Second Quantization

Applications

Trond Saue Laboratoire de Chimie et Physique Quantiques CNRS/Universit´ede Toulouse (Paul Sabatier) 118 route de Narbonne, 31062 Toulouse (FRANCE) e-mail: [email protected]

Insight from Crawford & Schaefer acknowledged. Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 1 / 37 Where we stopped last time

Second quantization starts from field operators ψ†(1), ψ(1) sampling the electron field in space. It provides a very convenient language for the formulation and implementation of quantum chemical methods. Occupation number vectors (ONVs) are defined with respect to some (orthonormal) M orbital set {ϕp(r)}p=1 Their occupation numbers are manipulated using creation- and annihilation operators, † âp and âp, which are conjugates of each other. The algebra of these operators is summarized by anti-commutator relations

h † † i h † i âp, âq = 0; [âp, âq] = 0; âp, âq = δpq + + + and reflects the fermionic nature of electrons.

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 1 / 37 What about spin ?

By convention, the z-axis is chosen as spin-axis such that the electron spin functions 2 |s, ms i are eigenfunctions of ˆs and ˆsz 2 ˆs |s, ms i = s (s + 1) |s, ms i ; ˆsz |s, ms i = ms |s, ms i It is also convenient to introduce step operators ˆs+ = ˆsx + iˆsy and ˆs− = ˆsx − iˆsy p ˆs± |s, ms i = s (s + 1) − ms (ms ± 1) |s, ms ± 1i 1 1 1 1 1 Electrons are spin- 2 particles with spin functions denoted |αi = 2 , 2 and |βi = 2 , − 2 . The action of the spin operators is summarized by

2 ˆs ˆsz ˆs+ ˆs− 3 1 |αi 4 |αi 2 |αi 0 |βi 3 1 |βi 4 |βi − 2 |βi |αi 0

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 2 / 37 Spin in second quantization

We may separate out spin from spatial parts of the creation- and annihilation operators, giving

h † † i h † i 0 0 0 âpσ, âqσ0 = 0; [âpσ, âqσ ]+ = 0; âpσ, âqσ0 = δpqδσσ ; σ, σ = α or β + + We may also separate out spin in the electronic Hamiltonian. For the (non-relativistic) one-electron part we obtain

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 3 / 37 Spin in second quantization

For the (non-relativistic) two-electron part we obtain

1 X X 0 0 † † Hˆ = hϕ σϕ τ|gˆ|ϕ σ ϕ τ ia a a 0 a 0 2 2 p q r s pσ qτ sτ rσ pqrs στσ0τ 0

1 X X 0 0 † † = hϕ ϕ |gˆ|ϕ ϕ ihσ|σ ihτ|τ ia a a 0 a 0 2 p q r s pσ qτ sτ rσ pqrs στσ0τ 0

1 X X † † = hϕ ϕ |gˆ|ϕ ϕ iδ 0 δ 0 a a a 0 a 0 2 p q r s σσ ττ pσ qτ sτ rσ pqrs στσ0τ 0

1 X X X † † = hϕ ϕ |gˆ|ϕ ϕ ie ; e = a a a a 2 p q r s pq,rs pq,rs pσ qτ sτ rσ pqrs στσ0τ 0 στ Operator algebra

† † † † † † † apσaqτ asτ 0 arσ0 = −apσaqτ arσasτ = apσarσaqτ asτ − δqr δστ apσasτ .. shows that epq,rs = Epr Eqs − δrqEps

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 4 / 37 Hartree-Fock theory in second quantization Reference ONV and orbital classes

In ﬁrst quantization language the Hartree-Fock method employs a single Slater determinant as trial function. M In second quantization we start from some orthonormal orbital basis {ϕp}p=1 , which deﬁnes our Fock space, and build a reference ONV in that space

† † † |0i = â1â2 ... âN |vaci = |1, 1, 1, 1, 0,..., 0i | {z } | {z } N M−N

For further manipulations it is useful to introduce orbital classes:

I occupied orbitals: i, j, k, l,... I virtual (unoccupied) orbitals: a, b, c, d,... I general orbitals: p, q, r, s,...

