Development in Cumulants

The work of Mukherjee and Kutzelnigg, and their general normal ordering [93–96], has led to the widespread use of the cumulant expansion. This expansion allows the reduced density matrices (RDMs) to be expanded in terms of cumulants. The n-particle cumulant is the irreducible, connected part of the reduced density matrix, which can-not be described by lower order cumulants (or density matrices) or by disconnected products of cumulants. The remainder of the reduced density matrix is described by lower order cumulants and density matrices.

The reduced density matrices (γ_1...n^1...n) can be expanded in cumulants (λ^1...n_1...n) [97][96]

γ_t^p = λ^p_t (2.17) The physical meaning of the n-particle cumulants is not fully understood. For a refer-ence consisting of a single determinant, only the first order cumulant is non-zero, all higher order cumulants are zero. It is therefore logical that the higher order cumulants have been linked with describing electron correlation in multi-determinant theories.

Kutzelnigg and Mukherjee [95] stated that the n-particle cumulant directly describes n-particle correlations.

Cumulants have been suggested to be the logical choice in density matrix based many body theories because they are extensive quantities, (unlike the reduced density ma-trices), in that they cannot be described by products of disconnected cumulants and hence truncation of the cumulant series at any point does not affect the extensivity of the method. The truncation of the series is desirable for computational reasons, hence work has been done to look at the affect of the truncation of the series on its accu-racy in describing the RDM. Perturbation theory with a single reference shows that the higher order cumulants decrease in importance, meaning truncation of the

cumu-lant expansion of the RDM would have a negligible affect. Thus, since it has been shown that the 1 and 2 particle cumulants, in general, account for the majority of the higher order RDMs, it has become a common practice to neglect the third order cumu-lant and higher in the cumucumu-lant expansion of the RDMs. As mentioned, this still re-tains extensivity. Cumulant approximations of this nature are central to the Contracted Schrodinger (CSE) theories [98–102] and are used in the Canonical Transformation (CT) theory of Yanai and Chan [62–64], and both state-specific MRCC [103] and in-ternally contracted MRCC [103–105]. Even though neglect of higher order cumulants does not affect extensivity, and whose affect on the accuracy of the RDM description has been deemed small when a single reference determinant is used, the question of the validity of the approximation in the multi-determinantal wave function case is still unanswered.

Recent work [106] has again questioned whether neglecting higher order cumulants has a significant affect by explicity calculating the higher order cumulants in a few test cases and observing their magnitude. This work shows no decisive conclusion, because whether or not the cumulants decay at higher orders, and are therefore less important, seems to be system specific. Shamasundar[107] justifies the use of cumulant approximations by stating that in general states with multireference character, higher order cumulants will have less importance, unless extended electron delocalisation has occurred, in which case, higher order cumulants would be needed in the RDM decomposition. Based on perturbation theory arguments and the accuracy of methods involving them, approximate cumulant decompositions of RDMs are used here.

Make the approximation that the third and higher-order cumulants (as defined in [97]) are zero. In that case, the density matrices may be re-expressed as products of lower order density matrices,

γ_mⁱ = hi^†mi γ_mn^ij = hi^†j^†nmi γ_mno^ijk = hi^†j^†k^†onmi

=Aˆ_abc^{ijk 1}₄γ_mn^ij γ_o^k− ¹₃γ_mⁱ γ_n^kγ_n^l

(2.21) γ^ijkl_mnop = hi^†j^†k^†l^†ponmi

Spin-Orbital Derivation of MR3VCC 61

=Aˆmnop^ijkl 1

32γ_mn^ij γ_op^kl− ¹₈γ_mn^ij γ_o^kγ_p^l + ¹₈γ_mⁱ γ_n^jγ_o^kγ_p^l + ¹₈γ_mn^ij γ_o^kγ_p^l +₂₄¹ γ_mⁱ γ_n^jγ_o^kγ_p^l
(2.22)

=Aˆmnop^ijkl 1

32γ_mn^ij γ_op^kl+ ¹₆γ_mⁱ γ_n^jγ_o^kγ_p^l

(2.23) Now starting from (2.9):

N = ¹₂γ_op^ij K_ab^ijT_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^opAˆmnop^ijkl (₃₂¹γ_mn^ij γ_op^kl+ ¹₆γ_mⁱ γ_n^jγ_o^kγ_p^l)

+ 2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^opAˆmnop^ijkl (₃₂¹γ_mn^ij γ_op^kl+ ¹₆γ_mⁱ γ_n^jγ_o^kγ_p^l) (2.24)

