Second Quantization

Home , First quantization, Fock space, Slater determinant

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

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

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 . • R=1 P R RQ P Q The second quantization" operators are projections of the • exact operators onto a basis of spin orbitals. For an incomplete basis, the second quantization representation depends on when the projection is made. Heisenberg uncertainty principle

c c Position and momentum do not commute: [x , px] = iN. • c c Note that x and px contain sums over N electrons, and only the observables of the same electron do not commute.

What happens with [x,ˆ pˆ ] in second quantization? • x It follows that •

[xˆ, pˆx] = xP R(px)RQ (px)P RxRQ aP† aQ * { − }+ #P Q #R In a complete basis, we ﬁnd: [x,ˆ pˆ ] = iNˆ. • x

Expectation values

We are interested in the expectation value of a general • one- and two-electron Hermitian operator Ωˆ with respect to a normalized reference state 0 . | !

ˆ 1 Ω = Ω0 + ΩP QaP† aQ + 2 ΩP QRSaP† aR† aSaQ #P Q P#QRS 0 = c k , 0 0 = 1, 0 Ωˆ 0 = ? | ! k| ! " | ! " | | ! #k We write the expectation value as follows: • 1 0 Ωˆ 0 = Ω + Ω 0 a† a 0 + Ω 0 a† a† a a 0 " | | ! 0 P Q" | P Q| ! 2 P QRS" | P R S Q| ! #P Q P#QRS Density matrices

One-electron density-matrix elements: •

D¯ = 0 a† a 0 P Q " | P Q| ! Two-electron density-matrix elements: •

d¯ = 0 a† a† a a 0 P QRS " | P R S Q| !

Properties of the one-electron density matrix

D¯ is an M M positive semideﬁnite, Hermitian matrix. • × A diagonal element is referred to as the occupation number • ω¯ of the spin orbital φ (x) in the electronic state 0 , P P | ! 2 ω¯ = D¯ = 0 a† a 0 = 0 Nˆ 0 = k c P P P " | P P | ! " | P | ! P | k| #k Occup. numbers are real numbers between zero and one, • 0 ω¯ 1 ≤ P ≤ The trace of the density matrix is equal to the number of • electrons,

TrD¯ = ω¯ = 0 Nˆ 0 = 0 Nˆ 0 = N P " | P | ! " | | ! #P #P Natural spin orbitals

Since D¯ is a Hermitian matrix, we may diagonalize it with a • unitary matrix U, D¯ = Uη¯U†

The eigenvalues are real numbers 0 η¯ 1, known as • ≤ P ≤ natural-orbital occupation numbers. The sum of the these numbers is again equal to the number of electrons. The eigenvectors U constitute the natural spin orbitals. •

Properties of the two-electron density matrix

The elements of the two-electron density matrix d¯ are not • all independent,

d¯ = d¯ = d¯ = d¯ P QRS − RQP S − P SRQ RSP Q

We deﬁne the two-electron density matrix T¯ with elements •

T¯P Q,RS = d¯P RQS with P > Q, R > S

The diagonal elements ω¯ are pair-occupation numbers, • P Q

ω¯ = T¯ = 0 a† a† a a 0 = 0 Nˆ Nˆ 0 P Q P Q,P Q " | P Q Q P | ! " | P Q| ! = k k c 2 P Q| k| #k Operator rank

In the manipulation of operators and matrix elements in • second quantization, we often encounter commutators and anticommutators,

[A,ˆ Bˆ] = AˆBˆ BˆAˆ − ˆ ˆ ˆ ˆ ˆ ˆ [A, B]+ = AB + BA

The anticommutation relations of creation and annihilation • operators can be used to simplify commutators and anticommutators of strings of operators.

The particle rank of a string is the number of elementary • operators divided by two (e.g., the rank of a creation operator is 1/2 and the rank of a ON operator is 1).

Rank reduction

Rank reduction is said to occur when the rank of a • commutator or anticommutator is lower than the combined rank of the operators involved,

aP† aP + aP aP† = 1 The rank of the operator products is 1, the rank of the anticommutator is 0. Simple rule: • Rank reduction follows upon anticommutation of two strings of half-integral rank and upon commutation of all other strings.

