A Whirlwind Introduction to Coupled Cluster Response Theory. 1 Time-Independent Coupled Cluster Theory

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1 A Whirlwind Introduction to Coupled Cluster Response Theory T. Daniel Crawford, Virginia Tech, Blacksburg, Virginia, U.S.A. 1 Time-Independent Coupled Cluster Theory If the Hamiltonian is independent of time, the corresponding Schrödinger equation is Ĥ Ψ = E Ψ 1 the electronic wave function may be conveniently parametrized using the exponential form Ψ CC = e ˆT 0, 2 where 0 is the reference determinant usually Hartree-Fock, ˆT is a cluster operator that generates linear combinations of substituted determinants from 0 : ˆT 0 = ˆT 0 = t τ 0 = t. 3 where denotes sets of singly, doubly, etc. substituted determinants, τ is a second-quantized excitation operator, the corresponding wave function amplitude/coefficient is t. Insertion of Eq. 2 into Eq. 1 gives Ĥe ˆT 0 = E CC e ˆT 0, 4 where E CC is still exact in a complete orbital basis with untruncated ˆT. In the conventional coupled cluster method, the energy wave function ampltidues are obtained by a projective approach rather than a variational one by multiplying both side by the inverse of the exponential, projecting onto the reference determinant, or substituted determinants, viz. e ˆT Ĥe ˆT 0 = E CC 0, 5 E CC = 0 e ˆT Ĥe ˆT 0, 6 0 = e ˆT Ĥe ˆT 0, 7 where both equations require that the Slater determinants form an orthonormal set. The coupled cluster equations are typically expressed in terms of the non-hermitian similarity transformed Hamiltonian, H e ˆT Ĥe ˆT, 8 1

2 which may be interpreted as an effective Hamiltonian whose eigenspectrum is identical to the original Hamiltonian if ˆT remains complete. This operator becomes convenient for several reasons, not least because it may be expressed as a finite commutator expansion, H = H + Ĥ, ˆT + 1 Ĥ, ˆT, 2 ˆT +..., 9 which truncates at the quartic terms in ˆT due to the two-electron nature of the electronic Hamiltonian. In terms of H, we obtain the governing equations for the energy the cluster amplitudes E CC = 0 H = H An alternative view of the non-variational coupled cluster equations arises through the variational optimization of Lagrangian function, L CC = 0 H 0 + λ H 0, 12 where the λ serve as the linear multipliers. If we recognize that = 0 τ, 13 then we may define a cluster de-excitation operator analogous to ˆT as 0 ˆΛ = 0 ˆΛ = 0 λ τ = λ. 14 This allows us to simplify the expression for the Lagrangian to obtain L CC = 0 H ˆΛ H 0 = Λ H Variational optimization of L CC with respect to the Λ amplitudes gives 0 = L CC λ = H 0, 16 which identical to the defining equation for the ˆT amplitudes in Eq. 11. Similar optimization of L CC with respect to the ˆT amplitudes gives 0 = L ˆΛ CC = H, τ 0, 17 t which is the defining equation for the ˆΛ amplitudes. The structure of the Lagrangian the fact that the similarity transformed Hamiltonian is non-hermitian leads naturally to the identification of left-h coupled cluster wave function, Ψ CC = e ˆT, 18 which is clearly distinct from its right-h counterpart in Eq. 2. 2

3 2 Time-Dependent Coupled Cluster Theory A time-dependent Hamiltonian requires us to use the corresponding time-dependent Schrödinger equation, Ĥ Ψ = i d Ψ atomic units 19 As noted above, the coupled-cluster parametrization of the electronic wave function has distinct right-h states, Ψ CC = e ˆT t 0 e iɛt, 20 left-h states Ψ CC = t e ˆT t e iɛt, 21, as before, the cluster operators generate linear combinations of substituted determinants from the left, ˆT t 0 = ˆT t 0 = t tτ 0 = t t, 22 or from the right, 0 ˆΛt = 0 ˆΛ t = 0 λ tτ = λ t, 23 where the amplitudes, t t λ t carry the time dependence. In addition, the wave function phase, ɛt, depends on time, though for now we take the reference determinant to be timeindependent no orbital response. This parametrization leads to distinct right- left-h Schrödinger equations, respectively, Ĥe ˆT t 0 e iɛt = i d e ˆT t 0 e iɛt, 24 t e ˆT t e iɛt Ĥ = i d t e ˆT t e iɛt. 25 Explicit time differentiation of the amplitudes phase factor leads to for the right-h Schrödinger equation e T e iɛ Ĥ = Ĥe ˆT 0 e iɛ = i d ˆT e ˆT 0 e iɛ e ˆT 0 dɛ eiɛ, 26 i 0 dˆλ e ˆT e iɛ + i d ˆT e ˆT e iɛ ˆT dɛ e e iɛ 27 3

