Second-Quantization Methods for Fermions

Many-electron Slater determinant wavefunctions are speciﬁed in terms of orthonormal 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 represented 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 number 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 integral 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:

(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 anticommutation 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, therefore 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.

8 How are the creation and destruction operators used in practical applications?

1. In perturbative expansions of N-electron wavefunctions or when attempt-

ing to optimize the spin-orbitals φi, Slater determinants are constructed starting from some ”reference determinants” by replacing certain spin- orbitals by other spin-orbitals. This replacement is achieved by using the

† replacement operator ci cj.

2. Computation of the expectation values of one- and two-electron operators.

The electron hamiltonian is represented as a sum of one- and two-electron terms:

0 H = Ho + H (17) where: XN Ho = h(xi) (18) i=1 N N 0 1 X X H = v(xi, xj) (19) 2 i6=j j Rewrite the hamiltonian for the electrons in terms of electron creation and destruction operators: If the total many-electron wave function is represented as a linear combination of Slater determinants in the basis of one-electron wavefunctions {| φii}, the single-particle operator Ho can be expressed with the help † of ci and cj as: X † X † Ho = hφi | h | φjici cj = hijci cj (20) i,j i,j Z ∗ hij = hφi | h | φji = φi (x)h(x)φj(x)dx . (21) The summation in eqn. (20) runs over all values of i and j.

The operator for the electron-electron interaction H0 acquires the form:

0 1 X † † H = hφiφj | v | φkφlici cjclck (22) 2 i,j,k,l

9 ZZ ∗ ∗ hφiφj | v | φkφli = φi (x1)φj (x2)v(x1, x2)φk(x1)φl(x2)dx1dx2 . (23) and the summation runs over all values of i, j, k, l. We point out that the order

of the destruction operators is clck in eqn. (22), whereas in the matrix element we have the reverse order, namely l comes after k.

10 To prove this we compare the matrix elements of the one- and two-electron operators eqs. (18) and (19), respectively, with those of eqs. (20) and (22) for any arbitrary combination of two N-electron single determinantal wavefunctions

Φa and Φb. We remind that if the matrix elements of two operators in a given basis are identical, the two operators are identical. The proof is performed in the simplest possible cases: in the case of a one-electron system for Ho and a two-electron system for H0.

1. Matrix elements of Ho for a one-electron system The two Slater determinants consist each one of just a single one-

electron spin-orbital, | φli or | φki, describing just one electron

with coordinate x1:

Φa = Det | φl(x1) |=| φl(x1)i (24)

Φb = Det | φk(x1) |=| φk(x1)i . (25)

The oﬀ-diagonal matrix element of Ho (eqn. (18)) equals:

hΦb | Ho | Φai = hΦb | h(x1) | Φai = hφk | h(x1) | φli (26)

On the other hand the matrix element of the operator eqn. (20) equals:

= hφk | h(x1) | φli (27)

because

cj | Φai = cj | φli = 0 for j 6= l

11 otherwise

cj | Φai = cj | φli =| vaci for j = l

and:

† ci | vaci =| φki = Φb for i = k

Just the terms with i = k and j = l survive. This illustrates

that the two representations of the one-electron operator Ho yield the same matrix elements, therefore they are identical for any one- electron system.

2. Matrix elements of H0 for a two-electron system

In this case the two Slater determinants are constructed out of two one-electron basis wavefunctions 1 Φ = Det | φ (x )φ (x ) |= √ [φ (x )φ (x ) − φ (x )φ (x )] a p 1 q 2 2 p 1 q 2 q 1 p 2 1 Φ = Det | φ (x )φ (x ) |= √ [φ (x )φ (x ) − φ (x )φ (x )] . b m 1 n 2 2 m 1 n 2 n 1 m 2 For the matrix element of the two-electron operator eqn. (19) we have:

