Second Quantization Formalism⇤

Juan Carlos Paniagua

Departament de Ciència de Materials i Química Física & Institut de Química Teòrica i Computacional (IQTC-UB) Universitat de Barcelona

December, 2014 - March 2nd,2017

Contents

1 Introduction 1

2 The Fock space 2

3 Electron creation and annihilation operators 2 3.1 Number operators ...... 4 3.2 Anticommutation rules ...... 4

4 The many-electron hamiltonian in second quantization 5 4.1 Restricted spin-orbitals ...... 7

5 Non-ﬁxed-particle systems 7

6 Particles and holes 7

1 Introduction

Quantum chemists we are seldom concern with phenomena involving variations in the number of particles, so they do not need to resort to quantum ﬁeld theories to lay the foundations of their work (one exception are certain spectroscopic phenomena with require a quantum description of electromagnetic radiation). The standard quantum mechanic theory for material particles is then a suﬃcient basis for their purposes. In this

⇤This document was writen en 2014 and revised in 2017 for the subject Mathematical Foundations of Quantum Mechanics of the Máster Interuniversitario en Química Teórica y Modelización Computacional. It is published under an Attribution 4.0 Creative Commons International License (CC BY 4.0 ). You can download it from the Dipòsit Digital of the Universitat de Barcelona (http://hdl.handle.net/2445/107727). You are free to copy and redistribute the material, remix, transform, and build upon it for any purpose, provided that you give appropriate credit, supply a link to the license, and indicate if changes were made. See https://creativecommons.org/licenses/by/4.0/ for further details. 2 Juan Carlos Paniagua

theory, the wave function norm remains constant over time, which implies the conservation of the number of particles of every type. However, even in this context it is often convenient to use some tools of quantum ﬁeld theories —in particular, the creation and annihilation operators— to state certain mathematical developments, particularly in the study of solid-state and molecular electronic structure. This way of formulating the theory is known as second quantization formalism.1

2 The Fock space

Creation and annihilation operators are applications that, when applied to a state of an n-particle system, produce a state of an (n + 1)-andan(n 1)-particle system, respectively. Therefore they act in a broader Hilbert space that those considered so far, which is known as the Fock space ( ). If all of the variable-number F particles are of the same type the Fock space is the direct sum of every fixed-particle-number space.2 In the particular case of an electron system (or any system made of identical fermions) the Fock space is: a2 an = ⌦ ⌦ F H0 H1 H1 ···H1 ··· where and are, respectively, the zero-electron and one-electron Hilbert spaces. is a one-dimensional H0 H1 H0 space containing a normalized vector 0= that represents a state with no electrons (the vacuum state), which |i is different from the zero vector (0). Let us choose a normalized discrete basis set , , in . The { 1 ··· i ···} H1 set n of all n-electron Slater determinants { I } n I ( 1I nI ) ⌘ ··· an is then a normalized basis of ⌦ ,andthecollectionofallthesebasisforeveryvalueof ↵ n H1 0, 1 , 2 n I I ···{ I }··· is a normalized basis of . F When referring to state vectors as elements of the Fock space an occupation number representation is often n used, in which each basis vector I is identified by a sequence of occupation numbers ni that take the value 1 n for the spin-orbitals present in I and 0 for all the other:

n = n , n , with n = n I | 1 ··· i ···i i i X For instance, if are the ﬁrst n spin-orbitals of the one-electron basis set, then n = ( ) = 1 ··· n 0 1 ··· n 1, 1, 0, 0, . The vacuum state has all the occupation numbers equal to zero: = 0, 0, . In the | ··· ···i |i | ··· ···i ↵ n case of boson systems the occupation numbers can take any natural value (including zero). | {z } 3 Electron creation and annihilation operators

