Two Electron Case

|0i = ¹₂C^mn|mni ≡ ¹₂C^mnm^†n^†|vaci (2.54)

Spin-Orbital Derivation of MR3VCC 69

γ_jⁱ = h0|i^†j|0i (2.55)

= C^ilC^jl (2.56)

γ_kl^ij = h0|i^†j^†lk|0i (2.57)

= C^ijC^kl (2.58)

λ^ij_kl= γ_kl^ij − γ_kⁱγ_l^j+ γ_lⁱγ_k^j (2.59)

= C^ijC^kl− C^imC^kmC^jnC^ln+ C^imC^lmC^jnC^kn (2.60) Assume, without loss of generality, that we have a natural orbital basis, ie that γ is diagonal:

γ_kⁱ = (c⁽ⁱ⁾)²δ_ik (2.61)

C^ij = c⁽ⁱ⁾δ_j,[i] (2.62)

γ_kl^ij = c⁽ⁱ⁾c^(k)δ_j,[i]δ_l,[k] (2.63) Restricting the excitation operators making up ˆT to exclude the null space,

T_ab^ij = c⁽ⁱ⁾δ_j,[i]T_ab (2.64)

Normalisation:

1 =X

ij

1

2(C^ij)² = 1 2

X

i

(c⁽ⁱ⁾)² (2.65)

In the special case of just two orbitals, then c⁽¹⁾ = c⁽²⁾ = 1, otherwise all coefficients are less than 1.

NC = ¹₂γ_op^ij K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op c⁽ⁱ⁾c^(o)δ_j,[i]δ_p,[o]c^(m)c^(k)δ_n,[m]δ_l,[k]

+c⁽ⁱ⁾c^(m)δ_j,[i]δ_n,[m]c^(o)c^(k)δ_p,[o]δ_l,[k]

−8c⁽ⁱ⁾c^(m)δ_j,[i]δ_p,[m]c^(o)c^(k)δ_n,[o]δ_l,[k]

−8λ^ij_pnγ_m^kγ^l_o− 8γ_pⁱγ_n^jλ^kl_mo+ 8λ^kj_mnγ_o^lγ_pⁱ + 8λ^li_opγ_m^kγ_n^j + 32λ^il_mpγ_o^kγ_n^j

−4λ^ij_pnλ^kl_mo+ 4λ^kj_mnλ^li_op+ 8λ^il_mpλ^kj_on

(2.66)

= ¹₂γ_op^ij K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op c⁽ⁱ⁾c^(o)δ_j,[i]δ_p,[o]c^(m)c^(k)δ_n,[m]δ_l,[k]

+c⁽ⁱ⁾c^(m)δ_j,[i]δ_n,[m]c^(o)c^(k)δ_p,[o]δ_l,[k] zero, because, as noted earlier, the linked 3rd order energy terms (B+C) are equal and opposite to the unlinked terms for the 2 electron case. For the multireference case, there are more than 2 orbitals. If the exact answer for 2 electrons is simply a sum of terms 3B and 3C, then the sum above still containing the unlinked term must be zero and would show that the neglected terms (products of cumulants) are not required to be correct for this example system.

Evaluation of the approximate 3rd order energy derived above using the cumulant ex-pansion must be evaluated for the model system.

Evaluation for the model system

The total 3rd order energy including all linked term and unlinked terms is 0 for 2 electrons, however, the linked form of the energy is non-zero. If the approximate form of the energy derived above is to be relied upon, then it must also show the same

Spin-Orbital Derivation of MR3VCC 71

behaviour for the 2-electron, 2-reference model system define din the Appendix.

It can be shown that the evaluation of the approximate 3rd order total energy for the model system does indeed give 0, as desired, and the linked energy is shown to be non-zero.

