1 r r!( r, " V xc ( i j 1 r! ( r) 2,...) i r) E[#], #(r) Picture courtesy of Nic Harrison
11 The exchange-correlation functional ( r) = ( r) + ( r) + ( r) + ( r) KS E " $! #% T " $! #% E " $! #% " $! #% " $! # ext EC Erest % The functional form of the kinetic energy is unknown! known! self-interaction correction, exchange, correlation unknown! ( r) = T ( r) + ( r) + E ( r) + ( r) KS E " $! #% " $! #% " $! #% " $! #% " $! # S Eext C Ex c % Kinetic energy of the system of independent particles known! The exchange-correlation universal functional contains: unknown! The difference in the kinetic energy between the real system and the independent particle system The electron-electron repulsion interactions excluding the Coulomb interaction of the independent-particle system We need a guess for E xc!
12 The homogeneous electron gas The model used is extremely simple: a homogeneous gas of electrons. For the non-interacting gas the kinetic and exchange energy per particle can be computed the single particle wavefunctions are simply plane waves. 3 T[!] (3" )!( ) 10 E xc 2 5 [ ] 2 3 = # r 3 dr [!]!( )" [! ( )] # $ r r dr xc [ ] = [ ] + [ ]! "! "! " xc x c # x [ ] = C ( ( r) ) 1 3! " " J. Slater, 1951 x The exact dependence of! c ["] for the homogeneous electron gas can be computed by Quantum Monte Carlo simulations (Ceperley-Alder, 1980) % xc
13 The Local Density Approximation (LDA) For the ground state energy and density there is an exact mapping between the many body system and a fictitious non-interacting system: LDA Exc [!] = #!( r) " xc(!( r)) dr # # The energy functional is approximated as a local functional of the energy. Picture courtesy of Andreas Savin r % xc
14 The exchange-correlation hole The pair density determines the total energy: does the LDA reproduce the pair density? The exchange-correlation hole is the conditional probability, the probability of finding an electron at r 2 given that there is an electron at r 1 ( r, r )!! ( r, r ) = "!( r ) xc !( r1 ) It is the hole the electron at r 1 digs for itself in the surrounding density. The exchange-correlation hole has some properties. For example, it should normalise to exactly one electron, because the conditional probability for electrons of the same spin as electron 1 integrates to N " -1 instead of N " : #! xc ( r1, r2 ) dr2 = " 1
15 Why does the LDA work? The exchange-correlation hole indeed sums to -1 in LDA! The exchange-correlation hole is poorly estimated in LDA. However. LDA generates a reasonable estimate of the spherical-averaged exchange-correlation hole. Gunnarsson et al. 1979
16 The LDA energy densities in direct space The difference between the exact (V-QMC) and LDA energy density in bulk silicon (au) Exchange Correlation Hood et al., Phys. Rev. B 57, 8972 (1998) The errors in the exchange and corelation energy densities tend to cancel!
17 The Generalized Gradient Approximation (GGA) GGA Exc [!] = $!( r) " xc (!( r), #!( r) ) dr " r! xc
18 Families of approximations to E xc xc xc [ ( )]! =! " r LDA! xc =! xc #& "( r), %"( r) $ ' $ 2 2 % xc = xc ' ( r), & ( r),& ( r), & i ( i!! " " " + # ) * GGA meta-gga [ ( )] [ ( )]! = "! + #! $ r + %! $ r hybrid HF LDA GGA xc x x x Hybrid functionals incorporate a part of HF exchange ( exact exchange )
19 Performance of several functionals Brucite, Mg(OH) 2 Hybrids Basis set: 8-511G* (Mg), 8-411G* (O), 311G* (H) Differences between calculated and experimental results: a,c: lattice parameters, Å &V%: volumes, relative difference (calc-exp)/exp*100 OH: bond length of OH group, Å v: anharmonic stretching of the OH oscillator, cm -1 Table courtesy of Raffaella Demichelis
