Ц basis: cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z, . .... 6-31G∗ basis; in atomic units
(1 Hartree=27.21 eV) (Gill, 1993) .... 44 DFT, 1 WFT (MP2), good basis sets.
ELECTRONIC STRUCTURE FROM FIRST PRINCIPLES: WAVEFUNCTION THEORY (WFT) VS. DENSITY FUNCTIONAL THEORY (DFT)
Peter Saalfrank Universit¨ at Potsdam
¨ THE MOLECULAR SCHRODINGER EQUATION N electrons (mass me, charge −e) NA nuclei (masses MA, charges +ZAe, A = 1, . . . , NA) • Time-independent molecular Schr¨ odinger equation ³
´
Tˆe + Vee + Vek + Tˆk + Vkk Ψ(r, R) = EΨ(r, R)
r := r = (r 1, r2, . . . r N )
electron coordinates
R := R = (R1, R2, . . . RNA ) PN h ¯2 ˆ Te = − 2me i=1 ∆i P NA 1 h ¯2 ˆ Tk = − 2 A=1 M ∆A A PN PN 2 Vee = i=1 j>i 4π²e0 rij PNA PN ZA e2 Vek = − A=1 i=1 4π²0 R iA P N A P N A ZA ZB e 2 Vkk = A=1 B>A 4π²0 R
nuclear coordinates
E
total molecular energy
Ψ(r, R)
total molecular wavefunction
AB
kinetic energy operator electrons kinetic energy operator nuclei electron-electron repulsion (rij = |ri − rj |) electron-nuclear attraction nuclear-nuclear repulsion
THE BORN-OPPENHEIMER APPROXIMATION Electrons much faster than nuclei =⇒ separate nuclear from electronic motion =⇒ • Born-Oppenheimer approximation, giving: ➊ An electronic Schr¨odinger equation ³ ´ Tˆe + Vee + Vek Ψe(r; R) = Ee(R) Ψe(r; R) {z } | ˆe H
Ee(R) = electronic energy (parametrically dependent on R) Ψe(r; R) = electronic wavefunction (parametrically dependent on R)
➋ A nuclear Schr¨odinger equation
Tˆk + Vkk (R) + Ee(R) Ψk (R) = EΨk (R) {z } | V (R)
V v
E1
R
Ev0 R0
V (R) = potential energy surface =⇒ molecular structure nuclear wavefunction, with Ψ = Ψk · Ψe Ψk =
R
ELECTRONIC STRUCTURE METHODS ˆ e Ψe,n(r; R) = Ee,n(R) Ψe,n(r; R) H n = 0 (ground state), n > 0 (excited states) electronic structure
wavefunction based
semi− empirical HMO, EHT, CNDO, AM1, ....
ab initio
density based
first principles
Hartree− Fock (HF)
DFT Kohn−Sham
post−HF
many−body corrections
CI, MPn, CASSCF, ....
GW
semi− empirical EMT, EAM LDA, GGA, B3LYP, ....
ELECTRONIC STRUCTURE METHODS • The “wavefunction” world
• The “density” world
ˆ e|Ψei Ee = Ee[Ψe(r1, . . . , r N )] = hΨe|H variational principle: opt
Ee
= minΨe Ee[Ψe(r1, . . . , r N )]
Ee = Ee[ne(r)] variational principle: opt
Ee
= minne Ee[ne(r)]
=⇒ 3N-dimensional variational problem!
=⇒ 3-dimensional variational problem!
John Pople
Walter Kohn
THE WAVEFUNCTION WORLD: WAVEFUNCTIONS • Orbitals: Eigenfunctions of 1-electron Hamiltonians χ(x) = ψ(r) · γ(ω) |{z} |{z} spatial
spin function
ˆ 1) χa(x1) = εa χa(x1) h(x
γ(ω) = α(ω) or β(ω) ω = spin coordinate (±1/2 h ¯) x = (r, ω) Example: H atom (atomic units) ˆh(x1) = − 1 ∆1 − 1 2 r1 ψa(r1) = ψnlm(r1) = s, 2px, . . . orbitals 1 (n = 1, 2 . . . ) εa = ε n = − 2 2n
• N-electron wavefunctions: Slater determinants ¯ ¯ ¯ χ1(x1) χ2(x1) · · · χN (x1) ¯ ¯ ¯ . ¯ ¯ 1 ¯ χ1(x2) χ2(x2) . ¯ = |χ1(1), . . . , χN (N )i Ψ (x1, . . . , xN ) = √ ¯ .. .. ... ¯ N! ¯ ¯ ¯ χ1(xN ) χ2(xN ) · · · χN (xN ) ¯ ➊ antisymmetric
➋ Pauli principle
➌ normalized
➍ approximate
. Every Slater determinant represents an electron configuration, e.g. (1s 22s22p2) for C
THE WAVEFUNCTION WORLD: HARTREE-FOCK • The Hartree-Fock equations
• Use single determinant |Ψ0i = |χ1(1), . . . , χN (N )i as trial wavefunction for ground state ˆe • Use full electronic Hamiltonian H ˆ e|Ψ0i = min! • Determine orbitals χi from variational principle E0HF = hΨ0|H =⇒ Hartree-Fock equations χi(1) fˆ(1) χi(1) = εHF i
; i = 1, 2, . . . , N
• Properties
• Coupled single-electron equations, with Fock operator 1 fˆ(1) = − ∆1 − 2
NA X ZA A=1
r1A
+
N ³ X b=1
|
´
ˆ b(1) Jˆb(1) − K | {z } | {z } Coulomb {z exchange }
HF potential
Vˆ HF (1)
• Nonlinear equations =⇒ iterative (self consistent field, SCF) solution • Self-interaction free
THE ROOTHAAN-HALL METHOD: LCAO-MO • LCAO-MO Spatial orbitals (MOs) are expanded in a set of K “atomic orbitals” {φ ν } (AOs): ψj (r) =
K X ν=1
Cνj · φν (r)
=⇒ linear variation problem for Cνj , εj
; ν = 1, 2, . . . , K
closed- and open-shell variants
hφµ |φν i
Cνj Sµν |{z}
ν=1
hφµ |fˆφν i
Fµν = εj |{z}
ν=1
Cνj
K X
K X
• Roothaan-Hall equations
• Basis sets {φν }
➊ Slater functions: e−ζr ➋ Slater-type linear comb. of Gaussians: STO-3G, 6-31G∗, . . . ; cc-pVDZ, cc-pVTZ, . . .
2 /2 ➌ Plane waves: eikr , cutoff Vc = kmax
CORRELATED METHODS
• The error ladder of ab initio quantum chemistry
: A
10 ?: 2 LM >K: 8A
JIH $ "%#$ - E FG >: 4 8 A :
1>0 ? JIH 8: : : Q : 9 P%O%N E FG %#$$ !"
; 2 LH 7 K: A
10 ?: 2 LM K - S(& ER E %(U
A
10 ?0: A > >V: A8
JIH: Q: - #&
*) !(
2 : >?: AA A I >8 >:
T10 1 >: I 2 / 8 A 2 56 >45: JIH : 8 A Q: 10
! (&'
A :
1> : 2I / >: A 2 4556 >: B: 8 Q: W) - X +, E ! "+
> : : >> =