[a† , a a ] = [a† , a ] a a [a† , a ] = δ a δ a P R S P R + S − R P S + P R S − P S R Useful operator identities

[A,ˆ BˆCˆ] = [A,ˆ Bˆ]Cˆ + Bˆ[A,ˆ Cˆ]

[A,ˆ BˆCˆ] = [A,ˆ Bˆ] Cˆ Bˆ[A,ˆ Cˆ] + − +

[A,ˆ BˆCˆ] = [A,ˆ Bˆ]Cˆ + Bˆ[A,ˆ Cˆ] = [A,ˆ Bˆ] Cˆ Bˆ[A,ˆ Cˆ] + + + −

For example:

[a† , a a† a ] = [a† , a ] a† a a [a† , a† a ] P R S T + P R + S T − R P S T = δ a† a a [a† , a† ] a a† [a† , a ] P R S T − R P S + T − S P T + 0 1 = δP RaS† aT + δP T aRaS†

Normal-ordered second-quantization operators

A normal-ordered string of second-quantization operators • is one in which we ﬁnd all annihilation operators standing to the right of all creation operators. As an example, consider the string a a† a a† , • P Q R S

a a† a a† = δ a a† a† a a a† P Q R S P Q R S − Q P R S = δ δ δ a† a δ a† a + a† a a† a P Q RS − P Q S R − RS Q P Q P S R = δ δ δ a† a δ a† a + δ a† a a† a† a a P Q RS − P Q S R − RS Q P P S Q R − Q S P R

All of the strings in the rearrangement are in normal order. • None of them contribute to the vacuum expectation value, • vac a a† a a† vac = δ δ " | P Q R S| ! P Q RS Normal-ordered operators (continued)

Consider the two wavefunctions c and d : • | ! | ! c = c Xˆ vac , d = d Xˆ vac | ! k k| ! | ! k k| ! #k #k The matrix element •

c a a† a a† d = c∗ d vac Xˆ †a a† a a† Xˆ vac " | P Q R S| ! k k" " | k P Q R S k" | ! #k,k"

ˆ † † † ˆ can be evaluated by rearranging XkaP aQaRaSXk" into normal order.

Contractions

A contraction between two arbitrary elementary operators, • for example between aP and and aQ† is deﬁned as

a a† = a a† a a† P Q P Q − { P Q}

where the notation a a† indicates the normal-orderer { P Q} string. Thus, the contraction between the operators is simply the • original ordering of the pair minus the normal-ordered pair. The notation . . . introduces a sign ( 1)p, where p is the • { } − number of permutations required to bring the operators into normal order. Contractions (continued)

Examples with two elementary operators: •

aP aQ = 0, aP† aQ† = 0, aP† aQ = 0,

a a† = a a† a a† = a a† + a† a = δ P Q P Q − { P Q} P Q Q P P Q

An example with more than two elementary operators: •

a a a† a† = δ a a† P R Q S − P Q R S A sign change occurs for every permutation that is required • until the contracted operators are adjacent to one another.

Full contractions

A string of operators is fully contracted, if all operators are • pairwise contracted. Only full contractions contribute to the vacuum expectation value.

An example with two contractions: •

a a a† a† = a a a† a† = a a† δ = δ δ P R Q S − P R S Q − P Q RS − P Q RS

Rule of thumb: the sign of a full contraction is negative if • the number of crossings is odd, else positive. Wick’s theorem

Wick’s theorem provides a recipe by which an arbitrary string of annihilation and creation operators, ABC...XY Z, may be written as a linear combination of normal-ordered strings. Schematically, Wick’s theorem is:

ABC...XY Z = ABC...XY Z { } + ABC...XY Z { } singles#

+ ABC...XY Z + ... { } doub#les where “singles”, “doubles”, etc. refer to the number of pairwise contractions.