4 for the left-h, the notational time dependence has been supressed for simplicity. We may express these equations in terms of the similarity-transformed Hamiltonian, H e ˆT Ĥe ˆT 28 to obtain H 0 e iɛ = i d ˆT 0 eiɛ 0 dɛ eiɛ 29 e iɛ dˆλ H = i 0 e iɛ + i d ˆT e iɛ dɛ e iɛ. 30 Furthermore, since the exponential phase factor contains no coordinate dependence, multiplication by its complex conjugate yields, H 0 = i d ˆT 0 0 dɛ 31 dˆλ H = i 0 + i d ˆT dɛ. 32 Note that the time-derivatives of the cluster operators retain their excitation/de-excitation character because the differentiation only affects the corresponding amplitudes, i.e., d ˆT = τ dˆλ = dλ τ. 33 Furthermore, as long as the orbital space is comprised of disjoint sets of occupied virtual orbitals, the cluster operators communte within excitation/de-excitation classes, e.g., ˆT, ˆT d ν = 0, ˆT, ˆT ν = 0, ˆΛ, ˆΛ dˆλ ν = 0,, ˆΛ ν = 0, 34 but ˆΛ, ˆT ν Projecting the right-h Schrödinger equation, Eq. 31, onto the reference determinant gives an equation for the time dependence of the phase factor because 0 H 0 = i 0 d ˆT dɛ = dɛ, 36 4

5 where we have taken advantage of the fact that the derivative of ˆT is still an excitation operator the set of Slater determinants are orthonormal. Similarly, projection of Eq. 31 onto substituted determinants yields the time dependence of the ˆT amplitudes: H 0 = i. 37 We may obtain corresponding results from the left-h Schrödinger equation, though to do so requires some additional manipulation. Projecting Eq. 32 onto the reference determinant gives dˆλ H 0 = i i d ˆT 0 dɛ The left-h side may be manipulated by taking advantage of the fact that the set of Slater determinants is complete thus provides an identity 1 = Inserting this between ˆΛ Ĥ gives ˆΛ H 0 = 0 H ˆΛ H 0 = 0 H ˆΛ 0 0 H ˆΛ H 0 = 0 H 0 + i 0 ˆΛ. 40 where the final step arises due to Eq. 37. The first term on the right-h side of Eq. 38 is zero because the derivative of ˆΛ is still a de-excitation operator, the Slater determinants form an orthonormal set, while the final term collapses to only the derviative of the phase factor for similar reasons. This leaves only the second term of the right-h side, which may be simplified as follows: i d ˆT d ˆT 0 = i 0 ˆΛ 0 = i 0 ˆΛ d ˆT 0 = i 0 ˆΛ, 41 which exactly cancels the corresponding term from the left-h side. Thus, we find that 0 H 0 = dɛ, 42 5

6 which is identical to the result obtained from the right-h Schrödinger equation. Projecting the left-h Schrödinger equation onto the substituted determinants requires similar manipulation, dˆλ H = i 0 + i d ˆT dɛ = i dλ = i dλ + i ν d ˆT dɛ + i 0 ˆΛ 0 ˆΛ 0 ˆΛ ν τ ν + 0 ˆΛ 0 H 0, 43 where we have used the definition of the ˆT operator as well as Eq. 36 in the last step. A simpler, commutator-based expression may be obtained by adding a key term to both sides, viz. dλ H Λ τ H 0 = i + i ν Then, recognizing that the equation may be further simplified as dλ H, τ 0 = i + ν = τ 0, τ ν, τ = 0, 0 τ H 0 = 0, 0 ˆΛ ν τ ν + 0 ˆΛ 0 H Λ τ H ˆΛτ ν ν H ˆΛ 0 H 0 0 Λτ H 0 = i dλ + ν 0 ˆΛτ ν ν H ˆΛτ 0 0 H 0 0 Λτ H 0 = i dλ + 0 ˆΛτ H 0 0 Λτ H 0, 45 where we have used the resolution of the identity Eq. 39 in the last step. Now we can see that the last two terms on the right-h side cancel exactly, leaving a simpler final expression, which is the time-dependence of the λ amplitudes. dλ H, τ 0 = i, 46 6

