Coupled-Cluster Theory The single-reference coupled-cluster (CC) theory, employing the exponential wave function ansatz |Psi0> = exp(T) |Phi> = exp(T1+T2+...) |Phi>, where T1, T2, etc. are the singly excited (1-particle-1- hole), doubly excited (2-particle-2-hole), etc. components of the cluster operator T and |Phi> is the single- determinantal reference state (e.g., the Hartree-Fock determinant), is widely recognized as one of the most accurate methods for describing ground electronic states of atoms and molecules. CC approaches provide the best compromise between relatively low computer costs and high accuracy. They are particularly effective in accounting for the dynamical correlation effects. For example, the CCSD(T) approach, which is a No**2 * Nu**4 (or N**6) procedure in the iterative CCSD steps and a No**3 * Nu**4 (or N**7) procedure in the non-iterative steps related to the calculation of triples (T3) energy corrections, is capable of providing results of the CISDTQ or better quality (CISDTQ is an iterative No**4 * Nu**6 or N**10 procedure) when closed-shell molecules are examined. Here and elsewhere in this section, No and Nu are the numbers of correlated occupied and unoccupied orbitals. Symbol N designates a measure of the system size in the following sense: N=2 means a simultaneous increase of the number of correlated electrons and basis functions by a factor of two. Unlike single- and multi-reference CI methods and some variants of multi-reference perturbation theory, all standard CC methods, such as CCSD or CCSD(T), provide a size extensive description of molecular systems, i.e. no loss of accuracy occurs due to the mere increase of the system size when CC calculations are performed. Thanks to numerous advances in both the formal aspects of CC theory and the development of efficient computer codes, the single-reference CC approaches, such as CCSD and CCSD(T), are nowadays routinely used in calculations for non-degenerate closed- and open-shell electronic ground states of atomic and molecular systems with up to 50 or so correlated electrons and up to 200-300 or so basis functions. The application of the local correlation formalism within the context of CC theory enables one to extend the applicability of the CCSD(T) and similar CC approaches to systems with approximately 100 light atoms (hundreds of correlated electrons and > 1000 basis functions). Generalizations of CC theory to open-shell, quasi-degenerate, and excited states are possible, via the multi-reference, renormalized, extended, equation-of- motion, and response CC formalisms, and some of these extensions (for example, the equation-of-motion CC methods for excited states) have become as popular as the multi- reference CI, multi-reference perturbation theory, or CASSCF methods. We should also add that CC theory is a fundamental many-body formalism, whose applicability ranges from electronic structure of atoms and molecules and nuclear physics to extended systems, phase transitions, condensed matter theory, theories of homogeneous electron gas, and relativistic quantum field theory, to mention a few examples. Examples of applications of quantum chemical CC methods in ab initio calculations for atomic nuclei using modern nucleon-nucleon interactions by Piecuch and co-workers are listed in the reference section below. A number of review articles have been written over the years and it is difficult to cite all of them here. We recommend that users of GAMESS planning to use CC/EOMCC methods read one or more reviews listed below: "Coupled-cluster theory" J. Paldus, in S. Wilson and G.H.F. Diercksen (Eds.), Methods in Computational Molecular Physics, NATO Advanced Study Institute, Series B: Physics, Vol. 293, Plenum, New York, 1992, pp. 99-194. "Applications of post-Hartree-Fock methods: a tutorial." R.J. Bartlett and J.F. Stanton, in K.B. Lipkowitz and D.B.Boyd (Eds.), Reviews in Computational Chemistry, Vol. 5, VCH Publishers, New York, 1994, pp. 65-169. "Coupled-Cluster Theory: Overview of Recent Developments" R.J. Bartlett, in D.R. Yarkony (Ed.), Modern Electronic Structure Theory, Part I, World Scientific, Singapore, 1995, pp. 1047-1131. "Achieving chemical accuracy with coupled-cluster theory" T.J. Lee and G.E. Scuseria, in S.R. Langhoff (Ed.), Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, Kluwer, Dordrecht, The Netherlands, 1995, pp. 47-108. "Coupled-cluster Theory" J. Gauss, in Encyclopedia of Computational Chemistry, P.v.