Wick’s theorem (continued)

Applying Wick’s theorem to aP aQ† aRaS† yields:

a a† a a† = a a† a a† + a a† a a† P Q R S { P Q R S} { P Q R S}

+ a a† a a† + a a† a a† { P Q R S} { P Q R S}

+ a a† a a† { P Q R S}

= a† a† a a δ a† a + δ a† a − Q S P R − P Q S R P S Q R δ a† a + δ δ − RS Q P P Q RS

This result is identical to that obtained using the anticommutation relations. Wick’s theorem (continued)

Another example:

a† a a a† = a† a a a† + a† a a a† + a† a a a† P Q S R { P Q S R} { P Q S R} { P Q S R} = a† a† a a + δ a† a δ a† a P R Q S SR P Q − QR P S This result is also easily obtained using the anticommutation relations:

a† a a a† = a† a δ a† a a† a P Q S R P Q SR − P Q R S = a† a δ a† δ a + a† a† a a P Q SR − P QR S P R Q S

Application of Wick’s theorem

Consider the two one-electron states T = a† vac and • | ! T | ! U = a† vac . The matrix element T a a† a a† U is | ! U | ! " | P Q R S| ! evaluated by retaining only the fully contracted terms,

T a a† a a† U = vac a a a† a a† a† vac " | P Q R S| ! " | T P Q R S U | !

= vac a a a† a a† a† + a a a† a a† a† " |{ T P Q R S U } { T P Q R S U }

+ a a a† a a† a† + a a a† a a† a† vac { T P Q R S U } { T P Q R S U }| !

= δ δ δ + δ δ δ δ δ δ δ δ δ T U P Q RS T Q P S RU − T Q P U RS − T S P Q RU Generalized Wick’s theorem

The generalized Wick’s theorem provides a recipe by which we can evaluate a product of two normal-ordered strings,

ABC... XY Z... = ABC...XY Z... { }{ } { } + ABC...XY Z... { } singles#

+ ABC...XY Z... + ... { } doub#les Contractions need only be evaluated between normal-ordered strings and not within them.

Application of the generalized Wick’s theorem

Let us consider the product of the the strings aP† aQ and aR† aS, which both are in normal order:

a† a a† a = a† a a† a + a† a a† a P Q R S { P Q R S} { P Q R S} = a† a† a a + δ a† a − P R Q S QR P S

Of course, the same result is also obtained by inserting the anticommutation relation a a† = δ a† a into the product Q R QR − R Q aP† aQaR† aS. Fermi vacuum

In conﬁguration-interaction or coupled-cluster theories, it is • more convenient to deal with the N-electron reference determinant HF than with the true vacuum state vac . | ! | ! The evaluation of matrix elements using Wick’s theorem • were very tedious if one had to include the whole set of creation operators to generate HF from the true vacuum, | !

HF = a†a† a† a† ... vac | ! I J K L | !

We alter the deﬁnition of normal ordering from one given • relative to the true vacuum to one given relative to the reference state HF (Fermi vacuum). | !

Fermi vacuum and particle–hole formalism

When working on the Fermi vacuum, a hole is created by • the operator aI while a particle is created by aA† . We refer to operators that create or destroy holes and • particles as quasiparticle operators (q-operators). That is, q-annihilation operators are those that annihilate holes and particles (e.g., aI† and aA), and q-creation operators are those that create holes and particles (e.g., aI and aA† ). A string of second-quantization operators is normal • ordered relative to the Fermi vaccuum if all q-annihilation operators are standing to the right of all q-creation operators. Contractions in the particle–hole formalism

The deﬁnition of normal ordering relative to the Fermi • vacuum (denoted as “:...:”) changes the application of Wick’s theorem only slightly. The only nonzero contractions take place between q-annihilation operators that stand to the left of q-creation operators,

a†a = a†a :a†a : = a†a + a a† = δ I J I J − I J I J J I IJ

a a† = a a† :a a† : = a a† + a† a = δ A B A B − A B A B B A AB

aA† aB = aI aJ† = 0 All other combinations involve mixed hole and particle • indices for which the Kronecker delta functions give zero.

Wick’s theorem in the particle–hole formalism

Consider the overlap between two doubly-substituted determinants:

CD AB = HF a† a† a a a† a† a a HF KL IJ " | K L D C A B J I | ! 2 3 4 3 3

= HF a† a† a a a† a† a a + a† a† a a a† a† a a " | K L D C A B J I K L D C A B J I

+ a† a† a a a† a† a a + a† a† a a a† a† a a HF K L D C A B J I K L D C A B J I | !