The annihilation operator ai of an electron in the spin-orbital i is conveniently defined in the occupation number representation as ⌫i b ai n1, ni, =( 1) ni n1, 1 ni, | ··· ···i | ··· ···i i 1 where ⌫ = n and ⌫ =0. The reason for the term “annihilation” will become clear by applying this i j=1 j 1 b definition to some particular cases: P a 1,n , n , = 0,n , n , 1 | 2 ··· i ···i | 2 ··· i ···i a1 0,n2, ni, =0 b | ··· ···i a2 0, 1, ni, = 0, 0, ni, b | ··· ···i | ··· ···i a2 1, 1, ni, = 1, 0, ni, b | ··· ···i | ··· ···i 1While in quantum electrodynamics the energy associated to the classical electromagnetic fields becomes a quantized observable, in the present formalism a certain type ofb quantization will emerge from the quantum wave functions, hence the term “second quantization”. 2One could wonder why to use direct products to build de Hilbert space of a many-particle system from the one-particle spaces and direct sums to express the Fock space in terms of fixed-particle spaces. In the first case we have a complex system that can be divided into different subsystems, and these are different from their union. On the other hand, the Fock space is the Hilbert space of a single system in which the number of particles is not a fixed parameter, as in standard quantum mechanics, but an observable that can take different values, may evolve in time, and may even not be well defined. In the Hilbert space containing the states of this system there are subspaces that correspond to different eigenvalues of the “number of particles operator”, in the same way that there are subspaces corresponding to different values of any other observable, and the direct sum of all of these subspaces gives the whole Hilbert space. Second Quantization Formalism (2014-2017) 3

That is, if the spin-orbital is occupied in n , n , then a annihilates an electron in that spin-orbital, i | 1 ··· i ···i i and it changes the sign of the many-electron state vector if i was in an even position among the occupied states. So, from an n-electron state we obtain an (n-1)-electron state. If is empty in n , n , the result b i | 1 ··· i ···i of applying ai to this many-electron vector is zero. When we represent the many-electron basis vector as a Slater determinant then the eﬀect of the annihilation operator ai overb a determinant containing i takes the form:

⌫i b ai ( j i k) =( 1) j i k ··· ··· ··· ··· E E b where ⌫i is again the position number of i minus 1, and i means that i is absent in the determinant. ⌫i is also the number of transpositions needed to bring i to the first position of the determinant, so that we can obtain the effect of ai by first bringing i to the first position and then dropping it from the determinant. The application of ai would give zero if i were not occupied in the determinant: b b ai j i k =0 ··· ··· E Likewise, the creation operator ai† of anb electron in the spin-orbital i is defined by

⌫i ai† n1,b ni, =( 1) (1 ni) n1, 1 ni, | ··· ···i | ··· ···i

Some examples reveal that thisb operator creates an electron in the spin-orbital i if this was empty, and introduces a change of sign if the creation takes place in an even position among the occupied spin-orbitals:

a † 0,n , n , = 1,n , n , 1 | 2 ··· i ···i | 2 ··· i ···i a1† 1,n2, ni, =0 b | ··· ···i a2† 0, 0, ni, = 0, 1, ni, b | ··· ···i | ··· ···i a2† 1, 0, ni, = 1, 1, ni, b | ··· ···i | ··· ···i Any many-electron basis vector canb be obtained from the vacuum state by successive application of creation operators: n1 ni n , n , = a † a † 0, 0, | 1 ··· i ···i 1 ··· i ···| ··· ···i ⇣ ⌘ ⇣ ⌘ In terms of Slater determinants the eﬀect of theb creation operatorb ai† over a product not containing i is:

⌫i ai† j i k =( 1) ( j b i k) ··· ··· ··· ··· E E and b ai† ( j i k) =0 ··· ··· E In the right hand side of the first equation we can bring i to the first position of the product by making ⌫i b transpositions, so that we also can say that ai† creates an electron in the spin-orbital i and in the first position of the determinant:

ai† j b i k = ( i j k) ··· ··· ··· E E Let us now show that ai† is theb adjoint of ai. From now on we will assume that the one-electron basis , , is orthonormal,butmostoftheresultsaregeneral.Wewanttoprovetheequality { 1 ··· i ···} b b n0 , n0 , a n , n , = a †(n0 , n0 , ) n , n , h 1 ··· i ···| i| 1 ··· i ···i i 1 ··· i ··· 1 ··· i ··· D E for any two sequences of occupation numbers n0 , n0 , and n , n , .Byusingtheabovedeﬁnition b { 1 ··· i ···}b { 1 ··· i ···} of ai† the right hand side member becomes

⌫i0 ( 1) (1 ni0 )n0 ,n1 1 n0 ,ni b 1 ··· i ··· since the many-electron basis vectors are orthonormal. Likewise, the left hand side member is

⌫i ( 1) nin ,n1 n ,1 n 10 ··· i0 i ···

This two expressions vanish unless n0 = n , n0 =1 n , in which case they coincide. 1 1 ··· i i ··· 4 Juan Carlos Paniagua

