Second-Quantization Methods for Fermions Many-electron Slater determinant wavefunctions are specified in terms of or- thonormal one-electron spin-orbitals φi(x) = ψi(r)χi(ξ); ξ: spin coordinate, x: for r and ξ, the spin function χi: either α or β, defined by: α(1) = 1, α(−1) = 0 β(1) = 0, β(−1) = 1 For a system involving N electrons the total N-electron wave function repre- sented by a single Slater determinant is: 1 Φ = Det | φ (x )φ (x ) . φ (x ) | (N!)1/2 1 1 2 2 N N φ (x ) φ (x ) . φ (x ) 1 1 1 2 1 N 1 φ2(x1) φ2(x2) . φ2(xN ) = (N!)1/2 . φN (x1) φN (x2) . φN (xN ) 1 X = (−1)pP φ (x )φ (x ) . φ (x ) 1/2 1 1 2 2 N N (N!) P where P is a permutation operator upon the electron coordinates, p is the num- ber of interchanges in P , and the sum is over N! different permutations. The determinant representation of the N-electron wave functions warrants the properties: 1. Antisymmetric total wave function: Pauli’s exclusion principle. 2. The determinantal wave function equals zero if it has two identical columns, i.e. if two electrons occupy the same spin-orbital: Fermi statistics. 1 3. The total wave function is normalized by the factor 1 provided the (N!)1/2 one-electron wave functions are normalized, so that: Z Z Z 2 X 2 2 2 | φi(x) | dx = | ψi(r) | | χi(ξ) | dr = | ψi(r) | dr = 1 ξ=±1 since the functions {ψi} are orthonormalized. Z ∗ ∗ φi (x)φj (x)dx = δij Theorem: If F is a symmetric operator and Φa and Φb are normalized determinantal wavefunctions, then: Z Z Φ∗F Φ dτ 0 = (N!)1/2 Φ∗F φ (x )φ (x ) . φ (x )dτ 0 b a b a1 1 a2 2 aN N 0 with dτ = dx1dx2 ... dxN and the prime indicates that the multiple inte- gral includes a sum over the two values of each of the spin variables. A symmetric operator is one which is symmetric in the permutation of electron coordinates, including spin, so that PF = F . Z 1 Z X Φ∗F Φ dτ 0 = Φ∗F (−1)pP φ (x )φ (x ) . φ (x )dτ 0 b a 1/2 b a1 1 a2 2 aN N (N!) P 1 X Z = (−1)pP (P −1Φ∗)F φ (x ) . φ (x )dτ 0 1/2 b a1 1 aN N (N!) P 1 X Z = P Φ∗F φ (x ) . φ (x )dτ 0 (N!)1/2 b a1 1 aN N ZP = (N!)1/2 Φ∗F φ (x ) . φ (x )dτ 0 b a1 1 aN N −1 ∗ p ∗ since P Φb = (−1) Φb. The value of the integral in line three is the same for every permutation P and there are N! permutations in all. The normalization of the N-electron wavefunction can be immediately ver- ified: Z Z Φ∗Φ dτ 0 = (N!)1/2 Φ∗φ (x ) . φ (x )dτ 0 a a a a1 1 aN N Z X = (−1)pP φ (x ) . φ (x ) φ (x ) . φ (x )dτ 0 a1 1 aN N a1 1 aN N P Z = | φ (x ) |2| φ (x ) |2 ... | φ (x ) |2 dτ 0 = 1 a1 1 a2 2 aN N owing to the orthonormality of {φ }. ai 2 4. The determinantal functions are orthogonal. Assume that Φa and Φb differ by at least one one-electron wavefunction, i.e.: φ = φ , i 6= j, ai bi φ 6= φ aj bj Z Z Φ∗Φ dτ 0 = (N!)1/2 Φ∗φ (x ) . φ (x )dτ 0 a b a b1 1 bN N Z X = (−1)pP φ (x ) . φ (x ) φ (x ) . φ (x )dτ 0 a1 1 aN N b1 1 bN N P Z Z Z = | φ (x ) |2 dx ... φ (x )φ (x )dx ... | φ (x ) |2 dx = 0 a1 1 1 aj j bj j j aN N N If Φa and Φb differ in more than one function, that is, if the sets {φa} and {φb} differ in more than one component, clearly the same result will follow. The determinantal wavefuctions thus form an orthonormal set and, if we assume that this set is complete, any N-electron wavefunction Ψ can be expanded in an infinite series of the type: X Ψ = BaΦa (1) a 3 Creation and destruction operators for fermions: anticommutation properties Slater determinantal wavefunctions involving orthonormal spin-orbitals φi(x) can be represented in terms of products of creation operators on the so-called vacuum ket state vector | vaci: † † † −1/2 crcs . ct | vaci ↔ (N!) Det | φt . φsφr |≡| φt . φsφr | (2) (See Appendix 1 for the discussion of the order of spin-orbitals in eqn. ( 2).) There is a one-to-one correspondence between the Slater determinants and the † † † product of crcs . ct operating on | vaci. Fermi statistics and the antisymmetric properties of many-electron wavefunctions: fundamental anticommutation relations among the creation operators † † † † † † [cr, cs]+ ≡ crcs + cscr = 0 (3) † † † † Pauli’s exclusion principle: crcr = −crcr = 0. The Fermion destruction operator ci, which is the adjoint of the creation oper- † ator ci , can be thought of as annihilating an electron in the spin-orbital φi and is defined to yield zero when operating on the vacuum ket vector: ci | vaci = 0 (4) An operator c is the adjoint of c† if: † ∗ hφi | c | φji = (hφj | c | φii) . (5) This means that instead of destroying an electron in | φji we may create an electron in | φii and then take the complex conjugate of the matrix element. 