7 3 Coupled Cluster Response Theory If the Hamiltonian is partitioned into a time-independent zeroth-order part a time-dependent perturbation, Ĥ = Ĥ0 + Ĥ1 t, 47 where the perturbation is a Fourier transform of a frequency-dependent operator, Ĥ 1 t = dωĥ1 ωe iωt, 48 then the cluster operators may be exped in orders of the perturbation, namely, ˆT = ˆT 0 + ˆT 1 + ˆT ˆΛ = ˆΛ 0 + ˆΛ 1 + ˆΛ Note that the zeroth-order ˆT 0 ˆΛ 0 amplitudes correspond to the time-independent quantities described in section 1, whose defining expressions are Eqs , respectively. Similarity transformation of the Hamiltonian gives e ˆT Ĥe ˆT = e ˆT 0 ˆT 1 ˆT 2... Ĥe ˆT 0 + ˆT 1 + ˆT Collecting the zeroth-order amplitudes together, we may define H in a manner consistent with Section 1, H e ˆT 0 Ĥe ˆT 0, 52 then define the fully transformed Hamiltonian as H = e ˆT 1 ˆT 2... He ˆT 1 + ˆT Note that H contains only zeroth- first-order terms with zeroth-order time-independent amplitudes, ˆT 0, H = H 0 + H 1, 54 whereas H contains all orders: H = H 0 + H1 + H2..., 55 where, for example, H 0 = H 0 = e ˆT 0 Ĥ 0 e ˆT 0, 56 H 1 = H 1 + H0, ˆT 1, 57 H 2 = H1, ˆT H0, ˆT 1, ˆT 1 + H0, ˆT

8 Amplitudes in each order may be obtained by similar expansion of the governing equations, Eqs , in conjunction with the commutator expansion of H. For the ˆT amplitudes, for example, the first few orders are derived from Eq. 37 to give i 2 i 1 i 0 = H0 0 = H 0 0 = 0, 59 = H1 0 = H H0, ˆT 1 0, 60 = H2 0 = H1, ˆT H0, ˆT 1, ˆT H0, ˆT 2 0. Similarly, for the ˆΛ amplitudes, i dλ2 i dλ1 i dλ0 61 = 0 H0, τ 0 = 0, 62 = ˆΛ 0 H1, τ ˆΛ 1 H0, τ 0, 63 = ˆΛ 0 H2, τ ˆΛ 1 H1, τ ˆΛ 2 H0, τ We may define the time-dependent perturbed amplitudes in terms of Fourier transforms of their frequency-dependent counterparts in analogy to the frequency-dependent perturbation of Eq. 48 as, for example, t 1 t = dω t 1 ω e iωt 65 t 2 t = dω 1 dω 2 t 2 ω 1, ω 2 e iω1t e iω2t. 66 These expressions are trivially differentiated to give, 1 = dω iω t 1 ω e iωt 67 2 = dω 1 dω 2 iω 1 iω 2 t 2 ω 1, ω 2 e iω 1t e iω 2t. 68 8

