The Hartree-Fock Method

Transcription

1 1/ December, 2015

2 Problem Statement First Quantization Second Quantization References Overview 1. Problem Statement 2. First Quantization 3. Second Quantization References 2/41

3 Problem Statement First Quantization Second Quantization References Motivation The time-independent Schrödinger equation is defined as Ĥ Ψpxq EΨpxq, (1) where Ĥ ist the Hamiltonian operator and E is the energy of the state Ψ Find an approximate solution of the time-independent Schrödinger equation for a many-body system 3/41

4 Problem Statement First Quantization Second Quantization References Examples of Many-Body Systems Multi-electron atoms Molecules (a) Oxygen (b) Water molecule 4/41

5 Problem Statement First Quantization Second Quantization References The Molecular Hamiltonian Operator Ĥ Kinetic energy (nucl. and el.) hkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkj K ÿnuclei i 1 2 N electrons ÿ 2 r 2m i ` i j 1 2 2m i 2 r j ` Coulomb repulsion (el.-el.) hkkkkkkkkkkkkkikkkkkkkkkkkkkj 1 1 Nÿ Nÿ e 2 4πɛ 0 2 r k r l k 1 ląk 1 Kÿ Nÿ Z m e 2 ` 1 1 Kÿ Kÿ Z o Z p e 2 4πɛ 0 r m 1 n 1 n R m 4πɛ loooooooooooooooomoooooooooooooooon 0 2 R o 1 pą0 o R k looooooooooooooomooooooooooooooon Coulomb attraction (el.-nucl.) Coulomb repulsion (nucl.-nucl.) (2) 5/41

6 Problem Statement First Quantization Second Quantization References Born-Oppenheimer Approximation Nuclei of atoms are often much heavier than electrons Consider nuclei as stationary to calculate properties of the electrons Hamiltonian operator for electrons: Ĥ el Kinetic energy (el.) hkkkkkkkkkkikkkkkkkkkkj ÿ N electrons j r 2m j ` i 4πɛ Kÿ Nÿ Z m e 2 4πɛ 0 r m 1 n 1 n R m looooooooooooooomooooooooooooooon Coulomb attraction (el.-nucl.) Coulomb repulsion (el.-el.) hkkkkkkkkkkkkikkkkkkkkkkkkj Nÿ Nÿ k 1 ląk e 2 r k r l (3) 6/41

7 Problem Statement First Quantization Second Quantization References Hilbert Space Single-body quantum system is described by a Hilbert space H of dimension dim H d N distinguishable particles are described by tensor product of N single-body Hilbert spaces H pnq H bn Nâ H (4) i 1 with dim H pnq d N (5) 7/41

8 Problem Statement First Quantization Second Quantization References Complexity of the Hilbert Space d N basis functions are needed to span the Hilbert space H pnq Exponential scaling of the Hilbert space dimension with number of particles is a big challenge Single fermion has a Hilbert space H C 2 with dim H 2 N fermions have Hilbert space H pnq C 2N with dim H pnq 2 N Basis for N 30 fermions is already of size 2 30, i.e., over one billion basis functions 8/41

9 Problem Statement First Quantization Second Quantization References Many-Fermion System I Use Hartree product ansatz for many-particle wave function: ψ pr 1,..., r N q «Nź φ i pr i q, (6) i 1 where φ i is a single-particle wave function Problem: Hartree product does not take into account antisymmetric properties of fermions, i.e., ˆP 12 ψpr 1, r 2 q ψpr 2, r 1 q φ 1 pr 2 qφ 2 pr 1 q φ 1 pr 1 qφ 2 pr 2 q, where ˆP 12 is the exchange operator 9/41

10 Problem Statement First Quantization Second Quantization References Many-Fermion System II Solution: Fermionic wave function has to be antisymmetric under particle exchange, thus the antisymmetrized wave function is defined as Ψ paq? 1 ÿ sgnppq ψ `r pp1q,..., r ppnq N! p (7) 1{? N! is the normalization factor Sum goes over all permutations p of N particles sgnppq `1 if p represents an even number of permutations, else sgnppq 1 10/41