= ¹₂γ_op^ij K_ab^ijT_ab^op

+ 2⁻⁹K_ab^ij T_cd^klT_cd^mnT_ab^opAˆmnop^ijkl (γ_mn^ij γ_op^kl) + ¹₆2⁻⁴K_ab^ijT_cd^klT_cd^mnT_ab^opAˆmnop^ijkl (γ_mⁱ γ_n^jγ_o^kγ_p^l) + 2⁻⁸K_ab^ij T_cd^klT_ca^mnT_bd^opAˆmnop^ijkl (γ_mn^ij γ_op^kl) + ¹₆2⁻³K_ab^ijT_cd^klT_ca^mnT_bd^opAˆmnop^ijkl (γ_mⁱ γ_n^jγ_o^kγ_p^l) (2.25)

= ¹₂γ_op^ij K_ab^ijT_ab^op

+ 2⁻⁹K_ab^ij T_cd^klT_cd^mnT_ab^opAˆmnop(4γ_mn^ij γ_op^kl− 16γ_mn^il γ_op^kj + 4γ_mn^kl γ_op^ij)

+¹₆2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^opAˆmnop(4γ_mⁱ γ_n^jγ_o^kγ_p^l − 16γ_mⁱ γ_n^lγ_o^kγ_p^j + 4γ_m^kγ_n^lγ_oⁱγ_p^j) + 2⁻⁸K_ab^ij T_cd^klT_ca^mnT_bd^opAˆmnop(4γ_mn^ij γ_op^kl− 16γ_mn^il γ_op^kj + 4γ_mn^kl γ_op^ij)

+¹₆2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^opAˆmnop(4γ_mⁱ γ_n^jγ_o^kγ_p^l − 16γ_mⁱ γ_n^lγ_o^kγ_p^j + 4γ_m^kγ_n^lγ_oⁱγ_p^j)

(2.26)

= ¹₂γ_op^ij K_ab^ijT_ab^op

+ 2⁻⁹K_ab^ij T_cd^klT_cd^mnT_ab^op(16γ_mn^ij γ_op^kl− 64γ_mp^ij γ_on^kl + 16γ_op^ijγ_mn^kl

− 64γ_mn^il γ_op^kj+ 256γ^il_mpγ_on^kj − 64γ_op^ilγ_mn^kj + 16γ_mn^kl γ_op^ij − 64γ^kl_mpγ_on^ij + 16γ_op^klγ_mn^ij )

+¹₆2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op(16γ_mⁱ γ_n^jγ_o^kγ_p^l − 64γ_mⁱ γ_p^jγ_o^kγ_n^l + 16γ_oⁱγ_p^jγ_m^kγ_n^l

− 64γ_mⁱ γ_n^lγ_o^kγ_p^j+ 256γ_mⁱ γ_p^lγ_o^kγ_n^j − 64γ_oⁱγ_p^lγ_m^kγ_n^j + 16γ_m^kγ_n^lγⁱ_oγ^j_p− 64γ_m^kγ_p^lγ_oⁱγ_n^j + 16γ_o^kγ_p^lγ_mⁱ γ_n^j)

+ 2⁻⁸K_ab^ij T_cd^klT_ca^mnT_bd^op(16γ_mn^ij γ_op^kl− 64γ_mp^ij γ_on^kl + 16γ_op^ijγ_mn^kl

− 64γ_mn^il γ_op^kj+ 256γ^il_mpγ_on^kj − 64γ_op^ilγ_mn^kj + 16γ_mn^kl γ_op^ij − 64γ_mp^kl γ_on^ij + 16γ_op^klγ_mn^ij )

+¹₆2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^op(16γ_mⁱ γ_n^jγ_o^kγ_p^l − 64γ_mⁱ γ_p^jγ_o^kγ_n^l + 16γ_oⁱγ_p^jγ_m^kγ_n^l

− 64γ_mⁱ γ_n^kγ_o^jγ_p^l + 256γ_mⁱ γ_p^kγ_o^jγ_n^l − 64γ_oⁱγ_p^kγ_m^j γ_n^l

+ 16γ_m^kγ_n^lγ_oⁱγ_p^j− 64γ_m^kγ_p^lγ_oⁱγ_n^j + 16γ_o^kγ_p^lγ_mⁱ γ_n^j) (2.27) Unfortunately this isn’t the best way to go since some of these terms mix not only terms 3B and 3C of VCC, but also the unlinked diagram. For example, the density matrix product γ_mp^il γ_on^kj multiplying K_ab^ij T_cd^klT_cd^mnT_ab^op. To analyse what this term contains, we look at what it corresponds to in the single reference case, by assuming we only have one reference function and evaluating this term.