3.1 Number operators The product

a †a n i i ⌘ i is known as occupation number operator of the spin-orbital i for reasons that will now become evident: b b b ⌫ ⌫ ⌫ a †a n , n , = a †( 1) i n n , 1 n , =( 1) i n ( 1) i (1 (1 n )) n , n , i i | 1 ··· i ···i i i | 1 ··· i ···i i i | 1 ··· i ···i 2 Since nbi canb only take the valuesb 1 and 0, ni = ni and n n , n , = n n , n , i | 1 ··· i ···i i | 1 ··· i ···i That is, n , n , is an eigenvector of n ,anditseigenvalueistheoccupationnumberofthestate . | 1 ··· i ···i b i i Occupation number operators are self-adjoint: b n0 , n0 , a †a n , n , = a (n0 , n0 , ) a n , n , = a †a (n0 , n0 , ) n , n , 1 ··· i ··· i i 1 ··· i ··· h i 1 ··· i ··· | i| 1 ··· i ···i i i 1 ··· i ··· | 1 ··· i ··· D E D E and they commute among themselves, since, for i = j, b b b 6 b b b

ai†aiaj†aj = ai†aj†ajai = aj†ajai†ai

Their eigenvalues univocally determineb ab completeb b b setb ofb stateb vectors,b b b sob that they are a CSCO. On the other hand they are idempotent (n 2 = n for the basis set n , n , ), so that they are projection operators. i i {| 1 ··· i ···i} ni projects onto the the subspace spanned by the many-electron basis vectors in which i is occupied. The sum of occupation numberb b operators for every spin-orbital is known as the electron number operator for obviousb reasons: n = ni i X b b n n , n , = n n , n , = n n , n , | 1 ··· i ···i i | 1 ··· i ···i | 1 ··· i ···i i X Diﬀerent n do not projectb onto orthogonal subspaces, since n n n , n , =0if n = n =1,so, i i j | 1 ··· i ···i 6 i j according to theorem 7, n is not a projection operator. 3 n In general,b linear combinations of Slater determinants , suchb asc Conﬁguration Interaction wavefunctions, are not eigenfunctions ofb the occupation number operators, but their expected value can still be used to assign an occupation number to each spin-orbital in the wave function, also referred to as the population of the spin-orbital:

n n n n n n ni n = ni = CI I ni CJ J = CI⇤CJ I ni J h i h | i * | + h | i XI XJ XIJ n n n b b b Since ni J =1 J if J contains i and vanishes otherwise, we can restrict the summation over J to the determinants containing that spin-orbital: b n n 2 n n = C⇤C = C 1 h ii I J h I | J i | I |  I,J i I i X3 X3 Thus, populations are, in general, less than 1 for multiconﬁgurational wavefunctions. Since all of the n-electron determinants are eigenfunctions of the electron number operator n with eigenvalue n,thesameapplieston .

3.2 Anticommutation rules b Electron creation and annihilation operators fulﬁll the following anticommutation rules:

ai, aj† = ij [ai, aj]+ = ai†, aj† =0 + + h i h i b b b b b b where the anticommutator is deﬁned as A, B AB + BA. + ⌘ Let us prove the ﬁrst rule. For i

⌫ ⌫ a a † + a †a n , n , = a ( 1) j (1 n ) n , 1 n , + a †( 1) i n n , 1 n , i j j i | 1 ··· i ···i i j | 1 ··· j ···i j i | 1 ··· i ···i ⌫j ⌫i ⇣ ⌘ =(1) (1 nj)( 1) ni n1, 1 ni, 1 nj, b b b b b | ··· ···b ···i +( 1)⌫i n ( 1)⌫j0 (1 n ) n , 1 n , 1 n , =0 i j | 1 ··· i ··· j ···i since ⌫0 = ⌫ 1,dependingonn being 0 or 1 respectively. For i = j j j ± i ⌫ a a † + a †a n , n , = a ( 1) i (1 n ) n , 1 n , + n n , n , i i i i | 1 ··· i ···i i i | 1 ··· i ···i i | 1 ··· i ···i ⇣ ⌘ 2⌫i 2 =(1) (1 ni) n1, ni, + ni n1, ni, = n1, ni, b b b b b | ··· ···i | ··· ···i | ··· ···i since n or 1 n must vanish. i i According to these rules, if we commute a pair of annihilation or creation operators we have to introduce a change of sign; that is, those pairs of operators anticommute:

a a = a a a †a † = a †a † i j j i i j j i If the two operators are of either type then b b b b b b b b

a a † = a †a i j ij j i so that they anticommute if they correspond to diﬀerent states: b b b b a a † = a †a for i = j i j j i 6 but ai does not anticommute (nor commute)b b with abi†: b

a a † =1 a †a b i i b i i An immediate consequence of these rules is that we cannot create two of them in the same state: b b b b 2 1 ai† = ai†, ai† =0 2 + ⇣ ⌘ h i in accordande with the fermionic character ofb the electrons.b b Exercise Use the second quantized form of Slater determinants to show that ( i j) ( k l) = ikjl iljk. h | i Hint:Writethedeterminantsascreationoperatorsactingonthevacuumstate;thenmovethecreationoperators from the left to the right-hand side of the scalar product; then move the resulting annihilation operators to the right until they operate directly on the vacuum state.

4 The many-electron hamiltonian in second quantization

We will now obtain an expression of the hamiltonian operator of a many-electron system in terms of creation and annihilation operators that is independent of the number of electrons in the system. This makes it quite convenient for some mathematical developments and, in particular, for inﬁnite systems such as solids. The non-relativistic electronic hamiltonian of a system with n electron and N nuclei is a sum of one-electron and two-electron terms: n n 1 n 1 nH = h(i)+ r i=1 i=1 j=i+1 ij X X X 2 N Z b where h(i)= ri A . We want to show that the second quantization formalism allows to put it in 2 A=1 riA the form b P 1 H = h a †a + g a †a †a a rs r s 2 rstu r s u t rs rstu X X b b b b b c b h = h g = (1) (2) 1 (1) (2) nH H where rs r s and rstu r s r12 t u .Tobeprecise, is the projection of onto the n-electronD subspaceE of ,alsoreferredtoastheD restrictionE of H to that subspace. The sums extend b F b b over the spin-orbitals of the one-electron basis ,therebeingnoreferenceton. { r} b 6 Juan Carlos Paniagua

n an To prove the preceding statement we will show that H and H (restricted to ⌦ )havethesamematrix H1 elements for a given n-electron basis set. Let us first consider the one-electron part of H. Its matrix element are: b b b ( k0 l0 ) hrsar†as ( k l) = hrs ar ( k0 l0 ) as ( k l) * ··· ··· + ··· | ··· rs rs X X ⌦ ↵ This vanishes unless r k0 l0 bandbs k l . If both n-electronb basis vectorsb are equal the only non- 2{ ··· } 2{ ··· } vanishing terms in the double sum correspond to r = s,andthecorrespondingdiagonalmatrixelementreduces to hrr ar ( k l) ar ( k l) = hrr ··· | ··· r k l r k l 2X{ ··· } ⌦ ↵ 2X{ ··· } b b If the two basis vectors differ in one spin-orbital —say a in the former is replaced by b in the latter— then the only surviving term in the double sum is the one with r = a and s = b;thatishab. If there are two or more differing spin-orbitals the matrix element vanishes. These results are the Slater-Condon rules for the n one-electron-type operator i=1 h(i). Let us now consider the two-electron terms of the second quantized hamiltonian. For a diagonal matrix P element we have