11 Problem Statement First Quantization Second Quantization References Slater Determinant The many-fermion wave function in Eq. 7 can be written as a Slater determinant: Ψ paq? 1 φ 1 pr 1 q... φ N pr 1 q.. N! (8) ˇˇˇˇˇˇ φ 1 pr N q... φ N pr N q ˇ The set of antisymmetrized Slater determinants forms the basis of the many-particle Hilbert space H pnq Example: For a two-electron system we would have Ψ paq pr 1, r 2 q 1? 2 rφ 1 pr 1 qφ 2 pr 2 q φ 1 pr 2 qφ 2 pr 1 qs (9) 11/41

12 Problem Statement First Quantization Second Quantization References Restrictions of the First Quantization Cumbersome to work with appropriately symmetrized many-body wave functions It is restricted to exactly N particles Quantum field theory: systems with variable particle numbers Solid state physics: infinite number of electrons 12/41

13 Problem Statement First Quantization Second Quantization References Fock Space N 0, 1,..., 8 indistinguishable particles are described by the Fock space which is defined as FpHq 8à S H bn (10) N 0 S` is the symmetrization operator used for bosons S is the anti-symmetrization operator used for fermions 13/41

14 Problem Statement First Quantization Second Quantization References The Occupation Number Basis I Let t φ 1 y,..., φ L yu be the basis of the single-particle Hilbert space H The Fock space basis consists of states (called Fock states) constructed by specifying the number of particles N α occupying the single-particle state φ α y, i.e., N 1, N 2,..., N L y (11) For fermions: N α P t0, 1u (Pauli exclusion principle) 14/41

15 Problem Statement First Quantization Second Quantization References The Occupation Number Basis II Fock space is the space of all occupation number states for all particle numbers N N particles Fermionic basis states 0 0, 0, 0,...y 1 1, 0, 0,...y, 0, 1, 0,...y, 0, 0, 1,...y, , 1, 0,...y, 0, 1, 1,...y, 1, 0, 1,...y, /41

16 Problem Statement First Quantization Second Quantization References Fermion Creation Operator â : α I In Fock space, the fermion creation operator â : α for the single-particle state φy α is introduced â : α increases the occupation number of N α by 1 if N α 0, e.g., â : 2 1, 0, 0,...y 1, 1, 0,...y (12) Fermion creation in a single-particle state that is already occupied destroys the state, e.g., â : 1 1, 1, 0,...y 0 (13) 16/41

17 Problem Statement First Quantization Second Quantization References Fermion Creation Operator â : α II Fermion creation operators have to anticommute, meaning! ) â α, : â : β 0 (14) Eq. 14 is needed, since Fock states are antisymmetric under interchange of pairs of fermions, i.e., â : αâ : β 0y â: βâ: α 0y (15) Further, Eq. 14 ensures that a state is destroyed if one tries to create a fermion in an already occupied state, since `â: α 2 0 (16) 17/41

18 Problem Statement First Quantization Second Quantization References Fock Basis in Second Quantization The Fock state N 1, N 2,..., N L y in the occupation number basis can be expressed in terms of creation operators: N 1, N 2,..., N L y Lź i 1 a : Ni i a : N1 1 a : N2 2 a L NL : 0y (17) Fermion creation operators anticommute, thus ordering of the operators matters The normal ordering that is used from now on is defined by Eq /41

19 Problem Statement First Quantization Second Quantization References Fermion Annihilation Operator â α Fermion annihilation operator â α decreases the occupation number of N α by 1 if N α 1, e.g., â 1 1, 0, 0,...y 0, 0, 0,...y (18) Fermion annihilation in a single-particle state that is not occupied destroys the state, e.g., â 2 0, 0, 0,...y 0 (19) Fermion annihilation operators anticommute, i.e., tâ α, â β u 0 (20) 19/41

