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 y(1); (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 f'p(r)gp=1 Their occupation numbers are manipulated using creation- and annihilation operators, y ^ap and ^ap, which are conjugates of each other. The algebra of these operators is summarized by anti-commutator relations h y y i h y i ^ap; ^aq = 0; [^ap; ^aq] = 0; ^ap; ^aq = 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 js; ms i are eigenfunctions of ^s and ^sz 2 ^s js; ms i = s (s + 1) js; ms i ; ^sz js; ms i = ms js; ms i It is also convenient to introduce step operators ^s+ = ^sx + i^sy and ^s− = ^sx − i^sy p ^s± js; ms i = s (s + 1) − ms (ms ± 1) js; ms ± 1i 1 1 1 1 1 Electrons are spin- 2 particles with spin functions denoted jαi = 2 ; 2 and jβi = 2 ; − 2 . The action of the spin operators is summarized by 2 ^s ^sz ^s+ ^s− 3 1 jαi 4 jαi 2 jαi 0 jβi 3 1 jβi 4 jβi − 2 jβi jαi 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 y y i h y i 0 0 0 ^apσ; ^aqσ0 = 0; [^apσ; ^aqσ ]+ = 0; ^apσ; ^aqσ0 = 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 ^ X X ^ 0 y H1 = h'pσjhj'qσ i^apσ^aqσ0 pq σ,σ0 X X ^ 0 y = h'pjhj'qihσjσ i^apσ^aqσ0 pq σ,σ0 X X ^ y = h'pjhj'qiδσσ0 ^apσ^aqσ0 pq σ,σ0 X ^ X y = h'pjhj'qiEpq; Epq = ^apσ^aqσ pq σ 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 y y H^ = h' σ' τjg^j' σ ' 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 y y = h' ' jg^j' ' ihσjσ ihτjτ ia a a 0 a 0 2 p q r s pσ qτ sτ rσ pqrs στσ0τ 0 1 X X y y = h' ' jg^j' ' 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 y y = h' ' jg^j' ' 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 y y y y y y y 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 f'pgp=1 , which defines our Fock space, and build a reference ONV in that space y y y j0i = ^a1^a2 ::: ^aN jvaci = j1; 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 y 1 X y y E = h0jH^ j0i = h0j h ^a ^a + V ^a ^a ^a ^a j0i + V pq p q 2 pq;rs p q s r nn pq pq;rs One-electron energy HF X y E1 = hpqh0j^ap^aqj0i pq I The operator ^aq tries to remove an electron to the right; this is only possible if orbital q is occupied. y I Likewise, the operator ^ap 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 y y E = V h0j^a ^a ^a ^a j0i 2 2 pq;rs p q s r pq;rs I Operators ^ar and ^as both try to remove an electron to the right; orbitals r and s must be occupied, but not identical y y I Operators ^ap and ^aq 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 y y y y o E = V h0j^a ^a ^a ^a j0i + V h0j^a ^a ^a ^a j0i 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 = fV − V g = h' ' k ' ' 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 [f' g] = h' jh^j' i + h' ' k ' ' i + V i i i 2 i j i j nn i ij .. and is minimized under the constraint of orthonormal orbitals h'i j'j i = δij This is normally done by the introduction of Lagrange multipliers HF HF X L [f'i g] = E [f'i g] − λij fh'i j'j i − δij g 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 f'pgp=1 X '~p = 'qcqp q and use the expansion coefficients fcqpg as variational parameters ? In order to preserve orthonormality the expansion coeffients must obey X X ∗ X ∗ h'~pj'~qi = h'r crpj's csqi = h'r j's i crpcsq = crpcrq = δpq rs rs | {z } r δrs ..which means that they must form a unitary (orthogonal) matrix for complex (real) orbitals: C yC = 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 1 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 1 X Ak exp(A) = k! k=0 We next consider the product 1 1 X X Am Bn exp(A) exp(B) = m! n! m=0 n=0 We rearrange to collect contribution of order k = m + n 1 k m k−m 1 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)y = exp Ay A unitary matrix is defined by U−1 = Uy which is obtained by using an anti-Hermitian A Ay = −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 y '~p = 'qUqp; U = exp (−κ); κ = −κ q I will now show that this corresponds to writing the optimized HF occupation-number vector as 0~ = exp (−κ^) j0i whereκ ^ is an orbital rotation operator with amplitudes κpq X y ∗ κ^ = κpq^ap^aq; 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 ~ y y y 0 = exp (−κ^) j0i = exp (−κ^) a1 a2 ::: aN jvaci Next, we insert exp (^κ) exp (−κ^) = 1 everywhere ~ y y y 0 = exp (−κ^) a1 exp (^κ) exp (−κ^)a2 exp (^κ) ::: exp (−κ^) aN exp (^κ) exp (−κ^) jvaci y y y y y = ~a1~a2 ::: ~aN exp (−κ^) jvaci ; ~ar = exp (−κ^) ar exp (^κ) First, we note that X y κ^ jvaci = κpqapaq jvaci = 0; pq 1 ) exp (−κ^) jvaci = 1 − κ^ + κ^2 − ::: jvaci = jvaci 2 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 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 y Using the BCH expansion with A = −κ^ and B = ar we get h i 1 h h ii ~ay = exp (−κ^) ay exp (^κ) = ay − κ,^ ay + κ,^ κ,^ ay − ::: r r r r 2 r h yi To evaluate the commutator κ,^ ar we use our rule h i h i h i A^B^; C^ = A^ B^; C^ − A^; C^