9 Comparing these expressions to the corresponding derivatives above inserting the Fourier transforms of all time-dependent quantities gives i 1 = = = dω ω t 1 ω e iωt = H1 0 = H dω H1 ωe iωt 0 dω H 1 ωe iωt 0 + H0, ˆT 1 0 H 0, dω ˆT 1 ωe iωt We may equate the integration kernels of each term to obtain ωt 1 ω = H 1 ω 0 + H0, ˆT 1 ω 0, 70 where ˆT 1 ω is a cluster operator built from the frequency-dependent first-order amplitudes H 1 ω is the similarity transformation of the frequency-dependent perturbation Ĥ1 ω using only the zeroth-order ˆT 0. This is the equation that must be solved to obtain the first-order, frequency-dependent cluster amplitudes. Using a similar approach for the second-order derivatives, we see i 2 = dω 1 = H1, ˆT 1 = 1 2 dω 1 dω 2 ω 1 + ω 2 t 2 ω 1, ω 2 e iω 1t e iω 2t H0, ˆT 1, ˆT 1 dω ˆP ω 1, ω 2 dω 1 dω H0 2, ˆT 1 ω H0, ˆT 2 H1 ω 1, ˆT 1 ω 2 e iω 1+ω 2 t 0 +, ˆT 1 ω 2 e iω 1+ω 2 t 0 + dω 1 dω H0 2, ˆT 2 ω 1, ω 2 e iω 1+ω 2 t 0, 71 where the permutation operator ˆP ω 1, ω 2 gives a sum of two terms: one as written plus a second with the frequencies reversed. By equating the integrs on both sides, we obtain ω 1 + ω 2 t 2 ω 1, ω 2 = 1 2 ˆP ω 1, ω 2 H1 ω 1, ˆT 1 ω H0, 2 ˆT 1 ω 1, ˆT 1 ω 2 + H0, ˆT 2 ω 1, ω We may apply the same technique to the ˆΛ amplitudes to give ωλ 1 ω = 0 ˆΛ 1 ω H0, τ ˆΛ 0 H1 ω, τ 0, 73 9

10 ω 1 + ω 2 λ 2 = 1 2 ˆP ˆΛ2 ω 1, ω 2 0 ω 1, ω 2 H0, τ ˆΛ 0 H2 ω 1, ω 2, τ + ˆΛ 1 ω 1 H1 ω 2, τ At long last, we are ready to define the coupled cluter response functions, starting from the Fourier expansion of the expectation value of a time-independent operator, Â,  =  d ω Â; Ĥ1 ω e iωt + dω 1 dω 2 Â; Ĥ1 ω 1, Ĥ1 ω 2 e iω1t e iω2t Replacing the general expectation value with its coupled cluster counterpart yields Ψ CC Â Ψ CC = t ˆΛt e T t Âe T t 0 = Ā 0, where we have temporarily re-introduced the time-dependence notation for clarity. Inserting the order-by-order expansion in Ĥ1 t for the double similarity transformation of  gives Ā 0 = Ā0, Ā0 Ā 1 =, ˆT 1, Ā 2 = 1 Ā0, 2 ˆT 1, ˆT Ā0 1 +, ˆT 2, etc. where the Ā1 terms are zero because of the time-independence of Â. Exping the coupled cluster expectation value in orders of Ĥ1, Ψ CC Â Ψ CC = 0 Ā ˆΛ 0 Ā 1 + ˆΛ 1 Ā ˆΛ 0 Ā 2 + ˆΛ 1 Ā 1 + ˆΛ 2 Ā Inserting Fourier transforms of each time-dependent quantity above equating the corresponding terms of each order in the general expectation value leads to expressions for the first few response functions, including the zeroth-order expectation value  0 = 0 Ā0 0, 77 the linear response function Â; Ĥ1 ω = ˆΛ Ā0 0, ˆT 1 ω + ˆΛ 1 ωā0 0, 78 the quadratic response function, Â; Ĥ1 ω 1, Ĥ1 ω 2 = 1 2 ˆP ω 1, ω 2 0 Ā 2 ω 1, ω ˆΛ 1 ω 1 Ā 1 ω 2 + ˆΛ 2 ω 1, ω 2 Ā

11 Recommended Reading 1 H. Koch P. Jørgensen, J. Chem. Phys., 93, Coupled cluster response functions. 2 O. Christiansen, P. Jørgensen, C. Hättig, Int. J. Quantum Chem., 68, Response functions from Fourier component variational perturbation theory applied to a timeaveraged quasienergy. 3 T. B. Pedersen H. Koch, J. Chem. Phys., 106, Coupled cluster response functions revisited. 11