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 5 / 37 Hartree-Fock theory in second quantization Hartree-Fock energy

HF X † 1 X † † E = h0|Hˆ |0i = h0| h â â + V â â â â |0i + V pq p q 2 pq,rs p q s r nn pq pq,rs One-electron energy HF X † E1 = hpqh0|âpâq|0i pq

I The operator âq tries to remove an electron to the right; this is only possible if orbital q is occupied. † I Likewise, the operator âp tries to remove an electron to the left; this is only possible if orbital p is occupied. I The final ONVs created left and right by these processes must be the same (to within a phase) for a non-zero inner product. We conclude HF X E1 = hii i

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 6 / 37 Hartree-Fock theory in second quantization Hartree-Fock energy

Two-electron energy HF 1 X † † E = V h0|â â â â |0i 2 2 pq,rs p q s r pq,rs

I Operators âr and âs both try to remove an electron to the right; orbitals r and s must be occupied, but not identical † † I Operators âp and âq both try to remove an electron to the left; orbitals p and q must be occupied, but not identical

I The ﬁnal ONVs created left and right by these processes must be the same (to within a phase) for a non-zero inner product. I There are two possibilites

HF 1 X n † † † † o E = V h0|â â â â |0i + V h0|â â â â |0i 2 2 ij,ij i j j i ij,ji i j i j i6=j The final expression is HF 1 X 1 X E = {V − V } = hϕ ϕ k ϕ ϕ i 2 2 ij,ij ij,ji 2 i j i j i6=j ij

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 7 / 37 Hartree-Fock theory in second quantization Stationarity condition

The Hartree-Fock energy is a functional of the occupied orbitals

HF X 1 X E [{ϕ }] = hϕ |hˆ|ϕ i + hϕ ϕ k ϕ ϕ i + V i i i 2 i j i j nn i ij .. and is minimized under the constraint of orthonormal orbitals

hϕi |ϕj i = δij This is normally done by the introduction of Lagrange multipliers

HF HF X L [{ϕi }] = E [{ϕi }] − λij {hϕi |ϕj i − δij } ij Is it possible to achieve minimization without constraints ?

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 8 / 37 Hartree-Fock theory in second quantization Parametrization

Suppose that we generate the optimized orbitals by transforming the initial orthonormal M set {ϕp}p=1 X ϕ˜p = ϕqcqp q

and use the expansion coeﬃcients {cqp} as variational parameters ? In order to preserve orthonormality the expansion coeﬃents must obey

X X ∗ X ∗ hϕ˜p|ϕ˜qi = hϕr crp|ϕs csqi = hϕr |ϕs i crpcsq = crpcrq = δpq rs rs | {z } r δrs ..which means that they must form a unitary (orthogonal) matrix for complex (real) orbitals: C †C = I 1 This adds 2 M(M + 1) constraints, and so we can not vary the coeﬃcients freely.

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 9 / 37 Hartree-Fock theory in second quantization Matrix exponentials

We can, however, circumvent these constraints by writing the matrix as an exponential of another matrix U = exp(A) You recall (I hope) that the exponential of a (complex or real) number is

∞ X ak exp(a) = ea = k! k=0 We have some simple rules, e.g.

eaeb = ea+b; ⇒ e−aea = 1

With matrices we have to be more careful, because, like operators, they generally do not commute.

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 10 / 37 Hartree-Fock theory in second quantization Matrix exponentials

In perfect analogy with numbers we deﬁne ∞ X Ak exp(A) = k! k=0 We next consider the product ∞ ∞ X X Am Bn exp(A) exp(B) = m! n! m=0 n=0 We rearrange to collect contribution of order k = m + n

∞ k m k−m ∞ k X X A B X 1 X k m k−m exp(A) exp(B) = = A B m! (k − m)! k! m k=0 m=0 k=0 m=0 With numbers, we obtain our desired result eaeb = ea+b by recognizing that k k X k m k−m k k! (a + b) = a b ; = m m m!(k − m)! m=0

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 11 / 37 Hartree-Fock theory in second quantization Matrix exponentials

With matrices this does not work, for instance (A + B)2 = A2 + AB + BA + B2 6= A2 + 2AB + B2 since, generally [A, B] 6= 0 However, [A, (−A)] = 0, so we can use this rule to obtain that exp(A) exp(−A) = I ; [exp(A)]−1 = exp(−A) It is also straightforward to show that exp(A)† = exp A†

A unitary matrix is deﬁned by U−1 = U† which is obtained by using an anti-Hermitian A A† = −A

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 12 / 37 Hartree-Fock theory in second quantization Exponential parametrization

We avoid Lagrange multipliers (constraints) by expressing the optimized orbitals as

X † ϕ˜p = ϕqUqp; U = exp (−κ); κ = −κ q

I will now show that this corresponds to writing the optimized HF occupation-number vector as

