M. A. Gusm~ao{ IF-UFRGS 1 FIP10601 { Text 10 Electronic Hamiltonian in second quantization When discussing the Hartree and Hartree-Fock approximations as well as DFT, we used wavefunctions that depended on coordinates of all the electrons in the system. Since solids are macroscopic, such wave functions, in general, cannot be determined. Even in the independent-electron approximation the functions would be Slater determinants of dimension equal to the number of electrons. In the usual formalism of many-body theory, it is convenient to avoid the coordinate representation, as we will see in the following. Occupation number representation Although Slater determinants are not solutions of the interacting-electron problem, they provide a basis for a useful representation. In such a basis, the noninteracting part of the Hamiltonian is diagonal. We will use this basis to rewrite the complete Hamiltonian, but in doing so we will avoid to work explicitly with functions of coordinates. Taking into account that the single-particle states appearing in Slater determinants are known (Bloch states for a given lattice potential), one just needs to inform which ones are involved, i.e. occupied by electrons, in a given global state of the system. The individual states are characterized by a wavevector k and a spin quantum number σ, being denoted by jkσi in the corresponding (abstract) Hilbert space of a single electron. In coordinate/spin representation, we have hrjkσi = k(r)χσ, which can be generalized to include a band index (for simplicity, we will restrict this discussion to a single band). Then, the basis vectors of an abstract space with states of arbitrary electron numbers, known as Fock space, may be represented as jn n n :::i ; (1) k1σ1 k2σ2 k3σ3 where nkσ is the occupation number of the state jkσi, i.e., the number of electrons in that state. Due to the fermionic nature of electrons, this number can only assume the values 0 (zero) or 1 (one). It is important to point out that, by construction, there is a one-to-one correspondence between a particular basis vector of the Fock space and a Slater determinant built with the individual Bloch states jk σ i for which n = 1. i i kiσi Second-quantization formalism The Fock space contains states with any number of particles. Physical states of a system of N electrons are vectors belonging to an N-particle subspace, i.e., spanned by basis 2 M. A. Gusm~ao{ IF-UFRGS vectors containing N occupation numbers equal to 1 and all the others equal to zero. However, it is convenient to define operators that allow us to go to subspaces with different particle numbers. Such operators (for obvious reasons) are called single-particle creation and annihilation operators. We can view a general N-particle state as being obtained from the vacuum (zero-particle state) by successive application of N creation operators. y Denoting by ckσ the creation operator of an electron in the single-particle basis state jkσi, we have, for a generic basis state in Fock space, nk σ nk σ nk σ jn n n :::i = cy 1 1 cy 2 2 cy 3 3 ::: j0i ; (2) k1σ1 k2σ2 k3σ3 k1σ1 k2σ2 k3σ3 where j0i ≡ j000 :::i is a simplified notation for the vacuum. The order of creation operators in Eq. (2) is relevant. This is due to the aforementioned correspondence with Slater determinants. Each pair of indices kσ is associated with a row of the Slater determinant. So, an exchange of two creation operators corresponds to an exchange of two rows of the determinant, with a consequent sign change. We say, then, that two creation operators anticommute, which means that their anticommutator is null: y y y y y y fckσ; ck0σ0 g ≡ ckσck0σ0 + ck0σ0 ckσ = 0 : (3) An immediate consequence of this relationship is that 2 y ckσ = 0 ; (4) expressing Pauli's Exclusion Principle. y It is easy to see that the Hermitian conjugate ckσ of the creation operator ckσ is the cor- responding annihilation operator. It suffices to observe what happens to a single particle. In this case we have y ckσj0i = jkσi ; (5) and the dual relation is hkσj = h0jckσ ; (6) which means that ckσ is a creation operator to the left. Now, taking into account the normalization of vectors, we have y hkσjkσi = h0jckσckσj0i = 1 : (7) This may be viewed in two ways: y y hkσjckσj0i = h0j0i = 1 ) hkσjckσ = h0j ; (8) h0jckσjkσi = h0j0i = 1 ) ckσjkσi = j0i : (9) y So, we see that ckσ annihilates on the left while ckσ annihilates on the right. On the other hand, orthogonality