Aspects of size extensivity in unitary group adapted multi-reference coupled cluster theories: the role of cumulant decomposition of spin-free reduced density matrices

Abstract

We present in this paper a comprehensive study of the various aspects of size extensivity of a set of unitary group adapted multi-reference coupled cluster (UGA-MRCC) theories recently developed by us. All these theories utilize a Jeziorski–Monkhorst (JM) inspired spin-free cluster Ansatz of the form $|\varPsi \rangle = \sum\nolimits_\mu \varOmega _\mu |\phi _\mu \rangle c_\mu$ with $\varOmega _\mu =\{\exp ({T_\mu })\}$, where $T_\mu$ is expressed in terms of spin-free generators of the unitary group $U(n)$ for n-orbitals with the associated cluster amplitudes. $\{...\}$ indicates normal ordering with respect to the common closed shell $core$ part, $|0\rangle$, of the model functions, $\{\phi _\mu \}$ which is taken as the vacuum. We argue and emphasize in the paper that maintaining size extensivity of the associated theories is consequent upon (a) connectivity of the composites, $G_\mu$, containing the Hamiltonian $H$ and the various powers of $T$ connected to it, (b) proving the connectivity of the MRCC equations which involve not only $G_\mu$s but also the associated connected components of the spin-free reduced density matrices (RDMs) obtained via their cumulant decomposition and (c) showing the extensivity of the cluster amplitudes for non-interacting groups of orbitals and eventually of the size-consistency of the theories in the fragmentation limits. While we will discuss the aspect (a) above rather briefly, since this was amply covered in our earlier papers, the aspect (b) and (c), not covered in detail hitherto, will be covered extensively in this paper. The UGA-MRCC theories dealt with in this paper are the spin-free analogs of the state-specific and state-universal MRCC developed and applied by us recently.We will explain the unfolding of the proof of extensivity by analyzing the algebraic structure of the working equations, decomposed into two factors, one containing the composite $G_\mu$ that is connected with the products of cumulants arising out of the cumulant decomposition of the RDMs and the second term containing some RDMs which is disconnected from the first and can be factored out and removed. This factorization ultimately leads to a set of connected MRCC equations. Establishing the extensivity and size-consistency of the theories requires careful separation of truly extensive cumulants from the ones which are a measure of spin correlation and are thus connected but not extensive. We have discussed in detail, using diagrams, the factorization procedure and have used suitable example diagrams to amplify the meanings of the various algebraic quantities of any diagram. We conclude the paper by summarizing our findings and commenting on further developments in the future.

Appendix: Definition of spin-free RDMs and the associated spin-free cumulants

The $n$-particle density matrices are not size-extensive, but are instead product separable, just like the wave operator. However, one can factorize any $n$-body density matrix in spinorbital basis as anti-symmetrized products of 1-particle density matrices and ‘$cumulants$’ which can be recursively defined as the rank $n$ of $\gamma _n$ increases. For now, we denote by $u,v....$, etc., the active spinorbitals. For a CASSCF function $\varPsi _0$, all the core spinorbitals are fully filled in each $\phi _\mu$ of $\varPsi _0$, and hence, all the $\gamma _n$s corresponding to a $\phi _\mu$ will factor out as anti-symmetrized product of $\gamma _{n_a}$ and $\gamma _{n_c}$ with $(n_a+n_c)=n$, where ${n_a}$ and ${n_c}$ are the number of valence and core occupancies:

$$\begin{aligned} \gamma _n = \frac{1}{n_a!n_c!} \mathcal {A} [\gamma _{n_a} \gamma _{n_c}] \end{aligned}$$

(77)

Clearly, the indices in $\gamma _{n_a}$ are all active whereas those in $\gamma _{n_c}$ are all hole spinorbitals. $\gamma _{n_c}$ can always be written as an anti-symmetric product of the $n_c$ 1-body density matrix elements with hole labels:

$$\begin{aligned} \gamma _n = \mathcal {A} [\gamma _1\bigotimes \gamma _1\bigotimes \gamma _1\bigotimes ...{n_c\, \text { times}}] \end{aligned}$$

(78)

with appropriate hole orbital indices. We thus see that $\gamma _{n_c}$ is completely product separable but for a general $\phi _\mu$, which is a multi-determinant function, $\gamma _{n_a}$ is not necessarily so. One can separate the product separable portion of $\gamma _{n_a}$ via cumulant decomposition where the cumulants are a measure of the extent of correlation beyond what is contained in the factorizable portion including Fermi correlation. One can introduce cumulants recursively in the following manner:

$$\begin{aligned} {\gamma _2}^{uv}_{wx}&= \gamma ^u_w\gamma ^v_x - \gamma ^u_x\gamma ^v_w + \lambda ^{uv}_{wx} \nonumber \\&= \mathcal {A} [\gamma ^u_w\gamma ^v_x] + \lambda ^{uv}_{wx}\end{aligned}$$

(79)

$$\begin{aligned} {\gamma _3}^{uvw}_{xyz}&= \mathcal {A} [\gamma ^u_x\gamma ^v_y\gamma ^w_z] + \mathcal {A} [\gamma ^u_x\lambda ^{vw}_{yz}] + \lambda ^{uvw}_{xyz} \end{aligned}$$

(80)

and so on [80].