4 The destruction and creation operators fulfill the following two anticommuta- tion relations: [ci, cj]+ ≡ cicj + cjci = 0 (6) † † † [ci, cj]+ ≡ cicj + cjci = δij (7) For a set of non-orthonormal spin-orbitals {φi} the last equation is replaced by: † † † −1 −1 [ci, cj]+ ≡ cicj + cjci = hφi | φji = (Sij) (8) 5 Analysis of the anticommutation relations eqs. (3), (6) and (7) 1. The anticommutation relation eqn. (3) † † † † † † [ci , cj]+ ≡ ci cj + cjci = 0 means that if we first create an electron in the spin-orbital φi and then in φj the result is: † † −1/2 cjci | vaci ↔ (N!) Det | φiφj | , (9) whereas if an electron is first created in the spin-orbital φj and then in φi the result is: † † −1/2 ci cj | vaci ↔ (N!) Det | φjφi | . (10) The two state vectors differ by the permutation of two spin-orbitals, there- fore the two determinants have opposite signs. Hence, the interpretation of the anticommutation relation eqn. (3) is in terms of the permutational symmetry of determinants. 2. The content of the anticommutation relation eqn. (7): † † † [ci, cj]+ ≡ cicj + cjci = δij † † • i = j cici + ci ci = 1 † A ket in which φi is present: cici gives zero (Pauli’s exclusion principle). † ci ci yields: † † † † † ki † † † † † ci cict cu . ci . cw | vaci = (−1) ci cici ct cu . cw | vaci (11) † where ki is the number of creation operators standing to the left of ci in the original ket. If this is, according to eqn. (7), equal to the original ket, then we must have: † † † † ki † † † † † ct cu . ci . cw | vaci = (−1) ci cici ct cu . cw | vaci ki † † † † = (−1) ci ct cu . cw | vaci (12) 6 † cici , when operating on a ket that does not contain φi, leaves that ket unchanged † ci ci, when acting on a ket in which φi is present, leaves that ket alone. † ci ci, when operating on a ket in which φi is not present, gives zero. † Hence ci ci tells whether orbital φi occurs in a ket, i.e. whether φi is occupied by an electron or not: occupation number operator † ni = ci ci Total number operator N defined as: X N = ni i When operating on any ket N gives as its eigenvalue the total number of electrons in that ket. † † • i 6= j, eqn. (7) cicj + cjci = 0 implies that: – ci operating on any ket that does not contain φi yields zero, since † † † ki † † † cict cu . cw | vaci = (−1) ct cu . cwci | vaci = 0 (13) † † by repeated use of cicj = −cjci and ci | vaci = 0. † † – When the ket contains φi, the two terms cicj and cjci cancel each other. † † – When the ket contains φj, the two terms cicj and cjci yield zero. † – For cjci acting on a ket that contains φi, † † † † † ki † † † † cjcict cu . ci cw | vaci = (−1) cjct cu . cw | vaci † † † † = ct cu . cjcw | vaci (14) which is simply a new ket with φi replaced by φj. 7 3. The meaning of the anticommutation relation eqn. (6) [ci, cj]+ ≡ cicj + cjci = 0 (15) • The action of cicj (i 6= j) on a ket in which φi and φj are present: † † † † † (ki+kj) † † † † † cicjct cu . ci . cj . cw | vaci = (−1) cicjcjci ct cu . cw | vaci is reduced to: (ki+kj) † † † † † (−1) ci(1 − cjcj)ci ct cu . cw | vaci † according to eqn. (7). The term, involving cjcj, vanishes because cj | vaci = 0, and hence we have: (ki+kj) † † † † (ki+kj) † † † (−1) cici ct cu . cw | vaci = (−1) ct cu . cw | vaci • If instead we consider the action of cjci, we obtain: † † † † † (ki+kj−1) † † † † † cjcict cu . ci . cj . cw | vaci = (−1) cjcici cjct cu . cw | vaci (ki+kj−1) † † † = (−1) ct cu . cw | vaci (16) which is opposite in sign to the result of the operation of cicj. Thus the statement cicj +cjci = 0 simply means that the effect of the destruction displays Fermion statistics.For i = j eqn. (6) reads cici = −cici = 0, which also expresses Pauli’s exclusion principle and Fermi statistics.