1 1 ( k l) grstuar†as†auat ( k l) = grstu asar ( k l) auat ( k l) * ··· 2 ··· + 2 ··· | ··· rstu rstu X X ⌦ ↵ b b c b b b c b The terms in this sum vanish unless r = t k l and s = u k l or r = u k l and s = t 2{ ··· } 2{ ··· } 2{ ··· } 2 k l . Thus, this diagonal element reduces { ··· } 1 grsrs asar ( k l) asar ( k l) + grssr asar ( k l) aras ( k l) 2 ··· | ··· ··· | ··· r,s k l 2X{ ··· } ⌦ ↵ ⌦ ↵ 1 b b b b b b b b = g g = g g 2 rsrs rssr rsrs rssr r,s k l r k l s k l ,s>r 2X{ ··· } 2X{ ··· } 2{ X··· } where we have taken into account the anticommutativity of the annihilation operators. Similar deductions can be applied for non-diagonal matrix elements. The expressions resulting for all these matrix elements are the n 1 n 1 Slater-Condon rules for the two-electron-type operator i=1 j=i+1 rij . n The diagonal matrix element of H corresponding to theP Hartree-FockP Slater determinant 0 is the Hartree- Fock energy: b HF n n n n 1 n n E = H = h a †a + g a †a †a a 0 0 rs 0 r s 0 2 rstu 0 r s u t 0 rs rstu D E X D E X D E b occ occ n b b n 1 n b b c b n = h a †a + g a †a †a a ab 0 a b 0 2 abcd 0 a b d c 0 ab D E abcd D E X X c b n c b b b where the sums extend over the occupied spin-orbitals of 0.Althoughthenumberofelectronsdoesnot appear explicitly in this expression, it is implied in the list of occupation numbers of n = n , n , : 0 | 1 ··· i ···i n = i ni. In the above demonstration we have assumed that the spin-orbitals form a complete set. For computa- P r tional reasons a ﬁnite subset must be used, so that the second quantized hamiltonian is, in fact, the projection of the exact hamiltonian onto the subspace spanned by that subset. n n 1 n 1 h(i) Similar expressions to those obtained for i=1 and i=1 j=i+1 rij can be used to write the second quantized form of any one- or two-electron-type operator. P P P Exercise Use the anticommutation rules to show that a one-electron-type operator of an n-electron system, F = n i=1 f(i)= rs frsar†as,canbecastintotheformofatwo-electron-typeoperator: b P P 1 1 b bF = f a †a †a a = f a †a †a a n 1 rt su r s u t n 1 rt su r s u t rstu rstu X X b (these being restricted to the n-electron subspace).b b c b b b c b Second Quantization Formalism (2014-2017) 7

4.1 Restricted spin-orbitals

Usually the spin-orbitals r are chosen as products of an orbital r and a spin vector ↵ or . Then r h s =0 1 D E unless r and s have the same spin factor, and r s t u =0unless r and t on the one hand, and r12 b s and u on the other, have the same spin factor.D By carrying outE the scalar products of the spin factors we are left with scalar products involving only orbitals. Thus, for a closed-shell determinant the electronic hamiltonian takes the form: 1 H = h a †a + g a †a †a a rs r s 2 rstu r s⌧ u⌧ t rs rstu X X=↵, X ,⌧X=↵, b d d d c d c h = h g = (1) (2) 1 (1) (2) r s t u where rs r s , rstu r s r12 t u and the indexes , , and extend over the orbital basis.D As told before,E this basisD set must be truncatedE in practice to a ﬁnite number m,sothatwework on a 2m-dimensionalb subspace of 1. Then the sums over r, s, t and u in the preceding equation extend over H those m orbitals and the resulting second-quantized operator is an approximation to the true hamiltonian nH.

b 5 Non-ﬁxed-particle systems

It is clear that the second quantized operator of any observable in ﬁxed-particle quantum mechanics must contain an equal number of creation and annihilation operators, so that these should always appear in pairs of either type. However, single operators that create or annihilate photons are needed to study spectroscopic phenomena in which de quantum nature of light plays a relevant role, such as the spontaneous emission of radiation or the Raman scattering. Since photons have spin 1 they are bosons and the corresponding creation and annihilation operators are deﬁned otherwise (see, for instance, Quantum electrodynamics by José A. N. F. Gomes and Juan C. Paniagua, in Computational Chemistry: Structure, Interactions and Reactivity,ed.byS.Fraga.Studiesin Physical and Theoretical Chemistry, vol. 77 (B). Elsevier, Amsterdam (1992)).

6 Particles and holes

Electron creation and annihilation operators are sometimes referred to a Fermi vacuum (or Fermi sea )instead of the zero-electron vacuum. The Fermi vacuum is the independent-electron ground state, in which all the electrons occupy the lowest-energy spin-orbitals. The energy of the highest occupied spin-orbital is known as the Fermi level. The independent-particle excited states are identified by specifying their occupation number differences with respect to that state vector; that is, the holes created in the Fermi sea by the operators aa and the particles created above the Fermi level by operators such as ar†. That is, an operator that annihilates an electron below the Fermi level is viewed as a hole creation operator.Aholeactsasaparticle(aquasi-particlec ) with positive charge e,andaneutralpairformedbyanelectronandaholeinteractingbyelectrostaticattractionb is sometimes called an exciton. The language of particles and holes is common in solid-state theory, and it is also sometimes used for finite systems, particularly in the statement of some post-Hartree-Fock methods such as configuration interaction or coupled cluster.