Single, double, triple, and quadruple

MP4 4 Order Moller-Plesset Perturbation Theory (including Singles, Doubles, Triples and Quadruples by default) Energies only... 

The MPn method treats the correlation part of the Hamiltonian as a perturbation on the Hartree-Fock part, and truncates the perturbation expansion at some order, typically n = 4. MP4 theory incorporates the effect of single, double, triple and quadruple substitutions. The method is size-consistent but not variational. It is commonly believed that the series MPl, MP2, MP3,. .. converges very slowly. 

Coupled cluster is closely connected with Mpller-Plesset perturbation theory, as mentioned at the start of this section. The infinite Taylor expansion of the exponential operator (eq. (4.46)) ensures that the contributions from a given excitation level are included to infinite order. Perturbation theory indicates that doubles are the most important, they are the only contributors to MP2 and MP3. At fourth order, there are contributions from singles, doubles, triples and quadruples. The MP4 quadruples... 

One revolution of the worm advances the gear teeth in direct proportion to the number of the worm threads. The most commonly used worms have either one, two, three, or four separate threads and are referred to as single-, double-, triple-, and quadruple- thread worms. The number of threads is determined by counting the number of starts or entrances at the end of the worm. 

Fig. 2.27. Total collision probability and fractions of single, double, triple and quadruple collisions versus collision gas pressure. The transmission of the main beam /(is given on the right ordinate. Values are for an ion of collision cross section 5 x 10 cm and 1 cm collision path 10 Torr= 1.33 Pa. Reproduced from Ref. [9] with permission. John Wiley Sons, 1985.

CCSDTQ (CC singles, doubles, triples, and quadruples) (75-75), CCSDTQP (CC singles, doubles, triples, quadruples, and pentuples) (7P), etc. approaches are far too expensive for routine applications. For example, the full CCSDTQ method requires iterative steps that scale as ( g(/i )is the number of occupied (unoccupied) orbitals in the molecular orbital basis). This scaling restricts the applicability of the CCSDTQ approach to very small systems, consisting of 2 - 3 light atoms described by small basis sets. For comparison, CCSD(T) is an nln procedure in the iterative CCSD steps and an nl procedure in the non-iterative part related to the calculation of the triples (T) correction. In consequence, it is nowadays possible to perform the CCSD(T) calculations for systems with 10-20 atoms. The application of the local correlation formalism (80-82) enabled SchOtz and Werner to extend the applicability of the CCSD(T) approach to systems with 100 atoms (53, 83, 84). 

The convergence behaviour of the MP expansion is obviously crucial to the success of the method, and numerous investigations have been carried out [9, 10]. Neither MP2 nor MP3 are entirely satisfactory. These implicitly require the first-order wave function, which involves only double excitations. MP4, if carried out completely, involves single, double, triple and quadruple excitations. It is more expensive than CISD, say, but often produces better results. MP5 and higher orders are likely to be impractically expensive. 

Fig. 6.42 Schematic cross sections of co-rotating fully intermeshing twin screws for single-, double-, triple-, and quadrupled-flighted screws. [Reprinted by permission from M. L. Booy, Geometry of Fully Wiped Twin-Screw Equipment Polym. Eng. Sci., 18, 973 (1978).]...

Table 3.3 Mature kernel phenotype of normal and selected single, double, triple, and quadruple recessive maize genotypes ...

FIGURE 15-7 Single, Double, Triple, and Quadruple Bonds. 

S. A. Kucharski and R.. Bartlett, /. Chem. Phys., 97, 4282 (1992). The Coupled-Cluster Single, Double, Triple, and Quadruple Excitation Method. 

CCSDTQ (CC singles, doubles, triples, and quadruples) (75-7S), CCSDTQP (CC singles, doubles, triples, quadruples, and pentuples) (79), etc. approaches are fat too expensive for routine applications. For example, the full CCSDTQ method requires iterative steps that scale as number of... 

Few studies have employed configuration interaction with all singles, doubles, triples, and quadruples (CISDTQ) because the number of triple and quadruple substitutions grows very rapidly with the number of electrons and basis functions (cf. Table 3). CISDTQ results are most commonly reported in benchmark full Cl studies to indicate the fraction of the basis set correlation energy recovered by triples and quadruples.17,22,39,801234,241,264 Nevertheless, the CISDTQ method has occasionally been used for benchmarking in cases where the full Cl was not technically feasible, because the CISDTQ results are expected to be very close to full Cl for small molecules. For systems with eight electrons or less at their equilibrium geometries, the CISDTQ method recovers more... 

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