0˜ = exp (−κˆ) |0i

whereκ ˆ is an orbital rotation operator with amplitudes κpq

X † ∗ κˆ = κpqˆapˆaq; κpq = −κqp pq

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 13 / 37 Hartree-Fock theory in second quantization Exponential parametrization

We start by the expansion

˜ † † † 0 = exp (−κˆ) |0i = exp (−κˆ) a1 a2 ... aN |vaci Next, we insert exp (ˆκ) exp (−κˆ) = 1 everywhere

˜ † † † 0 = exp (−κˆ) a1 exp (ˆκ) exp (−κˆ)a2 exp (ˆκ) ... exp (−κˆ) aN exp (ˆκ) exp (−κˆ) |vaci † † † † † = ã1ã2 ... ãN exp (−κˆ) |vaci ; ãr = exp (−κˆ) ar exp (ˆκ) First, we note that

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 14 / 37 Hartree-Fock theory in second quantization Baker-Campbell-Hausdorﬀ expansion

Baker Campbell Hausdorﬀ

We next use the Baker-Campbell-Hausdorﬀ expansion

∞ 1 X 1 (k) exp(A)B exp(−A) = B + [A, B] + [A, [A, B]] + ... = [A, B] 2 k! k=0 Proof: We introduce f (λ) = exp(λA)B exp(−λA) and note that

I f (0) = B I f (1) = exp(A)B exp(−A) 0 1 00 I Taylor expand: f (1) = f (0) + f (0) + 2 f (0) + ...

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 15 / 37 Hartree-Fock theory in second quantization Transformed creation operator

† Using the BCH expansion with A = −κˆ and B = ar we get h i 1 h h ii ˜a† = exp (−κˆ) a† exp (ˆκ) = a† − κ,ˆ a† + κ,ˆ κ,ˆ a† − ... r r r r 2 r

h †i To evaluate the commutator κ,ˆ ar we use our rule

h i h i h i AˆBˆ, Cˆ = Aˆ Bˆ, Cˆ − Aˆ, Cˆ Bˆ + + ..which gives       h †i X h † †i X  † h †i h † †i  X † κ,ˆ ar = κpq apaq, ar = κpq ap aq, ar − ap, ar aq = κpr ap  + +  pq pq  | {z } | {z }  p  δqr 0 

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 16 / 37 Hartree-Fock theory in second quantization Transformed creation operator

We proceed to the next commutator h h †ii X h † i X † X 2 † κ,ˆ κ,ˆ ar = κpr κ,ˆ ap = κpr κqpaq = κ qr aq p pq q We start to see a pattern h i 1 h h ii ˜a† = a† − κ,ˆ a† + κ,ˆ κ,ˆ a† − ... r r r 2 r X 1 X = a† − κ a† + κ2 a† − ... r pr p 2 qr q p q X 1 = δ − κ + κ2 − ... a† pr pr 2 pr p p X † = ap {exp [−κ]}pr p

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 17 / 37 Hartree-Fock theory in second quantization Transformed creation operator

To connect to orbital rotations we recall the formula

a† = ψˆ†(r)ϕ (r)d3r p ˆ p ...from which we obtain

† X † X † 3 † 3 ˜a = a {exp [−κ]} = ψˆ (r)ϕ (r) {exp [−κ]} d r = ψˆ (r)ϕ ˜ (r)d r r p pr ˆ p pr ˆ r p p which provides the connection

˜ X 0 = exp (−κˆ) |0i ⇒ ϕ˜r = ϕp(r) {exp [−κ]}pr p

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 18 / 37 Density functional theory in second quantization

The central quantity of DFT is the (charge) density ρ (r) It is an observable and therefore expressible as an expectation value * N +

X 3 ρ (r) = −e Ψ δ (ri − r) Ψ i=1 In second quantization the charge density operator is

† 0 3 0 0 3 0 X D 3 0 E † ρˆ = −e ψˆ r δ r − r ψˆ r d r = −e ϕ δ r − r ϕ a a ˆ p q p q pq X † † = −e Ωpq (r) apaq;Ωpq (r) = ϕp (r) ϕq (r) pq Just as in Hartree-Fock we may choose an exponential parametrization for the Kohn-Sham determinant

˜0 = exp (−κˆ) |0i such that the charge density is parametrized as D E X † ˜ † ˜ X ρ˜(r, κ) = −e ϕp (r) ϕq (r) 0 apaq 0 = −e Ωpq (r) Dpq (κ) pq pq

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 19 / 37 Wave-function based correlation methods

Hartree-Fock theory is the starting point for wave-function based correlation methods in that E”exact” = E HF + Ecorr This is where second quantization really shows its teeth

