Multiple bond breaking

In an addition reaction, atoms are added to a double or triple bond. One bond of the multiple bond breaks so that two new bonds can form. To recognize an addition reaction, remember that two compounds usually react to form one major product. (Sometimes two isomers are formed.) The product has more atoms bonded to carbon atoms than the organic reactant did. A general example of addition to an alkene is given below. 

Alternative mechanisms consistent with the observations would require much higher energy transition states - transition states in which multiple bond breakings and makings occur simultaneously, or even transition states of impossible geometries. 

The method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

Of the methods listed above, only the noniterative CC approaches based on the partitioning of the similarity-transformed Hamiltonian (24-28) and the (C)R-CC approaches of refs 9,13-18,20,21, which employ the MMCC formalism (P. 13, 14, 18, 19, 21, 45, 106, 107), retain the simplicity and the black-box character of the standard CCSD(T) or CCSD(TQf) methods. One of the two goals of the present work is the development of a new class of the MMCC-based black-box methods for multiple bond breaking. 

Because of the apparent relationship between the QVMMCC or QMMCC methods and ECC/XCC approaches, we also explore the usefulness of the ECC theory in studies of multiple bond breaking, using N2 as an example. Instead of the strictly bi-variational ECC method of Arponen and Bishop 114-123), we... 

Since the MBPT(3)-like wave functions ( Pq) proved to be rather ineffective in constructing the highly accurate corrections for multiple bond breaking, we decided to consider an alternative approach, in which iTg) in eqs (16) or (22) has a CC-like exponential form. 

The LMMCC or CR-CCSD[T] and CR-CCSD(T) methods cannot describe multiple bond breaking (cf, e.g., ref 14) because of the absence of the quadratic... 

These terms are present in the wave function T(,) defining variant b of the CR-CCSD(TQ) theory (the CRCCSD(TQ),b approach of ref 14). In consequence, the CR-CCSD(TQ),b results for multiple bond breaking are considerably better than the CRCCSD[T] and CR-CCSD(T) results (14). The Tj and other bilinear terms in cluster amplitudes, such as Jso... 

Thus, we can expect further improvements in the description of multiple bond breaking by the QMMCC method. This statement parallels similar findings by Head-Gordon et al. (27, 28, 128, 129), who considered the quadratic variant of the ECC theory of Arponen and Bishop (114 -123), in which the energy is calculated by imposing the stationary conditions for the asymmetric, doubly connected, energy functional... 

The primary motivation behind the QMMCC approximations is the need to improve the CR-CCSD(TQ) description of more complicated types of multiple bond breaking, such as triple bond breaking in N2. The CRCCSD[T], CR-CCSD(T), and CR-CCSD(TQ) methods are capable of providing an excellent description of PESs involving single and double bond dissociation (9, 13, 15, 17-21, 111), but the CR-CCSD(T) and CR-CCSD(TQ) results for triple bond breaking are less accurate (14, 18, 21). This can be seen by analyzing the CR-... 

Where H is the similarity-transformed Hamiltonian, eq (14), with respect to two independent cluster operators T and Z or, more precisely, with respect to the excitation operator T and the deexcitation operator Z The advantage of eq (36) over the expectation value of the Hamiltonian with the CC wave function, which can also improve the results for multiple bond breaking (28, 127), is the fact that EcC(z,j is a finite series in T and Z. Unfortunately, the power series expansions of (Z,7), eq (36), in terms of T and Z contain higher powers of... 

Neither the QMMCC(2,3) method nor its CR-CCSD[T] and CR-CCSD(T) counterparts are capable of describing multiple bond breaking (cf ref 14). Thus, in the present work we focus on the QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) approximations. In particular, we consider the QMMCC(2,4) approach, which allows us to understand the significance of the CR-CCSD(TQ) methods, when compared with the lower-order LMMCC and higher-order QMMCC models. A direct comparison of the QMMCC(2,4) and CR-CCSD(TQ),b energy expressions indicates that the CR-CCSD(TQ),b approach can be viewed as a sli tly modified QMMCC(2,4) theory, in which the only bilinear term of the type, multiplying [M 2)+T M. ( )ym. eq (40) and (2 2 1 ) lowest-order (r2 ) term. This is... 

As mentioned in the Introduction, the renormalized (R) and completely renormalized (CR) CC approaches, which represent new classes of noniterative single-reference CC methods that are capable of removing the failing of the standard CCSD(T) and similar methods at larger internuclear separations, are based on the formalism of the method of moments of CC equations (MMCC) [11-13, 30-32,36, 75, 76]. Thus, we begin our description of the R-CC and CR-CC methods and their quadratic MMCC (QMMCC) extension for multiple bond breaking with a synopsis of the general MMCC theory. 

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