20 Problem Statement First Quantization Second Quantization References Definition of â : α and â α I â : α N 1, N 2,..., N α,...y δ Nα0p 1q Sα N 1, N 2,..., N α ` 1,...y (21) â α N 1, N 2,..., N α,...y δ Nα1p 1q Sα N 1, N 2,..., N α 1,...y (22) where S α ÿ N γ (23) γăα and δ ij 1 if i j and 0 otherwise 20/41

21 Problem Statement First Quantization Second Quantization References Definition of â : α and â α II From Eqs. 21 and 22 it follows that the antisymmetric property of fermions is fulfilled, since â : αâ β â β â : α pα βq (24) 21/41

22 Problem Statement First Quantization Second Quantization References One-Particle Operators I One-particle operator ˆV p1q consists of a sum of N identical operators ˆV i acting only on the Hilbert space of the i-th electron ˆV p1q Nÿ i 1 ˆV i (25) Example: the kinetic energy operator defined as ˆT Nÿ i 1 ˆp 2 i 2m i (26) 22/41

23 Problem Statement First Quantization Second Quantization References One-Particle Operators II ˆVi operating on a one-electron wave function φ α pr i, s i q it produces a superposition of one-electron wave functions ˆV i φ α pr i, s i q ÿ β V βα φ β pr i, s i q (27) with the amplitudes V βα xi, β ˆV i i, αy (28) 23/41

24 Problem Statement First Quantization Second Quantization References One-Particle Operators III Applying ˆV p1q on a Fock state a superposition of states is generated ˆV p1q α 1,..., α i,..., α L y (29) ÿ Lÿ V βαi α 1,..., α i Ñ β,..., α L y β i 1 (30) α 1,..., α i Ñ β,..., α L y denotes the state obtained from α 1,..., α i,..., α L y upon replacing φ αi by φ β α 1,..., α i Ñ β,..., α L y â : βâα i α 1,..., α i,..., α L y (31) 24/41

25 Problem Statement First Quantization Second Quantization References One-Particle Operators IV The one-particle operator ˆV p1q can now be expressed as ˆV p1q ÿ α,β V βα â : βâα (32) 25/41

26 Problem Statement First Quantization Second Quantization References Two-Particle Operators I Two-particle operator ˆV p2q consists of a sum of N identical operators ˆV ij acting on the Hilbert spaces of two electrons ˆV p2q 1 2 Nÿ i j ˆV ij (33) Example: Coulomb interaction term 1 ÿ e 2 2 r i j i r j (34) 26/41

27 Problem Statement First Quantization Second Quantization References Two-Particle Operators II Analogous to the derivation shown for the one-particle p2q operator, one obtains for the two-particle operator ˆV ˆV p2q 1 2 with the amplitudes ÿ α,β,γ,δ V α,β,γ,δ â : αâ : βâγâ δ (35) V γδαβ pxi, γ xj, δ q ˆV ij p j, αy i, βyq (36) 27/41

28 Problem Statement First Quantization Second Quantization References Hamilton Operator in Second Quantized Notation I Use basis set of L orbital wave functions tf i u The matrix elements of Ĥel (Eq. 3) are defined as ż t ij d 3 rfi prq ˆ 2 2m 2 ` V pr i q f j prq (37) ż ż V ijkl e 2 d 3 r d 3 r 1 fi 1 prqf j prq r r 1 f k pr 1 qf l pr 1 q (38) where t ij is a one-particle and V ijkl a two-particle operator, and V prq 1 4πɛ 0 Kÿ m 1 Z m e 2 r R m (39) 28/41

29 Problem Statement First Quantization Second Quantization References Hamilton Operator in Second Quantized Notation II Ĥ el can now be written in second quantized notation as Ĥ el ÿ ijσ t ij â : iσâ jσ ` 1 2 ÿ ijklσσ 1 V ijkl â : iσâ : kσ 1â iσ 1â jσ (40) 29/41