The Conﬁguration Interaction (CI) method employs a linear parametrization

X a † 1 X ab † † |CI i = 1 + Cˆ |HF i ; Cˆ = c a a + c a a a a + ... i a i 4 ij a b j i ia ijab The Coupled Cluster (CC) method employs an exponential parametrization

X a † 1 X ab † † |CCi = exp Tˆ |HF i ; Tˆ = t a a + t a a a a + ... i a i 4 ij a b j i ia ijab

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 20 / 37 Matrix elements

Calculating the matrix element of a one-electron operator Ωˆ for a two-electron system (N = 2):

† † † hvac|as ar apaqat au|vaci

Based on the relations

† † âp |vaci = 0, ∀âp; hvac| âp = 0; ∀âp

Our strategy will be to move creation operators to the left and annihilation operators to the right, that is, we bring the operator string on normal-ordered form.

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 21 / 37 Matrix elements

We start by using our commutator rule h i h i h i AˆBˆ, Cˆ = Aˆ Bˆ, Cˆ − Aˆ, Cˆ Bˆ + + .. to obtain

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 22 / 37 Matrix elements

We next develop an analogous commutator rule h i h i h i Aˆ, BˆCˆ = Aˆ, Bˆ Cˆ − Bˆ Aˆ, Cˆ + + .. such that

Ωmn = hϕr ϕs |Ωˆ|ϕt ϕui = Ωrt δsu − Ωruδst − Ωst δru + Ωsuδrt We quickly run out of steam; we need more powerful tools !

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 23 / 37 Let us bring out some bigger guns...

(Wick’ed guys)

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 24 / 37 Normal ordering Deﬁnition n o Writing an operator string Oˆ on normal-ordered form Oˆ corresponds to moving all creation operators to the left and all annihilation operators to the left as if they all anticommuted, e.g. n † † o † † {apaq} = apaq; apaq = apaq

n † o † n † o † apaq = apaq; apaq = −aqap

A more complicated example is

n † † †o n † † †o n † † †o † † † as ar apaqat au = apas ar aqat au = − apat as ar aqau = apat auas ar aq The vacuum expectation value of a normal-ordered operator string is zero D n o E ˆ vac O vac = 0

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 25 / 37 Contraction

A contraction is deﬁned as xy = xy − {xy} There are four possible combinations

† † † † n † † o † † † † apaq = apaq − apaq = apaq − apaq = 0

apaq = apaq − {apaq}v = apaq − apaq = 0 † † n † o † † apaq = apaq − apaq = apaq − apaq = 0 † † n † o † † apaq = apaq − apaq = apaq + aqap = δpq

The only non-zero contraction occurs when an annihilation operator appears to the left of a creation operator.

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 26 / 37 Wick’s theorem

An operator string may be written as a linear combination of normal-ordered strings.

ABC ... XYZ = {ABC ... XYZ} X + ABC ... XYZ singles ( ) X + ABC ... XYZ doubles + ...

Only fully contracted terms contribute to vacuum expectation values.

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 27 / 37 Wick’s theorem: example

Returning to our one-electron expectation value we ﬁnd that

† † † † † † hvac|as ar apaqat au|vaci = hvac|as ar apaqat au|vaci

† † † + hvac|as ar apaqat au|vaci

† † † + hvac|as ar apaqat au|vaci Signs of fully contracted contributions are given by (−1)k where k is the number of crossing lines. We again obtain

† † † hvac|as ar apaqat au|vaci = δrpδqt δsu − δrpδquδst − δspδqt δru + δspδquδrt

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 28 / 37 Matrix elements

We have seen that any matrix element over a string of creation- and annihilation operators can be expressed as a vacuum expectation value and then evaluated using Wick’s theorem, e.g. ˆ X † † † Ωmn = hm|Ω|ni = Ωpqhvac|as ar apaqat au|vaci pq However, with an increasing number N of electrons the operator strings become long and the evaluation tedious. We need even bigger guns

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 29 / 37 Let us look at the vacuum state

We have seen that the vacuum expectation value of a normal-ordered string is zero. A prime example is

X 1 X hvac|Hˆ |vaci = hvac| h â†â + V â†â†â â |vaci = 0 pq p q 2 pq,rs p q s r pq pq,rs

(we dropped Vnn) The vacuum state can be deﬁned as the “empty” state

|vaci = |0, 0, 0,..., 0i ,

..alternatively as the occupation-number vector for which

ˆap |vaci = 0; ∀ˆap

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 30 / 37 Particle-hole formalism