The linked energy is given below (¯1 indicates β spin and 1 α spin), E_3vcc = E_L + E_{U L} = 0

E_L =

− 8 K_3¯^1¯₃¹ cos⁴θ 2⁻³T_3¯^1¯₃¹T_3¯^1¯₃¹T_3¯^1¯₃¹

− 24 K_3¯^1¯₃¹ cos³θ sin θ 2⁻³T_3¯^1¯₃¹T_3¯^2¯₃²T_3¯^2¯₃²

− 24 K_3¯^1¯₃¹ cos²θ sin²θ 2⁻³T_3¯^1¯₃¹T_3¯^2¯₃²T_3¯^2¯₃²

− 8 K_3¯^1¯₃¹ cos θ sin³θ 2⁻³T_3¯^2¯₃²T_3¯^2¯₃²T_3¯^2¯₃²

− 8 K_3¯^2¯₃² sin⁴θ 2⁻³T_3¯^2¯₃²T_3¯^2¯₃²T_2¯^2¯₃¹

− 24 K_3¯^2¯₃² cos θ sin³θ 2⁻³T_2¯^2¯₃²T_3¯^1¯₃¹T_3¯^1¯₃¹

− 24 K_3¯^2¯₃² cos²θ sin²θ 2⁻³T_3¯^2¯₃²T_3¯^1¯₃¹T_3¯^1¯₃¹

− 8 K_3¯^2¯₃² cos³θ sin θ 2⁻³T_3¯^1¯₃¹T_3¯^1¯₃¹T_3¯^1¯₃¹ 

(2.72) The linked energy contains the contributions from 23 terms. However, terms involv-ing second order cumulants sum to zero (terms 4-12). As mentioned, in the sinvolv-ingle- single-reference theory for 2 electrons, the linked energy terms partially cancel to give a linked energy that is the same as term −3C or ¹₂3B. Similar cancellations occur here for the evaluation of the multi-reference model system, the linked energy in 2.72 can be seen to reduce to be the same as simple individual contributions. The evaluation of terms 1β and 2α are give below,

Term 1β γ_op^ij γ_mn^kl K_ab^ij 

2⁻⁴T_cd^klT_ca^mnT_bd^op

(2.73)

= − 8 K_3¯^1¯₃¹ cos⁴θ 2⁻³T_3¯^1¯₃¹T_3¯^1¯₃¹T_3¯^1¯₃¹

− 24 K_3¯^1¯₃¹ cos³θ sin θ 2⁻³T_3¯^1¯₃¹T_3¯^2¯₃²T_3¯^2¯₃²

− 24 K_3¯^1¯₃¹ cos²θ sin²θ 2⁻³T_3¯^1¯₃¹T_3¯^2¯₃²T_3¯^2¯₃²

− 8 K_3¯^1¯₃¹ cos θ sin³θ 2⁻³T_3¯^2¯₃²T_3¯^2¯₃²T_3¯^2¯₃²

− 8 K_3¯^2¯₃² sin⁴θ 2⁻³T_3¯^2¯₃²T_3¯^2¯₃²T_2¯^2¯₃¹

− 24 K_3¯^2¯₃² cos θ sin³θ 2⁻³T_2¯^2¯₃²T_3¯^1¯₃¹T_3¯^1¯₃¹

− 24 K_3¯^2¯₃² cos²θ sin²θ 2⁻³T_3¯^2¯₃²T_3¯^1¯₃¹T_3¯^1¯₃¹

− 8 K_3¯^2¯₃² cos³θ sin θ 2⁻³T_3¯^1¯₃¹T_3¯^1¯₃¹T_3¯^1¯₃¹ (2.74)

Term 2α γ_mn^ij γ_op^klK_ab^ij 

2⁻⁴T_cd^klT_cd^mnT_ab^op

(2.75)

= + 8 K_3¯^1¯₃¹ cos⁴θ 2⁻³T_3¯^1¯₃¹T_3¯^1¯₃¹T_3¯^1¯₃¹ + 24 K_3¯^1¯₃¹ cos³θ sin θ 2⁻³T_3¯^1¯₃¹T_3¯^2¯₃²T_3¯^2¯₃² + 24 K_3¯^1¯₃¹ cos²θ sin²θ 2⁻³T_3¯^1¯₃¹T_3¯^2¯₃²T_3¯^2¯₃² + 8 K_3¯^1¯₃¹ cos θ sin³θ 2⁻³T_3¯^2¯₃²T_3¯^2¯₃²T_3¯^2¯₃² + 8 K_3¯^2¯₃² sin⁴θ 2⁻³T_3¯^2¯₃²T_3¯^2¯₃²T_2¯^2¯₃¹ + 24 K_3¯^2¯₃² cos θ sin³θ 2⁻³T_2¯^2¯₃²T_3¯^1¯₃¹T_3¯^1¯₃¹ + 24 K_3¯^2¯₃² cos²θ sin²θ 2⁻³T_3¯^2¯₃²T_3¯^1¯₃¹T_3¯^1¯₃¹

+ 8 K_3¯^2¯₃² cos³θ sin θ 2⁻³T_3¯^1¯₃¹T_3¯^1¯₃¹T_3¯^1¯₃¹ (2.76) The results of the evaluation of these terms can be related to the linked energy (eq.

