Particle rank

The cumulants have the important property (and this holds for the cumulants of arbitrary particle rank) of being additively separable. For a noninteracting... 

On the r-h-s there should be excitation operators of the same and of lower particle ranks. 

Eor the explicit expression see Ref [20].) This is a hierarchy of equations, first proposed independently by Nakatsuji [35] and Cohen and Frishberg [36]. In Eq. (153) Yj is expressed through 72 and 73, in Eq. (154) 72 is expressed through 73 and 74, and so on. This hierarchy is of no direct practical use, because there is no justification for a truncation of the ji at some k. Valdemoro [4] found an ingenious way to approximate the for higher particle rank k in terms of those of lower k and was so able to achieve a tmncation of the hierarchy. This reconstruction... 

We could, of course, start from the CSE/t, express the y in terms of the k/t, and so arrive at a hierarchy of equations for the Xk that can be truncated at some particle rank k. We prefer, however, to derive such a hierarchy directly, bypassing that in terms of the y, as will be done in Section IV.C. 

While the (one-particle) Brillouin condition BCi has been known for a long time, and has played a central role in Hartree-Fock theory and in MC-SCF theory, the generalizations for higher particle rank were only proposed in 1979 [38], although a time-dependent formulation by Thouless [39] from 1961 can be regarded as a precursor. 

We expand the Hamiltonian and the IBCj in terms of a perturbation parameter p. in the spirit of M0ller-Plesset perturbation theory [34]. Details are found in Ref. [25]. We need not worry about the particle rank to which we have to go, since this is fully controlled by the perturbation expansion. We limit ourselves to a closed-shell state, such that the zeroth order is simply closed-shell Hartree-Fock. 

To zeroth order I2 and all of higher particle rank vanish, while y is idempotent and has eigenvalues 0 or 1. There is no first-order contribution to y or E. The only first-order contribution is in A2 ... 

In order not to overcharge the notation, we indicate the order in perturbation theory (PT) for a matrix element by a subscript (that we otherwise reserve for the particle rank), if the particle rank is obvious from the labels. For matrices like k2 we indicate the order of PT by a subscript such as The second-order energy is... 

Let us first discuss (i). The primary difficulty associated with the infinite BCH expansion comes from the fact that each term in the expansion generates operators of greater particle rank (i.e., involving a longer string of creation and annihilation operators) than the previous term. Thus it is necessary to assume some closure or truncation when constructing H. This is commonly cited as an obstacle... 

Unlike the density cumulant expansion, which can in principle be exact for certain states (such as Slater determinants), the operator cumulant expansion is never exact, in the sense that we cannot reproduce the full spectrum of a three-particle operator faithfully by an operator of reduced particle rank. However, if the density cumulant expansion is good for the state of interest, we expect the operator cumulant expansion to also be good for that state and also for states nearby. 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

Figure 3. A diagrammatic representation of the cumulant decomposition ([IV, A2]p 2)) for the three-particle operator drawn in Fig. 2. Four kinds of one- and two-particle operators are obtained. The double line is the contraction for the particle-rank reduction (closure), where the correlation is averaged with the effective field (i.e., density matrices).

The rewriting of commutators and anticommutators is guided by the simple rule that the particle rank of the operator should be reduced. The particle rank of an operator consisting of a string of p creation and q annihilation operators is 2 (p+q). A reduction in the particle rank by one can... 

It is seen that the particle rank is reduced by two as a result of the double commutator. 

Item Formula Molar Mass (g/mol) Number of Particles Rank... 

Again, note the reduction in particle rank.) The importance of this result is that when operating on any eigenfunction of (e.g., S 0> = hs s 1) 0 will yield a function whose eigenvalue r i/i is increased by h... 

Show by performing the following commutators and anticommutators that one achieves a lowering of the particle rank of the operators involved ... 

Operator representation in general, a normal-ordered operator splits into different terms having different excitation/de-excitation, hole/particle ranks. For example, the normal-ordered Hamiltonian ... 

Although the similarity-transformed Hamiltonian is quartic in the cluster amplitudes, the equations for the cluster amplitudes (13.2.32) need not contain all the amplitudes to this order. In Section 13.2.8, we use the cluster-commutation condition (13.2.36) to show that, for a general operator O of particle rank mo, the state... 

For each cluster operator f i, the particle rank and the excitation rank are both equal to n,-. If the commutator does not vanish, then its particle and excitation ranks are given by... 

In calculating the particle rank wq, we have added the particle ranks of all the operators and subtracted k since each commutator in (13.2.53) reduces the rank by 1, as discussed in Section 1.8. In calculating the excitation level 5, we have added the excitation ranks of all operators, noting that commutation does not change the excitation rank of the operators. 

The T1 transformation does not affect the particle rank of the Hamiltonian. Indeed, the only complication that occurs upon the transformation (13.7.20) is a loss of symmetry in the one- and two-electron integrals, with only the particle-permutation symmetry of the two-electron integrals retained as discussed in Section 13.7.4. Therefore, if we are prepared to work with integrals of reduced symmetry, the T1 transformation of the Hamiltonian (13.7.20) will simplify the subsequent manipulation of the coupled-cluster equations considerably, effectively reducing the complexity of the CCSD equations to that of the CCD equations. 

The singles and triples make their first appearance in the second-order equations (14.3.20) and are modified by higher-order corrections to the amplitudes. The quadruples do not enter to second order since the commutator of 4> and is a three-electron operator - see the discussion of excitation ranks and commutators in Section 13.2.8. In general, the nth-order excitations enter first to order n — 1 since the particle ranks of the commutators in the equations of order n — 1 are at most n. The only exception to this rule are the singles, which - because of the Brillouin theorem - enter the equations to second order. These results are summarized in Table 14.2. 

We now expand the similarity-transformed fluctuation potential in a BCH series as in (13.2.37). From a consideration of particle ranks (see the discussion in Section 13.2.8), only single and double commutators contribute to the perturbed energy, which may be written in the form... 

For the triples, inspection of particle ranks shows that only the middle term on the right-hand side of (14.4.44) ves a nonzero contribution. From (13B.2.2), we now obtain the following expression for the right-hand side ... 

Before considering the evaluation and simplihcadon of commutators and anticommutators, it is useful to introduce the concepts of operator rank and rank reduction. The (particle) rank of a string of creation and annihilation operators is simply the number of elementary operators divided by 2. For example, Ihe rank of a creation operator is 1/2 and the rank of an ON operator is 1. Rank reduction is said to occur when the rank of a commutator or anticommutator is lower than the combined rank of the operators commuted or anticommuted. Consider the basic anticommutation relation... 

The particle rank is reduced by 2. In manipulating and simplifying nested commutators, the Jacobi identity is often useful ... 

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