The spin-free form of the above cumulant decomposition requires careful handling. One may imagine that a straightforward spin-free form of the cumulant decomposition should be possible by an appropriate summation over spin indices for the same set of orbitals in the upper and the lower array at the same place in the string. However, some reflection indicates that a cumulant decomposition involving only spin-free one-body reduced densities and appropriate spin-free cumulants is not generally possible for a non-singlet $\phi _\mu$, with a fixed $M_s$ value. Unless some spin-density matrix elements are introduced, generation of a spin-free $\varGamma$ is not possible. To circumvent this difficulty, Kutzelnigg and Mukherjee defined an $M_s$ averaged ensemble density matrix [80]:

$$\begin{aligned} {\varGamma _n}^{u_1u_2...u_n}_{v_1v_2...v_n}= \frac{1}{2S+1}{\sum _{M_s=-S,\sigma _1...\sigma _n}^{S}}{}^{M_s}{\gamma _n}^{{u_1}_{\sigma _1}{u_2}_{\sigma _2}...{u_n}_{\sigma _n}}_{{v_1}_{\sigma _1}{v_2}_{\sigma _2}...{v_n}_{\sigma _n}} \end{aligned}$$

(81)

where $^{M_s}\gamma _n$ is the spinorbital n-body-reduced density matrix for a given $\varGamma$ with a given value of $M_s$. Since this is an equally weighted sum of the n-RDMs for all $M_s$ values, it does not depend on $M_s$ at all and is invariant under rotation of the quantization axis. Kutzelnigg and Mukherjee demonstrated that, with such spin-free density matrices, a cumulant decomposition is possible which does not involve spin densities.

A consequence of this definition above is that the spin-free $\gamma ^u_v$ can be defined as a sum of two spinorbital densities $\gamma ^{u_\uparrow }_{v_\uparrow }$ and $\gamma ^{u_\downarrow }_{v_\downarrow }$ and can be written as

$$\begin{aligned} \gamma ^{u_\sigma }_{v_\sigma }= \frac{1}{2S+1}{\sum _{M_s=-1/2}^{1/2}}{}^{M_s}\gamma ^{u_\sigma }_{v_\sigma } \end{aligned}$$

(82)

For such $M_s$-averaged quantities, $\gamma ^{u_\sigma }_{v_\sigma }$ for both up and down spins are equal and hence each equals $1/2\gamma ^u_v$. With such simplifications, and introducing spin-free cumulants $\varLambda _n$ in a similar manner:

$$\begin{aligned}\varLambda ^{u_1u_2...u_n}_{v_1v_2...v_n}&= \frac{1}{2S+1}{\sum _{M_s=-s,\sigma _1...\sigma _n}^{s}} {}^{M_s}\lambda ^{{u_1}_{\sigma _1}{u_2}_{\sigma _2}...{u_n}_{\sigma _n}}_{{v_1}_{\sigma _1}{v_2}_{\sigma _2}...{v_n}_{\sigma _n}} \nonumber \\ &\quad \equiv \sum _{\sigma _1...\sigma _n}\lambda ^{{u_1}_{\sigma _1}{u_2}_{\sigma _2}...{u_n}_{\sigma _n}}_{{v_1}_{\sigma _1}{v_2}_{\sigma _2}...{v_n}_{\sigma _n}} \end{aligned}$$

(83)

It follows that:

$$\begin{aligned} {\varGamma _2}^{uv}_{wx} \equiv {\varGamma }^{uv}_{wx} = \gamma ^u_w\gamma ^v_x - \frac{1}{2}\gamma ^u_x\gamma ^v_w + \varLambda ^{uv}_{wx} \end{aligned}$$

(84)

From now on, we would take the indices $u,v,...$, etc., as orbitals and not spinorbitals.

In the general case of a $k$-body spin-free RDM, the cumulant decomposition would require the concept of partitions $i$ of $k$ labels with $N_i$ as the length of the upper and lower strings appearing in the products of $\varLambda$s in this partition. For the uniformity of notation, we would introduce the notation $\varLambda _1$ for $\gamma$. The product of $\varLambda$s are all connected in each partition $i$ due to connectivity of type (C). All the terms contributing to the partition are connected among themselves in a sense which is best explained by an example. Let us consider a partition of length $4$ with labels as shown: $(^{u_1u_2u_3u_4}_{v_1v_2v_3v_4})$. Here the upper indices are labeled by the index $u$ and lower indices are labeled by the index $v$ and the numeral in the subscript indicates the column to which each index belongs. If two terms from all the products of cumulants generated have the connectivities as shown in Figs. 5 and 6, then the connectivities of these two terms can be inferred by following the type of pairings of the upper and lower indices.

