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Second Quantization Second quantization Wim Klopper and David P. Tew Lehrstuhl für Theoretische Chemie Institut für Physikalische Chemie Universität Karlsruhe (TH) C4 Tutorial, Zürich, 2–4 October 2006 Introduction In the first quantization formulation of quantum mechanics, • observables are represented by operators and states by functions. In the second quantization formulation, the wavefunctions • are also expressed in terms of operators — the creation and annihilation operators working on the vacuum state. Operators (e.g., the Hamiltonian) and wavefunctions are • described by a single set of elementary creation and annihilation operators. The antisymmetry of the electronic wavefunction follows • from the algebra of the creation and annihilation operators. The Fock space Let φ (x) be a basis of M orthonormal spin orbitals, • { P } where x represents the electron’s spatial (r) and spin (ms) coordinates. A Slater determinant is a normalized, antisymmetrized • product of spin orbitals, φ (x ) φ (x ) . φ (x ) P1 1 P2 1 PN 1 φ (x ) φ (x ) . φ (x ) 1 ˛ P1 2 P2 2 PN 2 ˛ ˛ ˛ φP φP . φP = ˛ . ˛ | 1 2 N | √N! ˛ . ˛ ˛ . ˛ ˛ ˛ ˛ φ (x ) φ (x ) . φ (x ) ˛ ˛ P1 N P2 N PN N ˛ ˛ ˛ ˛ ˛ In Fock space (a linear ˛vector space), a determinant ˛is • represented by an occupation-number (ON) vector k , | ! 1 φP (x) occupied k = k1, k2, . , kM , kP = | ! | ! 0 φP (x) unoccupied $ The inner product Inner product between two ON vectors k and m : • | ! | ! M k m = δ = δ " | ! k,m kP mP P!=1 Applies also to the product between states with different • electron numbers. Resolution of the identity: 1 = k k • k | !" | For two general vectors in Fock space: • " c = c k , d = d k , c d = c∗ d | ! k| ! | ! k| ! " | ! k k #k #k #k The 2M -dimensional Fock space The Fock space F (M) may be decomposed as a direct • sum of subspaces F (M, N), F (M) = F (M, 0) F (M, 1) F (M, M) ⊕ ⊕ · · · ⊕ F (M, N) contains all M vectors for which the sum of the • N occupation numbers is N. ' ( The subspace F (M, 0) is the true vacuum state, • F (M, 0) vac = 0 , 0 , . , 0 , vac vac = 1 ≡ | ! | 1 2 M ! " | ! Creation operators The M elementary creation operators are defined by • k a† k , k , . , 0 , . , k = Γ k , k , . , 1 , . , k P | 1 2 P M ! p | 1 2 P M ! a† k , k , . , 1 , . , k = 0 P | 1 2 P M ! P 1 − with the phase factor Γk = ( 1)kQ p − Q!=1 Anticommutation relations take care of the phase factor. • An ON vector can be expressed as a string of creation • operators (in canonical order) working on the vacuum, M kP k = (aP† ) vac | ! % & | ! P!=1 Annihilation operators The M elementary annihilation operators are defined by • a k , k , . , 1 , . , k = Γk k , k , . , 0 , . , k P | 1 2 P M ! p | 1 2 P M ! a k , k , . , 0 , . , k = 0 P | 1 2 P M ! a vac = 0 P | ! with the same phase factor as before. Again, anticommutation relations take care of the • phase factor. a† is the Hermitian adjoint to a . These operators are • P P distinct operators and are not self-adjoint (Hermitian). Anticommutation relations The anticommutation relations constitute the fundamental • properties of the creation and annihilation operators: [aP , aQ]+ = aP aQ + aQaP = 0 [aP† , aQ† ]+ = aP† aQ† + aQ† aP† = 0 [aP† , aQ]+ = aP† aQ + aQaP† = δP Q All other algebraic properties of the second quantization • formalism follow from these simple equations. The anticommutation relations follow from the definitions of • aP and aP† given on the previous slides. Occupation-number operators The occupation-number (ON) operator is defined as • ˆ NP = aP† aP Nˆ k = a† a k = δ k = k k P | ! P P | ! kp1| ! P | ! ON operators are Hermitian (Nˆ † = Nˆ ) and commute • P P among themselves, [NˆP , NˆQ] = 0. The ON vectors are simultaneous eigenvectors of the • commuting set of Hermitian operators NˆP . The ON operators are idempotent projection operators, • 2 Nˆ = a† a a† a = a† (1 a† a )a = a† a = Nˆ P P P P P P − P P P P P P The number operator The Hermitian number operator Nˆ is obtained by adding • together all ON operators, M M M Nˆ = Nˆ = a† a , Nˆ k = k k = N k P P P | ! P | ! | ! P#=1 P#=1 P#=1 Let Xˆ be a string with creation and annihilation operators • with more creation than annihilation operators (the excess being N