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P.A. Kollman, H.F. Schaefer III, P.R. Schreiner (Eds.) Wiley, Chichester, U.K., 1998, Vol. 1, pp. 615-636. "A Critical Assessment of Coupled Cluster Method in Quantum Chemistry" J. Paldus and X. Li, Adv. Chem. Phys. 110, 1-175 (1999), "EOMXCC: A New Coupled-Cluster Method for Electronically Excited States" P. Piecuch and R.J. Bartlett, Adv. Quantum Chem. 34, 295-380 (1999). "An Introduction to Coupled Cluster Theory for Computational Chemists" T.D.Crawford, H.F.Schaefer in K.B. Lipkowitz and D.B.Boyd (Eds.), Reviews in Computational Chemistry, Vol. 14, VCH Publishers, New York, 2000, pp. 33-136. "In Search of the Relationship between Multiple Solutions Characterizing Coupled-Cluster Theories" P. Piecuch and K. Kowalski, in J. Leszczynski (Ed.), Computational Chemistry: Reviews of Current Trends, Vol. 5, World Scientific, Singapore, 2000), pp. 1-104. "Recent Advances in Electronic Structure Theory: Method of Moments of Coupled-Cluster Equations and Renormalized Coupled-Cluster Approaches" P. Piecuch, K. Kowalski, I.S.O. Pimienta, M.J. McGuire, Int. Rev. Phys. Chem. 21, 527-655 (2002). "New Alternatives for Electronic Structure Calculations: Renormalized, Extended, and Generalized Coupled-Cluster Theories" P. Piecuch, I.S.O. Pimienta, P.-F. Fan, and K. Kowalski, in J. Maruani, R. Lefebvre, and E. Brandas (Eds.), Progress in Theoretical Chemistry and Physics, Vol. 12, Advanced Topics in Theoretical Chemical Physics, Kluwer, Dordrecht, 2003, pp. 119-206. "Coupled Cluster Methods" J. Paldus, in Handbook of Molecular Physics and Quantum Chemistry, edited by S. Wilson (Wiley, Chichester, 2003), Vol. 2, pp. 272-313. "Method of Moments of Coupled-Cluster Equations: A New Formalism for Designing Accurate Electronic Structure Methods for Ground and Excited States" P. Piecuch, K. Kowalski, I.S.O. Pimienta, P.-D. Fan, M. Lodriguito, M.J. McGuire, S.A. Kucharski, T. Kus, and M. Musial, Theor. Chem. Acc. 112, 349-393 (2004). "Noniterative Coupled-Cluster Methods for Excited Electronic States" P. Piecuch, M. Wloch, M. Lodriguito, and J.R. Gour, in Progress in Theoretical Chemistry and Physics, Vol. 15, Recent Advances in the Theory of Chemical and Physical Systems," edited by S. Wilson, J.-P. Julien, J. Maruani, E. Brandas, and G. Delgado-Barrio (Springer, Berlin, 2006), pp. XXX-XXXX, in press. "Bridging Quantum Chemistry and Nuclear Structure Theory: Coupled-Cluster Calculations for Closed- and Open-Shell Nuclei" P. Piecuch, M. Wloch, J.R. Gour, D.J. Dean, M. Hjorth- Jensen, and T. Papenbrock, in V. Zelevinsky (Ed.), Nuclei and Mesoscopic Physics: Workshop on Nuclei and Mesoscopic Physics WNMP 2004, AIP Conference Proceedings, Vol. 777, AIP Press, 2005, pp. 28-45. These reviews point to the other review articles and many original papers. The list of original papers relevant to CC/EOMCC methods implemented in GAMESS is provided below. available computations (ground states) The CC programs incorporated in GAMESS enable user to perform conventional LCCD, CCD, CCSD, CCSD[T] (also known as CCSD+T(CCSD)), CCSD(T), and CCSD(TQ) calculations, renormalized (R) and completely renormalized (CR) CCSD[T], CCSD(T), and CCSD(TQ) calculations, and calculations using the rigorously size extensive completely renormalized CR- CC(2,3) (or CR-CCSD(T)L) approach for closed-shell RHF references. Performance of the ground-state CC methods has been discussed in a number of places (cf. the review articles mentioned above and references listed at the end of the "Coupled-Cluster Theory" section). Methods such as, for example, CCSD(T), CR-CC(2,3), and CCSD(TQ) provide excellent results for molecules in or near the equilibrium geometries. Almost all CC methods are excellent in describing dynamical correlation, while being relatively inexpensive and easy to use. One must remember, however, that the conventional single-reference CC methods, such as CCSD(T), should not be applied to bond breaking, diradicals, and other quasi-degenerate states, particularly (but not only) when the RHF determinant is used as a reference. In some of the most frequent cases of electronic quasi-degeneracies, including single-bond breaking and diradicals, the CR-CCSD(T), CR-CCSD(TQ), and CR-CC(2,3)= CR-CCSD(T)L methods can be used instead. The recently