2.72) very simply. For the 2 electron, 2 reference model, the linked approximate 3rd order VCC energy can be written,

E_L= 1β = −2α (2.77)

It is shown above that in the single reference limit term 2α reduces to VCC term 3C, so it is in complete analogy that the linked energy for the model can be described completely by −2α, just as in single reference the linked energy for 2 electrons can be written as -3C. The other term here that gives the full linked energy is term 1β, which, in the single reference limit, reduces to diagram A. This is a slightly different trend to the single reference case.

The approximate 3rd order energy contains some terms involving 2nd order cumulants (terms 4-12), that sum to 0 for the model. They may be important for the general multireference case, but their importance is not probed here. One can look at their importance by re-analysing the 4th order density matrix and showing how good the approximations to it are when containing the different terms.

For the model case, term 1β or −2α are the only terms needed for the description of the linked energy. Accordingly attempts should be made at capturing these terms via a transformation of the pair amplitudes.

Spin-Orbital Derivation of MR3VCC 73

The Transformation

As stated, it is apt to capture either term 1β or the negative of term 2α via a transfor-mation, as either of these terms give the correct linked energy for the model 2-electron multireference system. A pragmatic choice of which term to capture is to choose term

−2α, as the pairing of virtual indices is simpler than that of term 1β. Also, another consideration is the comparison to single reference theory. Term −2α is the multiref-erence generalisation of the single refmultiref-erence 3rd order VCC diagram -3C. -3C is the term captured by LPF+1D, which has shown to perform well (despite being a simple approximation to VCC)[11, 30], and this term is known to be important in the general case. Without further analysis of the multireference terms, capture of −2α is a logical choice.

To restate, term −2α is chosen to be captured by a transformation because it is exact for the model system and reduces directly to LPF+1 when a single reference function is used. Possible transformations to capture further terms are outlined later but not considered in detail in this work.

The requirement is the transformed amplitude must capture the correct 1st order energy but also the negative 3rd order energy term 2α.

N = ¹₂γ_op^ij K_ab^ijT_ab^op − 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^opγ_mn^ij γ_op^kl (2.78)

= ¹₂K_ab^ij γ_{mn 2}^ij T_ab^mn (2.79)

An appropriate transformation matrix, U, can be defined,

U = 1 + ∆ (2.80)

U_op^mn = δ_op^mn + ∆^mn_op (2.81)

∆^mn_op = ¹₂η^mn_kl γ_op^kl (2.82) η_kl^mn = ¹₂T_cd^mnT_kl^cdγ_op^kl (2.83)

The definition of a general transformed amplitude follows;

qT_ab^mn= ¹₂ δ_op^mn + ¹₄T_cd^klT_cd^mnγ_op^kl
−q

2 T_ab^op (2.84)

qT_ab^mn= ¹₂ U⁻

q 2

mn op

T_ab^op (2.85)

The result of the transformation can be viewed using the binomial expansion (1 + x)⁻¹ = 1 − x + . . .

1

2K_ab^ij γ_mn^ij (U⁻¹T_ab^op) = ¹₂K_ab^ijγ_mn^ij T_ab^mn − 2⁻⁴K_ab^ij γ_mn^ij T_cd^mnT_cd^klγ_op^klT_ab^op+ . . . (2.86) This is the 1st order energy contribution plus the negative of term 2α as required, show-ing the transformation matrix does as required. This transformation can be constructed at a cost in complexity of no more than o⁴v², by first computing the intermediate matrix η before contraction with the density matrix to make ∆ to be used in the transformation matrix, U.

Computational scaling:

T_cd^mnT_kl^cdγ_op^kl −−−−→^o4v2 η_kl^mnγ_op^kl −−−→^o6 ∆^mn_op U_op^mn = δ_op^mn+ ∆^mn_op

The formation of this transformation matrix is no more complex than the limiting steps in both Coupled Cluster and Configuration Interaction, however, the number of internal orbitals is greatly increased in the multireference case, making the U matrix more demanding to construct, and invert.