X , which can be negative). Then, [Nˆ, Xˆ] = N X Xˆ Nˆ commutes with a number-conserving string for which • N X = 0. Excitation operators The simplest number-conserving operators are the • elementary excitation operators ˆ P XQ = aP† aQ Xˆ P applied to k : • Q | ! P < Q Xˆ P k = δ δ Γk Γk . , 1 , . , 0 , . Q | ! P 0 Q1 P Q| P Q ! P > Q Xˆ P k = δ δ Γk Γk . , 0 , . , 1 , . Q | ! − P 0 Q1 P Q| Q P ! P = Q Xˆ P k = k k Q | ! P | ! Wavefunctions represented by operators Let φ (x) be a basis of M orthonormal spin orbitals. • { P } An arbitrary wavefunction (within the space spanned by all • Slater determinants that can be formed using these M spin orbitals) can be written as M kP ˆ c = ck k = ck (aP† ) vac = ckXk vac | ! | ! % & | ! | ! #k #k P!=1 #k An excitation operator can be applied to the above • wavefunction to yield another function, P Xˆ c = c! Q | ! | ! One-electron operators In first quantization, one-electron operators are written as • N c c f = f (xi) #i=1 The second-quantization analogue has the structure • ˆ c f = fP QaP† aQ, fP Q = φP∗ (x)f (x)φQ(x)dx #P Q ) The order of the creation and annihilation operators • ensures that the one-electron operator fˆ produces zero when it works on the vacuum state. One-electron operators: Slater–Condon rules occupied k fˆ k = f k a† a k = k f f " | | ! P P " | P P | ! P P P ≡ II #P #P #I k and k differ in one pair of occupation numbers: | 1! | 2! k = k , k , . , 0 , . , 1 , . , k | 1! | 1 2 I J M ! k = k , k , . , 1 , . , 0 , . , k | 2! | 1 2 I J M ! k fˆ k = Γk2 Γk1 f " 2| | 1! I J IJ k and k differ in more than one pair of occupation numbers: | 1! | 2! k fˆ k = 0 " 2| | 1! Two-electron operators In first quantization, one-electron operators are written as • N c 1 c g = 2 g (xi, xj) i=j ## The second-quantization analogue has the structure • 1 gˆ = 2 gP QRSaP† aR† aSaQ P#QRS The two-electron integral is • c gP QRS = φP∗ (x1)φR∗ (x2)g (x1, x2)φQ(x1)φS(x2)dx1dx2 ) ) The molecular electronic Hamiltonian ˆ 1 H = hnuc + hP QaP† aQ + 2 gP QRSaP† aR† aSaQ #P Q P#QRS with 1 ZαZβ hnuc = 2 rαβ α=β ## 1 Zα h = φ∗ (x) ∆ φ (x)dx P Q P − 2 − r Q * α α + ) # 1 g = φ∗ (x )φ∗ (x ) φ (x )φ (x )dx dx P QRS P 1 R 2 r Q 1 S 2 1 2 ) ) 12 First- and second-quantization operators compared First quantization Second quantization one-electron operator: one-electron operator: → c → i f (xi) P Q fP QaP† aQ " " two-electron operator: two-electron operator: → 1 c → 1 2 i=j g (xi, xj) 2 P QRS gP QRSaP† aR† aSaQ # " " operator independent of operator depends on → → spin-orbital basis spin-orbital basis operator depends on operator independent of → → number of electrons number of electrons exact operator projected operator → → Matrix elements in 2nd quantization Let • c = c k = c Xˆ vac | ! k| ! k k| ! #k #k d = d k = d Xˆ vac | ! k| ! k k| ! #k #k Then, • c Oˆ d = c∗ d vac Xˆ †OˆXˆ vac " | | ! k k" " | k k" | ! #k #k" Matrix elements become linear combinations of vacuum • expectation values. Note that Xˆk and Oˆ consist of strings of the same elementary creation and annihilation operators. Products of operators in 2nd quantization Recall: The (finite) matrix representation P of the operator • product P c(x) = Ac(x)Bc(x) is not equal to the product of the matrices A and B, P = AB ' Similarly, the product of the operators Aˆ and Bˆ in second • quantization requires special attention, c c ˆ A = A (xi), A = AP QaP† aQ #i #P Q c c ˆ B = B (xi), B = AP QaP† aQ #i #P Q P c = AcBc, Pˆ = ? Products in 2nd quantization (continued) c c c c c c c P = A B = O + T = A (xi)B (xi) #i 1 c c c c + 2 [A (xi)B (xj) + A (xj)B (xi)] i=j ## ˆ ˆ ˆ 1 P = O + T = OP QaP† aQ + 2 TP QRSaP† aR† aSaQ #P Q P#QRS c c OP Q = φP∗ (x)A (x)B (x)φQ(x)dx ) TP QRS = AP QBRS + ARSBP Q Using the anticommutation relations ˆ 1 T = 2 (AP QBRS + ARSBP Q)aP† aR† aSaQ P#QRS = AP QBRSaP† aR† aSaQ P#QRS = A B (a† a a† a δ a† a ) P Q RS P Q R S − RQ P S P#QRS = AP QaP† aQ BRSaR† aS AP RBRS aP† aS * + − * + #P Q #RS #P S #R ˆ ˆ = AB AP RBRQ aP† aQ − * + #P Q #R Operators in 2nd quantization are projections The final result for the representation of P c in second • quantization is ˆ ˆ ˆ P = AB + OP Q AP RBRQ aP† aQ * − + #P Q #R In a complete basis: ∞ A B = O .
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