proposed CR-CC(2,3) approach seems particularly promising in this regard, although the CR-CCSD(T) and CR-CCSD(TQ) approaches are very useful as well. The CR-CC(2,3) method has costs similar to those characterizing the CCSD(T) approach, while providing the results of the very high, full CCSDT, quality for diradicals and single-bond breaking where CCSD(T) fails. At the same time, the accuracy of CR- CC(2,3) calculations is comparable to or, sometimes, even better than that obtained with the conventional CCSD(T) approach for closed-shell molecules near the equilibrium geometries. Just like CCSD(T), the CR-CC(2,3) approximation is rigorously size extensive, while working much better than CCSD(T) when non-dynamical correlation effects become large. CR-CC(2,3) (CCTYP=CR-CCL) is among the most attractive ground-state CC options in GAMESS, providing GAMESS users with the highly accurate energies in the closed-shell, single-bond breaking, and diradical regions of molecular potential energy surfaces, and a number of one-electron properties calculated at the CCSD level at a price of single, relatively inexpensive calculation of the CCSD(T) type. One of the interesting features of GAMESS that can be particularly useful in high accuracy calculations for closed-shell systems is the presence of the (TQ) corrections to CCSD energies among various ground-state CC options. This includes the factorized CCSD(TQ),b method suggested by Kowalski and Piecuch, which describes triples effects at the CCSD(T) level, using noniterative steps that scale as N**7 with the system size, while providing information about the dominant effects due to quadruply excited clusters. The CCSD(TQ),b method is closely related to its CCSD(TQf) predecessor proposed by Kucharski and Bartlett. In fact, if desired, one can extract the CCSD(TQf) energy from the information printed in the GAMESS output when CCTYP=CCSD(TQ) or CR-CC(Q) as follows: CCSD + [R1-CCSD(TQ),A ? CCSD] * [CCSD(TQ),A DENOMINATOR] (the R1-CCSD(TQ),A method in the GAMESS output represents one of the renormalized CCSD(TQ) approaches, termed R- CCSD(TQ)-1,a, which are discussed below). The differences between the CCSD(TQ),b and CCSD(TQf) methods are minimal and the accuracies and costs of both approaches are virtually identical. In particular, both methods use relatively inexpensive noniterative steps that scale as N**6 or N**7 with the system size to determine the quadruples corrections. The unique features of the ground-state CC code in GAMESS are the renormalized (R) and completely renormalized (CR) CCSD[T], CCSD(T), and CCSD(TQ) methods [see K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18-35 (2000), idem., ibid. 113, 5644-5652 (2000), and P. Piecuch and K. Kowalski, in J. Leszczynski (Ed.), Computational Chemistry: Reviews of Current Trends, Vol. 5, World Scientific, Singapore, 2000, pp. 1-104], and the most recent (Fall 2005), rigorously size extensive formulation of CR-CCSD(T), termed CR-CC(2,3) or CR-CCSD(T)L [see P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105-1 - 224105-10 (2005) and P. Piecuch, M. Wloch, J.R. Gour, and A. Kinal, Chem. Phys. Lett. 418, 467-474 (2006)]. All of these approaches are based on the more general formalism of the method of moments of coupled-cluster equations (MMCC; biorthogonal MMCC in the case of CR-CC(2,3)), developed by the Piecuch group at Michigan State University. They remove or considerably reduce the pervasive failing of the conventional CCSD[T], CCSD(T), and CCSD(TQ) approximations at larger internuclear separations and for diradical systems, while preserving the ease of use and the relatively low cost of the single-reference methods of the CCSD(T) or CCSD(TQ) type. In analogy to the CCSD[T], CCSD(T), and CCSD(TQ) methods, the R-CCSD[T], R-CCSD(T), R- CCSD(TQ)-n,x (n=1,2;x=a,b), CR-CCSD[T], CR-CCSD(T), CR- CC(2,3), and CR-CCSD(TQ),x (x=a,b) approaches are based on an idea of improving the CCSD results by adding a posteriori noniterative corrections to CCSD energies. These corrections employ the generalized moments of CCSD equations (projections of the Schroedinger equation for the CCSD wave function on the triply (T) or triply and quadruply (TQ) excited determinants) and are designed by extracting the leading terms that define the theoretical difference between the CCSD and full CI energies. The CR- CCSD[T], CR-CCSD(T), and CR-CC(2,3) approaches