30 Problem Statement First Quantization Second Quantization References The Hartree-Fock Approximation Approximation is based on the assumption of independent electrons N -fermion ground state wave function is represented as a single Slater determinant Ψ phfq? 1 φ 1 pr 1, σ 1 q... φ N pr 1, σ 1 q N!.. (41) ˇ φ 1 pr N, σ N q... φ N pr N, σ N q ˇ 30/41

31 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations I Closed-shell conditions are assumed, i.e., each orbital is occupied by both an electron with spin Ò and spin Ó Ψ phfq (Eq. 41) in second quantized form: Ψy phfq ź c µσ : 0y (42) µ,σ where c : µσ is orthogonal and creates an electron in the orbital φ µ pr, σq 31/41

32 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations II c : µσ is expanded in terms of creation operators â : nσ of our finite basis set: c : µσ Lÿ d µn â nσ : (43) n 1 32/41

33 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations II Bond-order matrix is introduced: P ij ÿ xψ phfq â iσâ : jσ Ψ phfq y 2 ÿ σ ν d νid νj (44) The kinetic term of Ĥel is simplified to ÿ t ij â iσâ : jσ ÿ P ij t ij (45) ijσ ij 33/41

34 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations III The interaction term becomes xψ phfq a : iσa : kσ 1â lσ 1â jσ Ψ phfq y ijkl # P ij P kl P il P kj, σ σ 1 P ij P kl, σ σ 1 (46) Thus, the interaction term of Ĥel simplfies to 1 ÿ ˆ V ijkl V ilkj P ij P kl (47) 34/41

35 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations IV Combining Eqs. 45 and 47 leads to the energy term E phfq xψ phfq H phfq y ÿ P ij t ij ` 1 ÿ ˆ V ijkl V ilkj P ij P kl (48) ij ijkl 35/41

36 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations V Minimize E phfq under the condition that the states φy µ are normalized: 1 xφ µ φ µ y ÿ i,j d µid µj S ij (49) with the overlap matrix S defined as ż S ij d 3 rfi prqf j prqap (50) S is the identity matrix for an orthonormal basis set 36/41

37 Problem Statement First Quantization Second Quantization References ij The Hartree-Fock Equations VI Introduce Lagrange multipliers to enforce the constraint we have to minimize ÿ P ij t ij ` 1 ÿ ˆ V ijkl V ilkj P ij P kl ÿ ÿ ɛ µ ijkl µ i,j d µid µj S ij (51) 37/41

38 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations VII Set the derivative with respect to d µi to zero to get the Hartree-Fock equations for a finite basis set: Lÿ pf ij ɛ µ S ij qd µj 0 (52) j 1 with f ij t ij ` ÿ kl ˆ V ijkl 1 2 V ilkj P kl (53) 38/41

39 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations VIII Eq. 52 is a nonlinear generalized eigenvalue problem of the form Frxsx λsx (54) where F is the potential matrix and S is the overlap matrix Eq. 54 can only be solved iteratively until convergence to a fixed point is achieved, since F depends on the solution x 39/41

40 Problem Statement First Quantization Second Quantization References The Hartree-Fock Equations IX The ground state energy E 0 can be found using E 0 Nÿ ν 1 ɛ ν 1 2 ÿ ijkl ˆ V ijkl 1 2 V ilkj P ij P kl (55) The second term in Eq. 55 has to be subtracted, since the two-electron integrals are counted double 40/41

41 Problem Statement First Quantization Second Quantization References References Thijssen, Jos. Computational Physics, Cambridge university press, 2007 Giuliani, Gabriele, and Giovanni Vignale. Quantum theory of the electron liquid, Cambridge university press, 2005 Troyer, Matthias. Computational quantum physics, University lecture, 2015 Esslinger, Tilman. Experimental and theoretical aspects of quantum gases, University lecture, 2014 Moore, Michael. Phy851, University lecture, /41