Fig. 5

One mode of connectivity corresponding to partition $i$ and length $N_i=4$ with four $\gamma$ matrix elements

Fig. 6

Another mode of connectivity corresponding to partition $i$ and length $N_i=4$ where the pairs ($u_3,u_4$) and ($v_1,v_4$) form the upper indices of a $\varLambda _2$

In Fig. 5, $u_1$ and $v_2$ are paired which indicates that $u_2$ and $v_1$ are also connected being the corresponding indices. Columns $1$ and $2$ are thus connected. Further, since $u_2$ and $v_3$, $u_3$ and $v_4$ and $u_4$ and $v_1$ are also connected, the entire composite is connected. The overall contribution of this term is $\frac{1}{8}\varLambda ^{u_1}_{v_2}\varLambda ^{u_2}_{v_3}\varLambda ^{u_3}_{v_4}\varLambda ^{u_4}_{v_1}$. The prefactor $1/8$ depends entirely on the type of connectivity of cross-contractions between the four $\varLambda$s. In Fig. 6, the pairing of $u_1$ and $v_2$ and $u_2$ and $v_3$ produce two $\varLambda$s of rank $1$, thereby connecting columns $1$,$2$ and $3$, and $u_3$ and $u_4$ shown in the box are a part of a two body $\varLambda$, automatically making all four indices connected. The overall factor of this term in given by$\frac{1}{2}\varLambda ^{u_1}_{v_2}\varLambda ^{u_2}_{v_3}\varLambda ^{u_3u_4}_{v_1v_4}$ The accompanying factor $1/2$ again depends on connectivity and rank of $\varLambda$s.

Let us use indices $j_i=1$ and $j_i=2$ to indicate the different patterns of connectivity belonging to the same partition $i$. The factors, accompanying every connectivity symbolized as $\kappa ^i_{j_i}$, are $\kappa ^i_{1}=-1/8$ and $\kappa ^i_{2}=-1/2$, respectively. The factor $\kappa ^i_{j_i}$ is unique to every $j_i$ and does not depend on the connectivity of other partitions. We have collected in Table 1 the quantities $j_i$ and $\kappa _{j_i}$ for partitions of length 2, 3, and 4. Instead of the diagrams like Figs. 5 and 6, we have used a compact algebraic notation to describe the connectivity. In this notation, the connections involving $1-$body RDM are denoted by $u_P-v_Q$ to indicate the connection of a $u$ in the $P$th column with a $v$ in the $Q$th column. Similarly, an n-tuple connection is indicated by $(u_{k_1},u_{k_2},...,u_{k_n})-(v_{l_1},v_{l_2},...,v_{l_n})$. This particular n-tuple will generate a $\varLambda ^{n}$ where for an upper index $k_1$ the matching is with lower index $l_1$. Using this notation, Fig. 5 has the connectivity [1–2, 2–3, 3–4, 4–1] and Fig. 6 has the connectivity [1–2, 2–3, (3 4)–(4 1)].

Table 1 List of connectivities and the factors $\kappa ^i_{j_i}$ for partitions $N_i$ of ranks 2, 3, and 4

Generally speaking, the total contribution $\Delta ^{N_i}$ of a string of operators of length $N_i$ can be compactly written as

$$\begin{aligned} \Delta ^{N_i}= \sum _{j_i=\text {pattern }{N_{k_i}}} {\varPi _{k_i}}_{(\sum _{k_i}N_{k_i}=N_i)} \varLambda ^{N_{k_i}} \kappa ^i_{j_i} \end{aligned}$$

(85)

It is now straightforward to write the cumulant decomposition of a $k-$body RDM. Let us assume that $\varGamma ^{(k)}$ has a set of upper and lower indices $u_1...u_k$ and $v_1...v_k$, respectively. We may then generate all possible terms of the cumulant decomposition by first partitioning the set of $u$ and $v$ in all possible manner with the restriction ${N_1}\ge {N_2}\ge {N_3}...$ starting with $(^{u_1}_{v_1})$. The value of the partition $N_i$, $\Delta^{N_i}$, will be labeled by unique ordered set of $u$ and $v$ and the overall contribution of the first term of the cumulant will be $\varPi _i \Delta ^{N_i}$, where the upper indices $u_1...u_k$ and lower indices $v_1...v_k$ appear consecutively in the $\varGamma$ with all possible connectivity in a given partition. Every pattern characterized by the set $\{N_i\}$ will be given a class symbol $cl$. The overall contribution for every class $cl$ will be generated by simultaneous interchange of the same upper and lower indices in all possible ways between the different partitions. Clearly the possible permutations depend on the class $cl$ and contribution of all the terms of the class $cl$ can be compactly written as $P_{cl} \Delta^{cl}$. The overall contribution of the $\varGamma ^{(k)}$ can be then compactly written as

$$\begin{aligned} \varGamma ^{(k)}=\sum _{cl} P_{cl} \Delta ^{cl} \end{aligned}$$

(86)

with the identity permutation of $P_{cl}$ as the unity operator given by ${\mathbb{1}}$.