are capable of eliminating the unphysical humps on the potential energy surfaces involving single bond breaking produced by the conventional CCSD[T] and CCSD(T) methods. They also significantly improve the poor description of diradical species (for example, diradical transition states and intermediates) by the CCSD[T] and CCSD(T) methods. What is important in practical applications, the CR-CCSD(T) and CR- CC(2,3) approaches are capable of providing a good balance between the dynamical and nondynamical correlation effects when the diradical and closed-shell structures have to be examined together. The rigorously size extensive CR-CC(2,3) method is particularly effective in this regard, although the older and somewhat less expensive CR-CCSD(T) approach is very useful as well. The R-CCSD[T] and R-CCSD(T) approaches may improve the CCSD[T] and CCSD(T) results at intermediate internuclear separations, but they usually fail at larger distances. The CR-CCSD[T], CR-CCSD(T), and CR-CC(2,3) methods are better in this regard, since they often provide a very good description of single bond breaking at all internuclear separations. This includes various cases of unimolecular dissociations and exchange and bond insertion chemical reactions, in which single bonds break and form. We DO NOT recommend applying the CR- CCSD[T], CR-CCSD(T), and CR-CC(2,3) approaches to multiple bond breaking, although some types of multiple bond stretching can be described by these methods very well if the relevant stretches of chemical bonds are not too large. In general, however, multiple bond dissociations require using the higher-order methods, such as the completely renormalized CCSD(TQ) and CCSDT(Q) approaches (the CR- CCSD(TQ) methods are available in GAMESS), the so-called MMCC(2,6) method, and the more recent generalized and quadratic MMCC methods, if the single-reference approach is preferred, or the multi-reference CC methods of the state- universal and state-specific type (some of the most promising approaches in these categories, including active- space and state-universal CC methods, will be included in GAMESS in the future). In particular, the CR-CCSD(TQ) approaches available in GAMESS are reasonably accurate in situations involving double bond dissociations and a simultaneous stretching or breaking of two single bonds. They may work reasonably well even when the triple bond stretching or breaking is examined, but the results for more complicated cases of bond breaking are not as good as those that one can obtain with the best multi-reference approaches. A detailed description of the R-CCSD[T], R- CCSD(T), CR-CCSD[T], CR-CCSD(T), CR-CC(2,3), R-CCSD(TQ), and CR-CCSD(TQ) approaches and other MMCC methods can be found in several papers by Piecuch and coworkers listed at the very end of the "Coupled-Cluster Theory" section. Unlike the newest CR-CC(2,3) approximation, the somewhat older R-CCSD[T], R-CCSD(T), CR-CCSD[T], CR-CCSD(T), R- CCSD(TQ), and CR-CCSD(TQ) methods are not strictly size extensive, i.e. there are unlinked terms in the MBPT (many- body perturbation theory) expansions of the renormalized and completely renormalized [T], (T), and (TQ) corrections to CCSD energies. This has little or no effect on bond breaking (on the contrary, the CR-CCSD[T], CR-CCSD(T), and CR-CCSD(TQ) potential surfaces are MUCH better than potential energy surfaces obtained in the standard and size extensive CCSD[T], CCSD(T), and CCSD(TQ) calculations), but lack of strict size extensivity may have an effect on the results of calculations for larger and extended systems. A lot depends on the values of T2 amplitudes and the chemical problem of interest. If the T2 amplitudes are small, then the overlap denominator expressions which define the renormalized [T], (T), and (TQ) corrections of the R- CCSD[T], R-CCSD(T), CR-CCSD[T], CR-CCSD(T), R-CCSD(TQ), and CR-CCSD(TQ) methods are close to 1, in which case there is no major problem. If the T2 amplitudes are large, then these denominators may become significantly greater than 1. This behavior of the R-CCSD[T], R-CCSD(T), CR-CCSD[T], CR- CCSD(T), R-CCSD(TQ), and CR-CCSD(TQ) denominator expressions is extremely useful for improving the results for bond breaking, since the denominators defining the renormalized [T], (T), and (TQ) corrections damp the unphysical values of the standard [T], (T), and (TQ) corrections