Let us consider the occupation-number vector |0i corresponding to some reference determinant Φ0, e.g. the Hartree-Fock determinant. As before we introduce orbital classes with respect to this reference

I occupied orbitals: i, j, k, l,... I virtual (unoccupied) orbitals: a, b, c, d,... We observe the following † † aa |0i = ai |0i = 0; ∀aa, ai

† I with respect to the reference aa and ai act as annihilation operators † I their conjugates aa and ai act as creation operators † I aa creates an electron (particle), whereas ai creates a vacancy (hole) Using Wick’s theorem, we will express all operators in terms of normal-ordering with respect to the new reference, the Fermi vacuum. This also changes the zero of energy.

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 31 / 37 The normal-ordered electronic Hamiltonian One-electron part

Using Wick’s theorem the one-electron part of the Hamiltonian becomes

ˆ X † X † n † o H1 = hpqapaq = hpq apaq + apaq 0 0 pq pq Recall that the only non-zero contraction appears when a annihilation operator appears to the left of a creation operator This only happens when both p and q refer to occupied orbitals, giving X X X Hˆ = h a†a + δ δ = h a†a + h 1 pq p q 0 pq p∈i pq p q 0 ii pq pq i

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 32 / 37 The normal-ordered electronic Hamiltonian Two-electron part

For the two-electron part 1 X Hˆ = V a† a† a a 2 pq,rs p q s r pq,rs non-zero contractions only occur if p or q refer to occupied orbitals such that the † † corresponding operators ˆap and ˆaq are annihilators with respect to the Fermi vacuum. Non-zero double contractions are † † apaqas ar = −δp∈i δps δq∈j δqr 0 † † apaqas ar = δp∈i δpr δq∈j δqs 0

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 33 / 37 The normal-ordered electronic Hamiltonian Two-electron part

Non-zero single contractions are

n † † o n † † o † apaqas ar = − apas aqar = −δp∈i δps aqar 0 0 0

n † † o n † † o † apaqas ar = apar aqas = δp∈i δpr aqas 0 0 0

n † † o † apaqas ar = δq∈i δqs apar 0 0

n † † o n † † o † apaqas ar = − apaqar as = −δq∈i δqr apas 0 0 0

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 34 / 37 The normal-ordered electronic Hamiltonian ..almost there From the non-zero double contractions we get

1 X 1 X HF V (δ δ δ δ − δ δ δ δ ) = (V − V ) = E 2 pq,rs p∈i pr q∈j qs p∈i ps q∈j qr 2 ij,ij ij,ji 2 pq,rs ij From the non-zero single contractions we get

1 X n † o n † o Vpq,rs δp∈i δpr aqas − δp∈i δps aqar 2 0 0 pq,rs

1 X n † o n † o + Vpq,rs δq∈i δqs apar − δq∈i δqr apas 2 0 0 pq,rs

1 X n † o 1 X n † o = Viq,is aqas − Viq,ri aqar 2 0 2 0 iq,s iq,r

1 X n † o 1 X n † o + Vpi,ri apar − Vpi,is apas 2 2 0 pi,r pi,s

X n † o = (Vpi,qi − Vpi,iq) apaq 0 pq,i,

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 35 / 37 The normal-ordered electronic Hamiltonian Final form

The ﬁnal form of the electronic Hamiltonian is ! ˆ HF X X n † o 1 X n † † o H = E + hpq + (Vpi,qi − Vpi,iq) apaq + Vpq,rs apaqas ar 0 2 0 pq i pq,rs HF = E + HˆN where appears the HF energy

HF X 1 X E = h0|Hˆ |0i = h + (V − V ) ii 2 ij,ij ij,ji i ij and the normal-ordered electronic Hamiltonian

ˆ X n † o 1 X n † † o ˆ ˆ HN = fpq apaq + Vpq,rs apaqas ar = H − h0|H|0i 0 2 0 pq pq,rs

This result can be generalized: Ωˆ N = Ωˆ − h0|Ωˆ|0i

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 36 / 37 Final words

The second quantization formalism provides a powerful language for the formulation and implementation of quantum chemical methods Matrix elements over second quantized operators split into integrals over the operator in the chosen orbital basis and a vacuum expectation value. For the formulation of wave-function based electron correlation methods second quantization becomes an indispensable tool. Further sophistication is provided by Wick’s theorem, the particle-hole formalism and ...

Trond Saue (LCPQ, Toulouse) Second quantization ESQC 2019 37 / 37