Introduction to Second-quantization II...Contents, lecture II Spin in second quantization, Form of...

Introduction to Second-quantization II

Jeppe Olsen

Lundbeck Foundation Center for Theoretical ChemistryDepartment of Chemistry, Aarhus

July 3, 2012

Jeppe Olsen (Aarhus) Second Quantization II July 3, 2012 1 / 33

Contents, lecture II

Spin in second quantization, Form of spin-free one- and two-electronoperators, sections 2.1-2.2.2

A bit of tensor operators, sections 2.3.1, 2.3.2, part of 2.3.4

Rotations using exponentials I: first quantization, section 3.1

Rotations using exponentials II: second quantization, sections 3.2, 3.3

Density matrices, section 2.7


Spin in second quantizationSpin-functions for one electron, first quantization

Two spin-functions, α(ms), β(ms)

ms can take the two values12 ,−12

α(12 ) = 1, α(−12 ) = 0β(12 ) = 0, β(−12 ) = 1

Spin properties, atomic units, so ~ = 1

Scz α =1

2α (Sc )2α =

3

4α

Scz β = −1

2β (Sc)2β =

3

4β

α is ’spin-up’, β is ’spin-down’


Spin in second quantizationSpin-functions for one electron, first quantization, cont’d

Integration over spin

Easy:∫dms f (ms) = f (

12 ) + f (−12)

α, β constitute an orthonormal basis1∫dms α

⋆(ms)α(ms) =∫dms β

⋆(ms)β(ms) = 12∫dms α

⋆(ms)β(ms) = 0

(easily checked from definitions)


Spin in second quantizationSpin orbitals in first and second quantization

First quantization: separate functions for spatial- and spin-parts, e.g.φp(r)α(ms)

Second quantization: Both spatial and spin-parts created by oneoperator for example a†pα

Ground state of H2: a†1σgα

a†1σgβ|vac〉

Anti-commutation relation

[a†pσ, aqτ ]+ = δpqδστ


Spin in second quantizationVarious types of operators

Spin-free operators, first quantization

Operator that does not change spin-functions

f cα = αf c f cβ = βf c (1)

Operators like kinetic energy and Coulomb-repulsion depend only onspatial coordinates → spin-free

Pure spin-operators, first quantization

Operators that does not change spatial functions

f cφp(r) = φp(r)fc (2)

Operators like Sc+,Sc−, (S

c)2 are of this type

Mixed operators, first quantization

Changes both spatial and spin functionsJeppe Olsen (Aarhus) Second Quantization II July 3, 2012 6 / 33

Spin in second quantizationOne-electron operators in SQ, general form

Do you remember?

ĥ =∑

PQ

hPQa†PaQ

hPQ =

∫

dx φ⋆P(x) hc(x) φQ(x)

Separation of spatial and spin parts

P → pσ, Q → qτ

ĥ =∑

pσqτ

hpσqτa†pσaqτ

hpσqτ =

∫

dr

∫

dms φ⋆p(r)σ

⋆(ms) hc(r,ms ) φq(r)τ(ms )


Spin in second quantizationExample of pure spin-operator in SQ: Ŝ+

Form

Ŝ+ =∑

pσqτ

(S+)pσqτ a†pσaqτ

Sc+ in FQ

Sc+β = α, Sc+α = 0

Sc+φp = φpSc+ (S

c+ is a pure spin-operator)

The integrals (S+)pσqτ

(S+)pσqτ =

∫

dr

∫

dms φ⋆p(r)σ

⋆(ms)Sc+(ms)φq(r)τ(ms )

=

∫

dr φ⋆p(r)φq(r)

∫

dms σ⋆(ms)S

c+(ms)τ(ms) = δpqδσαδτβ


Spin in second quantizationExample of pure spin-operator in SQ: Ŝ+, cont’d

((S+)pσqτ = δpqδσαδτβ)

So

Ŝ+ =∑

pσqτ

(S+)pσqτ a†pσaqτ

=∑

pσqτ

δpqδσαδτβa†pσaqτ

=∑

p

a†pαapβ

Corresponds to intuitive picture: changes β to α


Spin in second quantizationŜ,Ŝz , Ŝ

2

In a similar way one obtains

Ŝ− =∑

p a†pβapα

Ŝz = 1/2(∑

p a†pαapα − a†pβapβ)

We may then obtain Ŝ2

Ŝ2 = Ŝ+Ŝ− + Ŝz(Ŝz − 1)


Spin in second quantizationForm of spin-free one-electron operator in SQ

The integrals hpσqτ

hpσqτ =

∫

dr

∫

dms φ⋆p(r)σ

⋆(ms)hc(r)φq(r)τ(ms )