between the vacuum and single-particle states allows us to write hkσj0i = h0jckσj0i = 0 ) ckσj0i = 0 ; y y h0jkσi = h0jckσj0i = 0 ) h0jckσ = 0 : (10) M. A. Gusm~ao{ IF-UFRGS 3 We see that an annihilation operator annihilates the vacuum on the right while a creation operator annihilates the vacuum on the left. y The conjugation relationship between ckσ and ckσ allows to write, from Eq. (3), the anti- commutation relation for annihilation operators, fckσ; ck0σ0 g ≡ ckσck0σ0 + ck0σ0 ckσ = 0 ; (11) also showing that 2 (ckσ) = 0 ; (12) as should be expected. The above derivations can be easily extended to independent-particle basis vectors of a many-particle Fock space. We just replace what we called \vacuum" by a basis vector with zero particles in the individual state corresponding to the operator subscript. The complete set of anticommutation relations for creation and annihilation of electrons in Bloch states with specified spin is y y y fckσ; ck0σ0 g = 0 ; fckσ; ck0σ0 g = 0 ; fckσ; ck0σ0 g = δkk0 δσσ0 ; (13) which agrees with the generic algebra of fermion operators. The last relationship above can be verified by evaluating matrix elements of the anticommutator between states of zero or one particle. We define the occupation-number operator as y n^kσ ≡ ckσckσ : (14) It is easy to check that its eigenvalues are indeed 0 and 1, the only possible occupation numbers of a single-fermion state. Operator representation in Fock space It is convenient to express any operator in terms of creation and annihilation operators, since we know how the latter act on Fock-space vectors.. We begin by noticing that a y product like ck0σ0 ckσ has the same effect of a projector within a single-particle subspace, since applying it to any state of a single electron, written as a linear combination of Bloch states, yields the same result as applying the projector jk0σ0ihkσj. So, given that a generic single-particle operator A(1) may be written as (1) X X 0 0 0 0 (1) X X (1) 0 0 A = jk σ ihk σ jA jkσihkσj ≡ Ak0σ0;kσ jk σ ihkσj ; (15) kk0 σσ0 kk0 σσ0 we can represent it in the form (1) X X (1) y A = Ak0σ0;kσ ck0σ0 ckσ : (16) kk0 σσ0 4 M. A. Gusm~ao{ IF-UFRGS Note that, with the same meaning for A(1), Eq. (15) refers to a single-electron Hilbert space while Eq. (16) defines an operator in the Fock space. For operators that do not depend on spin, Eq. (16) simplifies to (1) X (1) y A = Ak0k ck0σ ckσ : (17) kk0σ The matrix elements can be evaluated in coordinate representation: Z (1) 3 ∗ (1) Ak0k = d r k0 (r)A k(r) : (18) Similarly, let us consider a generic two-particle operator B(2) (also spin-independent, for simplicity). The projectors now involve two-particle states, written as direct products of single-particle ones, and the development is similar: (2) X X 0 0 0 0 (2) B = jk1σ1ijk2σ2i hk1σ1jhk2σ2jB jk2σ2ijk1σ1i hk2σ2jhk1σ1j 0 0 k1k1σ1 k2k2σ2 X X (2) 0 0 = B 0 0 jk1σ1ijk2σ2ihk2σ2jhk1σ1j ; (19) k1k2;k2k1 0 0 k1k1σ1 k2k2σ2 with Z Z (2) 3 3 0 ∗ ∗ 0 (2) 0 B 0 0 = d r d r 0 (r) 0 (r )B k (r ) k (r) : (20) k1k2;k2k1 k1 k2 2 1 Comparing with the one-particle case, we see that a projector in a two-particle space is equivalent to the action of two annihilation and two creation operators. Thus, the Fock- space representation of the operator B(2) is (2) X X (2) y y B = B 0 0 c 0 c 0 c c : (21) k1k2;k2k1 k1σ1 k2σ2 k2σ2 k1σ1 0 0 k1k1σ1 k2k2σ2 Electronic Hamiltonian The general rules derived above to express one- and two-particle operators in Fock space allow to rewrite the electronic Hamiltonian [Eq. (1) of Text 9] in the form X y 1 X y y H = "k c c + Uk0 k0 ;k k c 0 c 0 0 c 0 c ; (22) kσ kσ 2 1 2 2 1 k1σ k2σ k2σ k1σ 0 kσ k1k1σ 0 0 k2k2σ where the matrix elements of the e-e interaction are given by Z Z 3 3 0 ∗ ∗ 0 0 0 Uk0 k0 ;k k = d r d r 0 (r) 0 (r )U(r − r ) (r ) (r) : (23) 1 2 2 1 k1 k2 k2 k1 M. A. Gusm~ao{ IF-UFRGS 5 With our choice of basis, the single-particle part of the Hamiltonian has a diagonal rep- resentation, involving the energies of Bloch states. Denoting by H(1) the one-electron 0 Hamiltonian, i.e., kinetic energy and lattice periodic potential (called Hl in the beginning of Text 9), we have (1) "k = hkσjH jkσi ; (24) which is spin-independent and can be evaluated in the coordinate representation as Z h¯2 " = d3r ∗(r) − r2 + V (r) (r) : (25) k k 2m k As remarked before, we are using, for simplicity, a notation without band indices, which is applicable when the relevant physical processes occur in a single band.