= (δ δ δ δ ) (δ δ δ δ ) IK JL − IL JK AC BD − AD BC Normal-ordered one-electron operator

The molecular electronic Hamiltonian reads: • ˆ 1 H = hnuc + hP QaP† aQ + 2 gP QRSaP† aR† aSaQ #P Q P#QRS Applying Wick’s theorem to its one-electron term yields: •

hP QaP† aQ = hP Q:aP† aQ: + hP Q: aP† aQ : #P Q #P Q #P Q = hP Q:aP† aQ: + hII #P Q #I

Normal-ordered two-electron operator

We rewrite a† a† a a as: • P R S Q

aP† aR† aSaQ = :aP† aR† aSaQ:

+ : aP† aR† aSaQ : + : aP† aR† aSaQ : + : aP† aR† aSaQ :

Hence, • 1 1 2 gP QRSaP† aR† aSaQ = 2 gP QRS:aP† aR† aSaQ: P#QRS P#QRS 1 + (g g ) :a† a : + (g g ) IIP Q − IP QI P Q 2 IIJJ − IJJI I#P Q #IJ The normal-ordered electronic Hamiltonian

We note that: •

h + 1 (g g ) = E (Hartree–Fock energy) II 2 IIJJ − IJJI HF #I #IJ h + (g g ) = f (Fock-matrix element) P Q IIP Q − IP QI P Q #I We obtain: • ˆ 1 H = fP Q:aP† aQ: + 2 gP QRS:aP† aR† aSaQ: + EHF #P Q P#QRS ˆ ˆ ˆ H = FN + VN + EHF

Hˆ = Fˆ + Vˆ = Hˆ E N N N − HF

Brillouin’s theorem

A Let = a† a HF be a singly-substituted determinant. • I A I | ! The matrix element HF Hˆ A can be computed using 3 5 " | | I Wick’3s theorem, 5 A HF Hˆ = HF Hˆ a† a HF = HF Fˆ a† a HF " | | I " | A I | ! " | N A I | ! 5 = f HF :a† a : a† a HF P Q" | P Q A I | ! #P Q

= f HF a† a a† a HF P Q" | P Q A I | ! #P Q

= fIA = 0 (if Brillouin condition fulﬁlled) First-order interacting space

AB Similarly, for = a† a HF we obtain: • IJ A I | !

1 = g HF a† a† a a a† a† a a 2 P QRS" | P R S Q A B J I P#QRS

+ aP† aR† aSaQaA† aB† aJ aI + aP† aR† aSaQaA† aB† aJ aI

+ a† a† a a a† a† a a HF = g g P R S Q A B J I | ! IAJB − IBJA

Spin in second quantization

So far, we have used the upper-case index P to count spin • orbitals of the form

φP (x) = φpσ(r, ms) = φp(r)σ(ms)

m is the spin coordinate and the spin function σ(m ) is • s s either α(ms) or β(ms). The theory of second quantization can also be formulated • using the composite index pσ. For example, the anticommutator between creation and annihilation operators can be written as

[a† , a ] = δ = δ δ pσ qτ + pσ,qτ pq στ

With lower-case indices p, we count spatial orbitals φ (r). • p Spinfree one-electron operators

Consider the following spinfree (or spinless) operator: • N c c ˆ f = f (ri), f = fpσ,qτ ap†σaqτ i=1 pσqτ # #

c fpσ,qτ = φp∗(r)σ∗(ms)f (r)φq(r)τ(ms)dr dms ) ) c = δστ φp∗(r)f (r)φq(r)dr = δστ fpq ) The sum over spin functions in the second quantization • operator fˆ can be accounted for in the singlet excitation operator ˆ Epq = ap†αaqα + ap†βaqβ, f = fpqEpq pq #

Spinfree two-electron operators

The spinfree two-electron operator

c 1 c g = 2 g (ri, rj) i=j ## gives

1 gˆ = 2 gpσ,qτ,rµ,sν ap†σar†µasνaqτ pqrs στµν # # 1 = 2 gpqrs ap†σar†τ asτ aqσ pqrs στ # # = 1 g e , with e = E E δ E 2 pqrs pqrs pqrs pq rs − qr ps pqrs # Pure spin operators The representation of ﬁrst-quantization operators f c that work in spin space only may be written in the general form

ˆ c f = φp∗(r)σ∗(ms)f (ms)φq(r)τ(ms)dr dms ap†σaqτ pσqτ # ) c = σ∗(ms)f (ms)τ(ms)dms ap†σapτ στ p # ) # c c c Consider the operators Sz, S+ and S (the latter are known as step-up and step-down operators or ladder− operators).