K_ab^ijT_cd^klT_cd^mnT_ab^op γ_mp^il γ_on^kj

Therefore the simple product of density matrices,γ_mp^il γ_on^kj, when combined with the integrals and amplitudes, contains terms corresponding to the third order terms B and C as well as an unlinked term. The unlinked term cannot be separated out from the product of density matrices, meaning that a transformation involving this product could not be constructed that would generate only linked diagrams, violating the rules for extensivity. If included in a transformation, this density matrix product term would also generate both 3B and 3C terms of 3rd order VCCD, however it has been shown

Spin-Orbital Derivation of MR3VCC 63

that generating 3B and 3C separately is advantageous [11].

It is better to delay the repackaging of the 2nd order and lower cumulants into den-sity matrices, until the fully unlinked term appears. Assuming neglect of 3rd order cumulant λ^ijk_mnoand higher, but keeping cumulants explicitly in the working,

γ_mⁱ = hi^†mi (2.29)

λⁱ_m = γ_mⁱ (2.30)

γ_mn^ij = hi^†j^†nmi

=Aˆmn^ij 1

4λ^ij_mn+¹₂λⁱ_mλ^j_n

= λ^ij_mn+ λⁱ_mλ^j_n− λⁱ_nλ^j_m (2.31) λ^ij_mn = γ_mn^ij − γ_mⁱ γ_n^j + γ_nⁱγ_m^j (2.32) γ_mno^ijk = hi^†j^†k^†onmi

=Aˆ_mno^{ijk 1}₄λ^ij_mnλ^k_o +¹₂λⁱ_mλ^k_nλ^l_n

(2.33) γ_mnop^ijkl = hi^†j^†k^†l^†ponmi

=Aˆmnop^ijkl 1

32λ^ij_mnλ^kl_op+¹₈λ^ij_mnλ^k_oλ^l_p+₂₄¹ λⁱ_mλ^j_nλ^k_oλ^l_p

(2.34)

N = ¹₂hi^†j^†poi K_ab^ij T_ab^op

+ 2⁻⁴hi^†j^†k^†l^†ponmi K_ab^ij T_cd^klT_cd^mnT_ab^op+ 2⁻³hi^†j^†k^†l^†nmpoi K_ab^ij T_cd^klT_ca^mnT_bd^op (2.35)

= ¹₂hi^†j^†poi K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^op
Aˆ_mnop^ijkl ₃₂¹λ^ij_mnλ^kl_op+¹₈λ^ij_mnλ^k_oλ^l_p+₂₄¹ λⁱ_mλ^j_nλ^k_oλ^l_p

(2.36)

= ¹₂hi^†j^†poi K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^op
Aˆmnop(

1

4λ^ij_mnλ^kl_op+¹₂λ^il_mnλ^kj_op

+¹₂λ^ij_mnλ^k_oλ^l_p− 2λ^il_mnλ^k_oλ^j_p+¹₂λ^kl_mnλⁱ_oλ^j_p +λⁱ_mλ^j_nλ^k_oλ^l_p

(2.37)

= ¹₂γ_op^ij K_ab^ijT_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^op
(

λ^ij_mnλ^kl_op− 4λ^ij_mpλ^kl_on+ λ^ij_opλ^kl_mn

−2λ^il_mnλ^kj_op+ 8λ^il_mpλ^kj_on− 2λ^il_opλ^kj_mn +2λ^ij_mnλ^k_oλ^l_p− 8λ^ij_mpλ^k_oλ^l_n+ 2λ^ij_opλ^k_mλ^l_n

−8λ^il_mnλ^k_oλ^j_p+ 32λ^il_mpλ^k_oλ^j_n− 8λ^il_opλ^k_mλ^j_n +2λ^kl_mnλⁱ_oλ^j_p− 8λ^kl_mpλⁱ_oλ^j_n+ 2λ^kl_opλⁱ_mλ^j_n

+4λⁱ_mλ^j_nλ^k_oλ^l_p− 32λⁱ_mλ^j_pλ^k_oλ^l_n+ 4λⁱ_oλ^j_pλ^k_mλ^l_n

(2.38)

= ¹₂γ_op^ij K_ab^ijT_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^op
( λ^ij_mnλ^kl_op+ 2λ^ij_mnλ^k_oλ^l_p+ 2λ^kl_opλⁱ_mλ^j_n+ 4λⁱ_mλ^j_nλ^k_oλ^l_p +λ^ij_opλ^kl_mn+ 2λ^kl_mnλⁱ_oλ^j_p+ 2λ^ij_opλ^k_mλ^l_n+ 4λⁱ_oλ^j_pλ^k_mλ^l_n

−4λ^ij_mpλ^kl_on− 8λ^ij_mpλ^k_oλ^l_n− 32λⁱ_mλ^j_pλ^k_oλ^l_n

−2λ^il_mnλ^kj_op− 8λ^il_mnλ^k_oλ^j_p +8λ^il_mpλ^kj_on+ 32λ^il_mpλ^k_oλ^j_n

−2λ^il_opλ^kj_mn− 8λ^il_opλ^k_mλ^j_n

−8λ^kl_mpλⁱ_oλ^j_n

) (2.39)