=

∫

dms σ⋆(ms)τ(ms)

∫

dr φ⋆p(r)hc (r)φq(r) = δστhpq

Introduce Epq =∑

σ a†pσaqσ = a

†pαaqα + a

†pβaqβ

(generator of the linear group)

The operator then become

ĥ =∑

pσqτ δστhpqa†pσaqτ =

∑

pq hpqEpq

hpq depends only on orbitals


Spin in second quantizationSpin-free Two-electron operator in SQ

General two-electron operator in SQ

ĝ =1

2

∑

pσ,qτ,rµ,sν

gpσ,qτ,rµ,sνa†pσa

†rµasνaqτ (3)

Simplification for spin-free two-electron operator

gpσ,qτ,rµ,sν = gpqrsδστ δµν

gpqrs =

∫

drdr′φ⋆p(r)φ⋆r (r

′)g c (r, r′)φq(r)φs (r′)

Gives spin-free two-electron operator

ĝ =1

2

∑

pqrs

gpqrs (∑

στ

a†pσa†rτasτaqσ)


Spin-free Two-electron operator in SQ, cont’d

A simple rewrite to obtain E-operators

ĝ =1

2

∑

pqrs

gpqrs (∑

στ

a†pσa†rτasτaqσ)

= −12

∑

pqrs

gpqrs(∑

στ

a†pσa†rτaqσasτ )

= −12

∑

pqrs

gpqrs(∑

στ

a†pσ(−aqσa†rτ + δqrδστ )asτ )

=1

2

∑

pqrs

gpqrs (∑

στ

a†pσaqσa†rτasτ − δqr

∑

σ

a†pσasσ)

=1

2

∑

pqrs

gpqrs (EpqErs − δqrEps) =1

2

∑

pqrs

gpqrsepqrs

Definition: epqrs = EpqErs − δqrEps = erspqJeppe Olsen (Aarhus) Second Quantization II July 3, 2012 13 / 33

Spin in second quantizationThe nonrelativistic Hamiltonian in SQ

Terms

One-electron part containing kinetic energy and nuclear attraction ←ĥ

Two-electron part containing electron-electron repulsion ← ĝSpin-free !

The form

Ĥ = ĥ + ĝ

=∑

pq

hpqEpq +1

2

∑

pqrs

gpqrs(EpqErs − δqrEps)

=∑

pq

hpqEpq +1

2

∑

pqrs

gpqrsepqrs


Spin tensor operators

Idea

We are used to talk about the spin-properties of wave functions

A wave function may this be a singlet, doublet, triplet ...

Introduce a similar concept for operators

Definition

A spin tensor operator of rank S is a set of 2S + 1 operatorsT̂ S,M ,M = −S ,S related as

[Ŝ±, T̂S,M ] =

√

S(S + 1)−M(M ± 1)T̂ S,M±1

[Ŝz , T̂S,M ] = MT̂ S,M

Spin-free operator?

Corresponds to S = 0 (aka singlet)


Spin tensor operatorsExamples

Doublet tensor operators(S=1/2)

The pair of creation operators {a†pα, a†pβ}. a†pα → M = 1/2.

The pair of annihilation operators {−apβ, apα}. −apβ → M = 1/2.

One-electron singlet operator(S=0)

Ŝ0,0pq =

1√2(a†pαaqα + a

†pβaqβ) =

1√2Epq

One-electron Triplet operator(S=1)

T̂ 1,1pq = −a†pαaqβ

T̂ 1,0pq =1√2(a†pαaqα − a†pβaqβ)

T̂ 1,−1pq = a†pβaqα


Rotations using exponentials I: first quantizationOrbital-rotations using exponential mappings

Problem

A set of orthonormal spin-orbitals is given φ

Obtain all orthonormal spin-orbitals φ̃ that can be obtained as linearcombinations of φ

φ̃ = φU

〈φ̃P |φ̃Q〉 = δPQ

We want a simple parameterization of all orthonormal spin-orbitals φ̃

Occurs in SCF and MCSCF optimization

( Note: P ,Q now refers to spin-orbitals)


Rotations using exponentials I: first quantizationMay the elements of U be used as independent parameters?