2 2 0 1 X X hΦb | H | Φai = hΦb | v(xi, xj) | Φai 2 i6=j j

= hΦb | v(x1, x2) | Φai

1 = h[φ (x )φ (x ) − φ (x )φ (x )] | v(x , x ) | 2 m 1 n 2 n 1 m 2 1 2

× [φp(x1)φq(x2) − φq(x1)φp(x2)]i

12 1 = [hφ φ | v | φ φ i − hφ φ | v | φ φ i 2 m n p q m n q p

−hφnφm | v | φpφqi + hφnφm | v | φqφpi]

= hφmφn | v | φpφqi − hφmφn | v | φqφpi (28) which follows from:

v(x1, x2) = v(x2, x1)

hφmφn | v | φpφqi = hφnφm | v | φqφpi

Using eqn. (22) for the two-electron operator H0 we arrive at:

1 X † † = hφiφj | v | φkφlihΦb | ci cjclck | Φai (29) 2 i,j,k,l The integral in the last line has a non-zero value only if:

† † ci cjclck | Φai = ± | Φbi, i.e. if l, k = p, q and i, j = m, n

• For the choice k = p, l = q, j = n, i = m we have:

† † † † cmcncqcpΦa = cmcncqcp | φpφq |

† † = cmcncq | φq |

† † = cmcn | vaci

† = cm | φn |

= | φnφm |= −Φb (30)

For this choice the second integral in eqn. (29) equals -1:

† † † † hΦb | ci cjclck | Φai = hΦb | cmcncqcp | Φai = −1

13 • For k = q, l = p, j = n, i = m we have:

† † † † hΦb | ci cjclck | Φai = hΦb | cmcncpcq | Φai = 1

• For k = p, l = q, j = m, i = n we have:

† † † † hΦb | ci cjclck | Φai = hΦb | cncmcqcp | Φai = 1

• For k = q, l = p, j = m, i = n we have:

† † † † hΦb | ci cjclck | Φai = hΦb | cncmcpcq | Φai = −1 which follow from the commutation relations for the electron creation and destruction operators.

Hence for the operator eqn. (22) the non-zero contributions are due to the combinations:

k = p, l = q, j = n, i = m

k = q, l = p, j = n, i = m

k = p, l = q, j = m, i = n

1 hΦ | H0 | Φ i = [hφ φ | v | φ φ i − hφ φ | v | φ φ i b a 2 m n p q m n q p

− hφnφm | v | φpφqi + hφnφm | v | φqφpi

= hφmφn | v | φpφqi − hφmφn | v | φqφpi (31) which is identical with eqn. (28). Both operator deﬁnitions for H0, eqs. (19) and (22), have identical matrix elements, therefore they are identical for any two-electron system.

14 The electron hamiltonian in the second quantization representation:

0 H = Ho + H

XN 1 XN XN H = h(xi) + v(xi, xj) i=1 2 i6=j j N N X † 1 X † † H = hφi | h | φjici cj + hφiφj | v | φkφlici cjclck i,j=1 2 i,j,k,l=1

Appendix 1: The order of creating two spin-orbitals in the vacuum ket does not matter at the beginning:

The two Slater determinants diﬀer by a factor -1 (a phase factor) due to the anticommutation relation between the two Fermion creation operators:

† † † † ci cj = −ci cj for i 6= j .

Once we agree upon a certain order of the spin-orbitals in the Slater determinant, resulting from the operation of the creation operators on the vacuum state vector, we should consistently keep it. When we do this, the anticommutation relation is implicit in the Slater determinant created, since the Slater determinant changes sign upon exchange of two columns.

References

[1] S. Raimes Many-electron Theory, North-Holland Publ. Co., Amsterdam- London,1972.

15 [2] S. Raimes The Wave Mechanics of Electrons in Metals, North-Holland Publ. Co., Amsterdam, 1963.

[3] P. Jørgensen and J. Simons Second Quantization-Based Methods in Quan- tum Chemistry, Academic Press, New York, 1981.