at larger internuclear separations or when the wave function gains a significant multi-reference character. The same applies to diradical species, where the standard [T], (T), and (TQ) corrections produce unphysical results and need damping that the renormalized methods provide. However, for larger many-electron systems (with 50 correlated electrons or more), the denominators defining the renormalized [T], (T), and (TQ) corrections may "overdamp" the [T], (T), and (TQ) energy corrections. On the other hand, the renormalized [T], (T), and (TQ) energy corrections are constructed using the cluster amplitudes resulting from the size extensive CCSD calculations. Moreover, it is often the case that the number of correlated electrons used in CC calculations for larger molecules (and only these electrons are used in constructing the renormalized [T], (T), and (TQ) corrections to CCSD energies) is much smaller than the total number of electrons. Thus, the consequences of the lack of strict size extensivity of the R-CCSD[T], R- CCSD(T), CR-CCSD[T], CR-CCSD(T), R-CCSD(TQ), and CR- CCSD(TQ) methods do not have to be serious for larger systems, particularly when one examines, for example, the relative energies of stationary points along the reaction pathways relative to the relevant reactants (see comments below). A number of interesting chemical problems involving smaller and medium size polyatomic diradical systems, including, for example, the Cope rearrangement of 1,5-hexadiene, the cycloaddition of cyclopentyne to ethylene, the isomerizations of bicyclopentene and tricyclopentane into cyclopentadiene, the thermal stereomutations of cyclopropane, and the relative energetics of dicopper systems relevant to molecular oxygen activation by copper metalloenzymes, where the standard CCSD(T) approach and, in some cases, the low-order multi- reference perturbation theory methods encounter serious difficulties, have been successfully examined with the CR- CCSD(T) approach, demonstrating that problems of size extensivity in CR-CCSD(T) calculations are of no major significance in molecules of these sizes. But one may have to be more careful when chemical systems have more than 50 correlated electrons. Extensive numerical tests indicate that lack of strict size extensivity has little (fraction of a millihartree or so) effect on the results of the CR- CCSD[T], CR-CCSD(T), and CR-CCSD(TQ) calculations for smaller systems. For larger systems, such as the glycine dimer described by the 6-31G basis set, the departure from rigorous size extensivity, as measured by forming the difference of the sum of the energies of isolated glycine molecules from the energy of the dimer consisting of glycine molecules at very large (200 bohr) distance, is ca. 3 millihartree (2 kcal/mol). The violation of strict size extensivity by the CR-CCSD(T) methods has been estimated at approximately 0.5 % of the total correlation energy (changes in the correlation energy if the relative energies along reaction pathways are examined), which is often a small price to pay considering the significant improvements that the renormalized CC methods offer for potential energy surfaces and diradicals and the ease with which the CR-CC calculations can be performed. IMPORTANT PRACTICAL ADVICE: In studies of reaction pathways with the CR-CCSD(T) approach, where reactants and products are connected by one or more transition states and intermediates and where there are two or more reactants, we STRONGLY RECOMMEND that the user of CR-CCSD(T) proceeds in a manner similar to multi- reference CI calculations. Thus, we advise to calculate the energies of transition states, intermediates, and products relative to reactants, using the total CR-CCSD(T) energy of a noninteracting complex formed by reactants (reactants separated by a large distance, say, 200 Angs.) as the reference energy of reactants rather than the sum of the CR-CCSD(T) energies of isolated reactants. This reduces the possible size extensivity errors in the CR-CCSD(T) calculations for larger systems to a minimum, since all species along a reaction pathway (including reactants, transition states, intermediates, and products) are treated then in the same, well balanced, manner. Similar remarks apply to the CR-CCSD(TQ) (and all R-CC) calculations. None of the above has to be done when the CR-CC(2,3) approach is employed, since