No, because:

The required orthonormality of φ̃ requires that U is unitary

U†U = 1

The unitary conditions on U gives constraints on its elements

The elements of U can therefore not be used as independentparameters

A parameterization of unitary matrices may be obtained in terms ofexponential matrices


Rotations using exponentials I: first quantizationExponential matrices

Definition

exp(A) =

∞∑

n=0

1

n!A

n

= 1+ A+1

2A

2 + · · ·

Straightforward extension of Taylor-expansion of exp(x), where x is anumber

Converges for any choice of finite A ( assuming finite dimension)


Rotations using exponentials I: first quantizationExponential matrices, cont’d

Relations

exp(A) exp(−A) = 1exp (A)† = exp(A†)

B exp(A)B−1 = exp(BAB−1)

exp(A+ B) = exp(A) exp(B) iff [A,B] = 0

The Baker-Campbell-Hausdorf (BCH) expansion

exp(A)B exp(−A) = B+ [A,B] + 12[A, [A,B]]

+ · · ·+ 1n!

[A, [A, · · · , [A︸︷︷︸

n

,B] · · · ]] + · · ·Jeppe Olsen (Aarhus) Second Quantization II July 3, 2012 20 / 33

Rotations using exponentials I: first quantizationUnitary matrices as exponentials of antihermitian matrices

Any unitary matrix may be written as the exponential of anantihermitian matrix

U = exp(−κ) κ† = −κ(κ is antihermitian, USED HERE)U = exp(iκ) κ† = κ(κ is hermitian)

Relies on two theorems1 exp(−κ) is unitary for κ antihermitian2 All unitary matrices can be written in this form

1: exp(−κ) is unitary

exp(−κ)† exp(−κ) = exp(κ) exp(−κ)= 1


Rotations using exponentials I: first quantizationUnitary matrices as exponentials of antihermitian matrices, cont’d

2: All unitary matrices may be written as exp(−κ)A unitary matrix U may be diagonalized U = VǫV†, VV† = 1

ǫ is a complex diagonal matrix and |ǫi | = 1|ǫi | = 1→ ǫi = exp(iδi ), δi is realTherefore U = V exp(iδ)V† = exp(−(−iVδV†))−iVδV† is antihermitian, (−iVδV†)† = iVδV†


Rotations using exponentials I: first quantizationUnitary matrices as exponentials of antihermitian matrices, cont’d

What is so great about U = exp(−κ)It as easy to obtain a independent set of parameters for anantihermitian matrix

1 Elements at or below the diagonal are independent parameters2 Obtain the elements above the diagonal as κQP = −κ⋆PQ ,P > Q

A parameterization of all sets of orthonormal orbitals

φ̃← U← κ← κPQ ,P > Q

The energy as a function of orbitals

E = E (φ̃) = E (κlow ), for example the SCF energy

The energy is now a function of a set of independent parameters

Standard methods like Newton method may be used to optimize E

Of some use for HF, essential for MCSCFJeppe Olsen (Aarhus) Second Quantization II July 3, 2012 23 / 33

Rotations using exponentials II: second quantizationRotation of creation-operators

Rotation of orbitals → rotation of creation operatorsRotated orbitals: φ̃P =

∑

Q φQUQP , U = exp(−κ)A creation operator creating on electron in spin-orbital φ̃P :ã†P =

∑

Q a†QUQP

An ONV of rotated spin-orbitalsã†P1ã†P2· · · |vac〉 =∑Q1Q2··· a

†Q1a†Q2· · · |vac〉UQ1P1UQ2P2 · · ·

Complicated form

An operator form of ã†

ã†P =

∑

Q a†Q(exp(−κ))QP

We will show that ã†P = exp(−κ̂)a†P exp(κ̂)

κ̂ =∑

PQ κPQa†PaQ (κ̂ is an antihermitian one-electron operator)

The exponential of an operator is defined as the exponential of matrixand has the same properties and relations.Jeppe Olsen (Aarhus) Second Quantization II July 3, 2012 24 / 33

Rotations using exponentials II: second quantizationProof of the identity exp(−κ̂)a†P exp(κ̂) =

∑Qexp(−κ)QPa

†Q

The BCH expansion gives

exp(−κ̂)a†P exp(κ̂) = a†P + [a

†P , κ̂] +

1

2[[a†P , κ̂], κ̂] + · · ·

Calculating the commutator gives

exp(−κ̂)a†P exp(κ̂)

= a†P −∑

Q

κQPa†Q +

∑

Q

1

2!(κ)2QPa

†Q + · · ·

=∑

Q

a†Q(1− κ+

1

2!(κ)2 + · · · )QP

=∑

Q

exp(−κ)QPa†QJeppe Olsen (Aarhus) Second Quantization II July 3, 2012 25 / 33

Rotations using exponentials II: second quantizationRotation of creation-operators

A compact form of ONV with rotated operators

|ÕNV 〉 = ã†P1 ã†P2· · · ã†PN |vac〉

= exp(−κ̂)a†P1 exp(κ̂) exp(−κ̂)a†P2

exp(κ̂)

· · · exp(−κ̂)a†PN exp(κ̂)|vac〉= exp(−κ̂)a†P1a

†P2· · · a†PN |vac〉

= exp(−κ̂)|ONV 〉

|ÕNV 〉 = exp(−κ̂)|ONV 〉Applying a single exponential operator rotates all the spin-orbitals , usedfor generating all sets of orthonormal orbitals in SCF and MCSCF.