= ¹₂γ_op^ij K_ab^ijT_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^op
( γ_mn^ij γ_op^kl

+γ_op^ijγ_mn^kl

−4γ_mp^ij γ_on^kl +4γ_mn^kj γ_op^li

+8λ^il_mpλ^kj_on+ 32λ^il_mpλ^k_oλ^j_n

(2.40)

N = ¹₂γ_op^ij K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ij T_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ij T_cd^klT_ca^mnT_bd^op

γ_op^ijγ_mn^kl +γ_mn^ij γ_op^kl − 4γ_mp^ij γ_on^kl

+4(λ^kj_mn+ 2γ_m^kγ_n^j)(λ^li_op+ 2γ_o^lγ_pⁱ) +8λ^il_mpλ^kj_on+ 32λ^il_mpλ^k_oλ^j_n

(2.41)

Spin-Orbital Derivation of MR3VCC 65

= ¹₂γ_op^ij K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ijT_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ijT_cd^klT_ca^mnT_bd^op

γ_op^ijγ_mn^kl +γ_mn^ij γ_op^kl − 4γ_mp^ij γ_on^kl

+16γ_m^kγ_n^jγ_o^lγ_pⁱ + 4λ^kj_mnλ^li_op+ 8λ^kj_mnγ_o^lγ_pⁱ + 8λ^li_opγ_m^kγ_n^j +8λ^il_mpλ^kj_on+ 32λ^il_mpγ^k_oγ_n^j

(2.42)

= ¹₂γ_op^ij K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ijT_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ijT_cd^klT_ca^mnT_bd^op

γ_op^ijγ_mn^kl +γ_mn^ij γ_op^kl − 4γ_mp^ij γ_on^kl

+4(γ_pn^ij − λ^ij_pn)(γ_mo^kl − λ^kl_mo)

+4λ^kj_mnλ^li_op+ 8λ^kj_mnγ_o^lγ_pⁱ + 8λ^li_opγ_m^kγ_n^j +8λ^il_mpλ^kj_on+ 32λ^il_mpγ^k_oγ_n^j

(2.43)

= ¹₂γ_op^ij K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ijT_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ijT_cd^klT_ca^mnT_bd^op

γ_op^ijγ_mn^kl +γ_mn^ij γ_op^kl − 4γ_mp^ij γ_on^kl

+4γ_pn^ijγ_mo^kl

−4λ^ij_pnγ_mo^kl − 4γ_pn^ijλ^kl_mo

+4λ^ij_pnλ^kl_mo + 4λ^kj_mnλ^li_op+ 8λ^kj_mnγ_o^lγ_pⁱ + 8λ^li_opγ_m^kγ_n^j +8λ^il_mpλ^kj_on+ 32λ^il_mpγ^k_oγ_n^j

(2.44)

= ¹₂γ_op^ij K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ijT_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ijT_cd^klT_ca^mnT_bd^op

γ_op^ijγ_mn^kl +γ_mn^ij γ_op^kl − 4γ_mp^ij γ_on^kl

−4γ_mp^ij γ_on^kl

−4λ^ij_pnλ^kl_mo+ 8λ^ij_pnγ_m^kγ_o^l− 4λ^ij_pnλ^kl_mo + 4γ_pⁱγ_n^jλ^kl_mo− 4γ_nⁱγ_p^jλ^kl_mo +4λ^ij_pnλ^kl_mo + 4λ^kj_mnλ^li_op+ 8λ^kj_mnγ_o^lγ_pⁱ + 8λ^li_opγ_m^kγ_n^j

+8λ^il_mpλ^kj_on+ 32λ^il_mpγ^k_oγ_n^j

(2.45)

= ¹₂γ_op^ij K_ab^ij T_ab^op

+ 2⁻⁴K_ab^ijT_cd^klT_cd^mnT_ab^op+ 2⁻³K_ab^ijT_cd^klT_ca^mnT_bd^op

γ_op^ijγ_mn^kl +γ_mn^ij γ_op^kl − 8γ_mp^ij γ_on^kl

+8λ^ij_pnγ_m^kγ_o^l + 4γ_pⁱγ_n^jλ^kl_mo− 4γ_nⁱγ_p^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

(2.46) This is an approximate form of the 3rd order VCCD energy. No simplifications have been made by considering only a single reference function, this form assumes a refer-ence of multiple functions but is completely general and hrefer-ence the form of an approx-imate VCC method should aim to approxapprox-imate this form as closely as possible.

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