Excitation rank

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

Rapid convergence of molecular properties with respect to the excitation rank is obtained with the systematic approximations of coupled-cluster (CC) theory... 

Following the customary terminology, we will call inactive holes the inactive occupied orbitals, doubly filled in every model CSF. The inactive particles will refer to aU the orbitals unoccupied in every CSF. Orbitals which are occupied in some (singly or doubly) but unoccupied in others are the active orbitals. In our spin-free form, the labels are for orbitals only, and not for spin orbitals. From the mode of definition, no active orbital can be doubly occupied in every model CSF. We want to express the cluster operator T, inducing excitations to the virtual functions, in terms of excitations of minimum excitation rank, and at the same time wish to represent them in a manifestly spin-free form. To accomplish this, we take as the vacuum—for excitations out of 4> — the largest closed-shell portion of it, For each such vacuum, we redefine the holes and particles, respectively, as ones which are doubly occupied and unoccupied in < 0 a-The holes are denoted by the labels. .., etc. and the particle orbitals are denoted as a, etc. The particle orbitals are totally unoccupied in any or are necessarily... 

The general problem of spin-adaptation using multiple vacuua depending upon the model function the component of the wave operator exp(T ) acts upon, is a nontrivial and rather involved exercise. Here we will consider the simplest yet physically the most natural tmncation scheme in the rank of cluster operators T, where each such operator is truncated at the excitation rank two. For generating the working equations for the spin-adapted theory in this case, it is useful to classify the various types of excitation operators leading to various virtual CSFs as ... 

One can so define a hierarchy of Cl methods, called CI-SD, CI-SDT, CISDTQ etc. where S, D, T, Q stands for singly, doubly, triply, quadruply excited configurations. This hierarchy has the advantage over MP-PT that it does converge to full Cl if one let the excitation rank go to the number of electrons. This is, however, just what one wants to avoid. 

If one truncates the Cl expansion one gets a result which is variational, but not extensive, one can truncate the CC-expansion (in the exponential) without loosing extensivity. It is then, however, practically impossible to be variational as well [125]. Like for Cl one can define CC-SD, CC-SDT, CC-SDTQ etc. The n-scaling of the computational demands is similar to that for the corresponding steps in the Cl hierarchy. Like for the Cl hierarchy the CC hierarchy also converges to full Cl if the excitation rank goes to n, but the convergence is definitely faster. 

That is, those that require the lowest excitation energies. Later, a psychological mechanism began to work supported by economics the high-energy excitations are numerous and, because of that, very expensive and they correspond to a high number of electrons excited. Due to this, a reasonable restriction for the number of configurations in the Cl expansion is excitation rank. We will come back to this problem later. 

The function ijfQ can be expanded in Slater determinants of various excitation rank (we use intermediate normalization) tj/Q = +exci tations. Then, by equalizing the two expressions... 

Similarly, for all two-, three-body,... excitations involving only inactive orbitals, there cannot be any factorized F, since there are only inactive orbitals in the pair (Ip,Ap). Hence, for all inactive excitations, of excitation rank i, we can write the matrix equation to be solved as... 

Performing CI calculations with the inclusion of all excitations (for the assumed value of M), i.e. ih full CI, is not possible in praetieal ealeulations due to the too long expansion. We are foreed to truncate the CI basis somewhere. It would be good to terminate it in sueh a way that all essential (the problem is what we mean by essential) terms are retained. The most significant terms for the correlation energy come from the double excitations since these are the first excitations coupled to the Hartree-Fock function. Fig. 10.6. Smaller, although important, eontributions come from other excitations (usually of low excitation rank). We certainly wish that it would be like this for large molecules. Nobody knows what the truth is. 

Due to the missing deexcitation part (i.e. that which lowers the excitation rank, e.g., from doubles to singles) the operators and T commute, hence the operators Uk and expCT) also commute ... 

If sa is positive, then A HF represents an excited determinant of excitation rank (level) sa relative to the Hartree-Fock state. Conversely, if sa is negative, then A HF) represents an excited determinant of excitation rank 5/i. ... 

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 excitation rank so of a given term in the operator O satisfies the condition... 

Since the total excitation rank of a nonzero commutator is equal to the sum of the excitation ranks of the individual operators, we have established the following range of excitation ranks of the commutator (13.2.60) ... 

For k = 2mo, the commutator has a sharply defined excitation rank t — mo in all other cases, the commutator possesses a range of excitation ranks (13.2.65). 

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. 

Next, we drop all terms containing in agreement with the 2/j -b 1 rule and all terms containing and in agreement with the 2n -b 2 rule. In addition, we observe that the fifth and sixth terms in (14.3.73) vanish since the commutators involve too high excitation ranks. We are then left with just two terms ... 

The operator in the exponential is called cluster operator and can be further subdivided according to excitation ranks... 

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