20 Determination of the band gap Calculated band structure for cubic KNbO 3 along the '-X direction of reciprocal space, as a function of the Hamiltonian. The values indicate the mixing parameter ( in the F-BLYP scheme. F. Corà, M. Alfredsson, G. Mallia, D.S. Middlemiss, W.C. Mackrodt, R. Dovesi, R. Orlando, Structure and Bonding 113 (2004) 171
21 Tayloring functionals: the case of ferroelectrics Most functionals fail to describe ferroelectric properties properly. Wu-Cohen modified the PBE functional to improve its performance for these materials. Excellent performance from hybrid functional B1, mixing HF exchange and Wu-Cohen GGA D. I. Bilc, R. Orlando, R. Shaltaf, G. M. Rignanese, J. Íñiguez, Ph. Ghosez, Phys. Rev. B 77, (2008)
22 The problem of weak interactions: Van der Waals LDA and GGA (and hybrid) functionals are unable to reproduce Van der Waals interactions properly. Graphite layers result to be unbound in most cases. Interaction energy (kj/mol) GGA 0 LDA HF hybrids PBE0(CPC) B3LYP(CPC) X3LYP(CPC) PW91(CPC) PBE(CPC) SVWN(CPC) HF(CPC) B3LYP-D* (CPC) Graphite layers are weakly bound with LDA, but trend is incorrect. c (Å) BSSE corrected B3LYP-D* Exp.: a = 2.46 (fixed) c = 6.71 Å BS: 6-31G(d) Courtesy of B. Civalleri
23 graphite layers BS: 6-31G(d) Interaction energy (kj/mol) Grimme s empirical correction S. Grimme, J. Comput. Chem., 2004, 25, 1463 and J. Comput. Chem., 2006, 27, 1787 Rcut ij C E s ' f ( R ) 6 R =! " " Disp 6 dmp ij, g 6 g ij ij, g B3LYP-D*(CPC) c (Å) f dmp j = j R = Ri + R vdw vdw vdw ij C Ci! C 1 ( Rij, g) =! " Exp. 1+ e B3LYP-D* ' d R / R ' 1 # $ % ij, g vdw & Exp.: in Å a = 2.46 c = 6.70 B3LYP-D*: a = c = Courtesy of B. Civalleri
24 Testing Grimme s correction 14 molecular crystals both dispersion bonded and hydrogen bonded C 2 H 2 CO 2 Propane NH 3 C 6 H 6 Formic acid 1,4-dichlorobenzene Formamide Naphthalene Urotropine Experimental sublimation energies at 298K available from published data (estimated error bar: ±4 kj/mol) Urea 1,4-dicyanobenzene Succinic anhydride Boric acid For some of them accurate low temperature structural data from NPD Courtesy of B. Civalleri
25 Cohesive energies: B3LYP vs B3LYP-D* (Grimme) BSSE corrected cohesive energy (kj/mol) BSSE corrected cohesive energies vs Experimental data Exp.: -&E=&H 0 sub (T)+2RT from data at 298K B3LYP-D Grimme B3LYP Exp < &E < -25 kj/mol Courtesy of B. Civalleri Experimental lattice energy (kj/mol) Cell fixed geometry optimization of the atomic positions at B3LYP/6-31G(d,p) B3LYP: MD=54.4 Empirical correction definitely improves cohesive energies Tendency of B3LYP-D Grimme to overestimate cohesive energy (MD=-6.0 & MAD=8.9) especially for HB molecular crystals Small basis sets suffer from large BSSE BSSE corrected data are less basis set dependent
26 Integration of the exchange-correlation functional O O The exchange-correlation density functional is integrated numerically on a mesh of points in atomic domains Mg Grid of Mg Grid of O Radial points: Gauss! formula (n r =number of radial points) O 001 plane of a unit cell of MgO O Angular points: Lebedev distribution (L=Lebedev accuracy parameter)
27 DFT integration: the grid size n r =55 L=13 n r =75 L=13 n r =75 L=16 default lgrid xlgrid size of grid electrons/cell E (hartree/cell)
28 DFT integration: grid pruning Pruning: Lebedev accuracy varies along radial intervals n r =55 L=13 n r =75 L=13 n r =75 L=16 unpruned grid (31878) (42956) (96038) pruned grid (11552) (20542) (41530)
29 DFT integration: use of symmetry default lgrid xlgrid size of grid t elapsed (s)
30 DFT integration: accuracy vs grid L 55 n r MgO Total energy (hartree/cell) L n r Time (s) required to compute the exchange-correlation contribution to the Fock matrix when symmetry is used Crystals with heavier atoms may require dense grids Consistency is more important than convergence to the most accurate result in geometry optimization or phonon calculation
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