Rotations using exponentials II: second quantizationRestrictions on κ̂

Hitherto: unrestricted κ̂

κ̂ =∑

PQ κPQa†PaQ with κ general antihermitian matrix → complete

general spin-orbital transformation1 Allows the spin-orbitals to become complex (elements of κ may be

complex)2 α- and β-spin-orbitals are allowed to have different spatial parts3 α- and β-spin-orbitals are allowed to mix

Two types of restrictions

1 Restrict rotations to allow only real orbitals

2 Restrict rotations to ensure that spatial parts of α− andβ-spin-orbitals are identical



Restriction to real orbitals

Restrict κ to Rκ containing only real elementsRκ is an antisymmetric matrix, RκQP = −RκPQ , RκPP = 0

κ̂ =∑

PQ

RκPQa†PaQ =

∑

P>Q

RκPQa†PaQ +

∑

PQ

RκPQa†PaQ +

∑

P>Q

RκQPa†QaP =

∑

P>Q

RκPQ(a†PaQ − a

†QaP)

Restrictions on spin-orbitalsRκpαqβ =

Rκpβqα = 0 (no mixing of α,β-orbitals)Rκpαqα =

Rκpβqβ =Rκpq (identical rotations of α,β-orbitals)

κ̂ =∑

p>q

Rκpq(a†pαaqα + a

†pβaqβ − a†qαapα − a

†qβapβ)

=∑

p>q

Rκpq(Epq − Eqp)Jeppe Olsen (Aarhus) Second Quantization II July 3, 2012 28 / 33


Real and spin-conserving rotations of ONV

|ÕNV 〉 = exp(−κ̂)|ONV 〉= exp(−

∑

p>q

Rκpq(Epq − Eqp))|ONV 〉

This is the form that will be used in MCSCF and SCF lectures -with smallnotational differences.


Density matrices: One-electron orbital density matrix

A wave function and a spinfree 1-e-operator

|0〉 =∑

k

Ck|k〉 Ω =∑

pq

ΩpqEpq (4)

Expectation value of spin-free 1-e operator

〈0|Ω̂|0〉 =∑

pq

Ωpq〈0|Epq |0〉 (5)

Consists of two parts

Integrals Ωpq : Depends on actual operator, independent of C

Matrix-elements 〈0|Epq |0〉: Independent of operator, depends on C


Density matrices: One-electron orbital density matrix,

cont’d

(〈0|Ω̂|0〉 =∑pq Ωpq〈0|Epq |0〉)

The one-electron orbital density matrix, D

Dpq = 〈0|Epq |0〉Matrix with dimensions M by M

All the information about C needed to calculate expectation values ofspin-free one-electron operators

Data reduction: millions/billions of parameters in C→ M by Mmatrix

Diagonalization of D → natural orbitals


Density matrices: The density in real space

One-electron multiplicative operator

〈0|Ω̂|0〉 =∑

pq

∫

dr φ⋆p(r)Ωc(r)φq(r) Dpq

=

∫

dr Ωc(r)

(∑

pq

Dpqφ⋆p(r)φq(r)

)

=

∫

dr Ω̂c(r)ρ(r) (6)

The density is ρ(r) =∑

pq Dpqφ⋆p(r)φq(r)

Function in real space, function of coordinates of one electron

Sufficient to evaluate expectation values of multiplicative1-electron-operators

D allows the calculation of expectation values of all one-electronoperators as it the density matrix

Of importance for DFT...


Density matrices: Two-electron orbital density matrix

A wave function and a spinfree 2-e-operator

|0〉 =∑

k

Ck|k〉 Ω =1

2

∑

pqrs

Ωpqrsepqrs (7)

Expectation value of spin-free 2-e-operator

〈0|Ω̂|0〉 = 12

∑

pqrs

Ωpqrs〈0|epqrs |0〉 =1

2

∑

pqrs

Ωpqrsdpqrs

d is the 2-e orbital density matrix

Contains all information about C needed to calculate expectationvalues of spin-free 2-electron operators

Dimension M4

Cannot be explicitly constructed and stored for large molecules

Is therefore usually constructed on the flight when needed


Introduction to Second-quantization II...Contents, lecture II Spin in second quantization, Form of...

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