Sc(m )α(m ) = 1 α(m ), Sc(m )β(m ) = 1 β(m ) z s s 2 s z s s − 2 s c c S (ms)α(ms) = β(ms), S (ms)β(ms) = 0 − − c c S+(ms)α(ms) = 0, S+(ms)β(ms) = α(ms)

Pure spin operators (continued)

ˆ 1 ˆ ˆ Sz = ap†αapα a† apβ , S+ = ap†αapβ, S = a† apα 2 − pβ − pβ p p p # 0 1 # # From these operators, it follows: ˆ ˆ ˆ ˆ S+† = S , S† = S+ − −

ˆ 1 ˆ ˆ 1 Sx = S+ + S = ap†αapβ + a† apα 2 − 2 pβ p 0 1 # 0 1 ˆ 1 ˆ ˆ 1 Sy = S+ S = ap†αapβ a† apα 2i − − 2i − pβ p 0 1 # 0 1 ˆ2 ˆ2 ˆ2 ˆ2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ S = Sx + Sy + Sz = S+S + Sz Sz 1 = S S+ + Sz Sz + 1 − − − 0 1 0 1

[Sˆ+, Sˆ ] = 2Sˆz − ROHF expectation value of Sˆ2 The expectation value of Sˆ2 with respect to a restricted open-shell Hartree–Fock (ROHF) reference state can easily be evaluated using Wick’s theorem in the particle–hole formalism.

ˆ ˆ HF S+S HF = HF ap†αapβa† aqα HF = Nα Nβ " | −| ! " | qβ | ! − pq # ˆ ˆ HF S S+ HF = HF a† apαaq†αaqβ HF = 0 " | − | ! " | pβ | ! pq # 1 1 HF Sˆ HF = HF a† a a† a HF = (N N ) " | z| ! 2 " | pα pα − pβ pβ | ! 2 α − β p # 2 1 1 2 HF Sˆ HF = HF a† a a† a + . . . HF = (N N ) " | z | ! 4 " | pα pα qα qα | ! 4 α − β pq # HF Sˆ2 HF = 1 (N N ) 1 (N N ) + 1 " | | ! 2 α − β 2 α − β 6 7

Mixed operators

Consider the (atomic) ﬁrst-quantization spin–orbit operator,

N N V c = V c (r , m ) = ξ(r ) "c(r ) Sc(m ) SO SO i si i i · si i=1 i=1 # # which in second quantization takes the form:

ˆ x ˆx y ˆy z ˆz VSO = VpqTpq + VpqTpq + VpqTpq pq with # 0 1 µ c Vpq = φp∗(r)ξ(r) *µ(r) φq(r) dr, (µ = x, y, z) ) and the triplet excitation operators

x 1 y 1 T = (a† a + a† a ), T = (a† a a† a ) pq 2 pα qβ pβ qα pq 2i pα qβ − pβ qα z 1 T = (a† a a† a ) pq 2 pα qα − pβ qβ One-electron density matrix

D = 0 E 0 = D¯ + D¯ , D = D∗ pq " | pq| ! pα,qα pβ,qβ pq qp Orbital occupation numbers, • D = ω = ω¯ + ω¯ , 0 ω 2 pp p pα pβ ≤ p ≤ Natural occupation numbers, •

D = UηU†, 0 η 2 ≤ p ≤

Two-electron density matrix

d = 0 e 0 = 0 a† a† a a 0 = d¯ pqrs " | pqrs| ! " | pσ rτ sτ qσ| ! pσ,qσ,rτ,sτ στ στ # # Pair occupation numbers, • d = ω = ω¯ , 0 ω 2(2 δ ) ppqq pq pσ,qτ ≤ pq ≤ − pq στ # The spin-density matrix

The spin-density matrix is deﬁned as • T 1 1 D = 0 a† a a† a 0 = (D¯ D¯ ) pq 2 " | pα qα − pβ qβ| ! 2 pα,qα − pβ,qβ The spin-density matrix measures the excess of the • density of alpha electrons over beta electrons. Similarly, the spin occupation number • ωT = 1 (ω¯ ω¯ ) p 2 pα − pβ measures the excess of alpha over beta electrons in φp. The trace of DT yields the total spin projection, • T 1 TrD = 0 a† a a† a 0 = 0 Sˆ 0 2 " | pα pα − pβ pβ| ! " | z| ! p #