CR-CC(2,3) is size extensive and the CR- CC(2,3) energy of A+B equals the sum of CR-CC(2,3) energies of A and B. The rigorously size extensive modifications of the CR- CC methods have recently (2005) been developed, using the idea of locally renormalized methods, such as LR-CCSD(T), which lead to size extensive results when localized orbitals are employed, and, in an alternative formulation, the idea of exploiting the left CC states combined with the so-called biorthogonal MMCC theory. The latter development seems particularly attractive. The resulting CR-CC(2,3) method, also called CR-CCSD(T)L, which combines the best features of CCSD(T) and CR-CCSD(T) and which we already mentioned above, satisfies the following criteria: (i) is at least as accurate as (sometimes more accurate than) CCSD(T) for nondegenerate ground states, (ii) provides highly accurate results for single-bond breaking and diradicals with the noniterative No**3 * Nu**4 steps similar to those of CCSD(T) and CR-CCSD(T),(iii) is more accurate than the CR-CCSD(T), LR-CCSD(T), and other non- iterative triples CC approaches, such as CCSD(2)T, which all aim at eliminating the failures of CCSD(T) in the diradical/bond breaking regions, and (iv) is rigorously size extensive without localizing orbitals. The criterion (ii) of a highly accurate description at the triples level of CC theory is defined here by the accuracy provided by the full CCSDT approach, which is almost exact in studies of diradicals and single-bond breaking, but also limited to very small systems with up to 2-3 light atoms due to very expensive iterative No**3 * Nu**5 steps that it uses. As demonstrated, for example, in recent studies of the relative energetics of the Cu2O2 systems with up to six ammonia ligands and thermal stereomutations of cyclopropane involving the trimethylene diradical as a transition state, CR-CC(2,3) has a wide range of applicability that includes larger polyatomic systems with up to 10-20 light and a few transition metal atoms. At the same time, CR-CC(2,3) provides a size extensive, highly accurate, and well balanced description of dynamical and nondynamical correlation effects in studies of single bond breaking and diradicals, particularly when the molecular systems involving a varying degree of diradical character along the relevant reaction pathways are examined. For all these reasons, the CR-CC(2,3) approach has been recently included in GAMESS. The CR-CC(2,3) method (invoked by typing CCTYP=CR-CCL in the input) seems to represent the most accurate non-iterative triples CC approximation formulated to date. Since the construction of the triples corrections to CCSD energies in CR-CC(2,3) calculations requires the determination of the left CCSD eigenstates, the CCPRP variable from $CCINP is automatically set at .TRUE. when variable CCTYP in $CONTRL is set at CR-CCL. As a result, by running the CR-CC(2,3) calculations, the user of GAMESS obtains a great deal of useful information in addition to excellent energetics (excellent as long as multiple bonds are not broken). This information includes the first-order reduced density matrices (printed in the PUNCH file), natural occupation numbers, and a variety of one-electron properties (e.g., electrostatic multipole moments) calculated at the CCSD level of theory. The ground-state CR-EOMCCSD(T) energies (cf. the next subsection), corresponding to CCTYP=CR-EOM calculations with NSTATE(1)=0,0,0,0,0,0,0,0, are printed as well. The CR-CC(2,3) approach has several variants, labeled with an additional letter, A-D (D means a full treatment of the perturbative denominators that are used to define triple excitation components, based on the diagonal matrix elements of the triples-triples block of the CCSD similarity transformed Hamiltonian; A means the crudest treatment of these denominators through bare orbital energies). Of all printed CR-CC(2,3) energies, the CR- CC(2,3),D value, which corresponds to the most complete variant of CR-CC(2,3), is the most accurate one and we STRONGLY RECOMMEND to use it in high accuracy calculations of molecular energetics. Because of the way the CR- CC(2,3),D approach is presently implemented in GAMESS, it is safer, for now, to use the simplified CR-CC(2,3),A or CR-CC(2,3),B models in numerical derivative calculations if there are orbital degeneracies (the aforementioned CCSD(2)T approach is equivalent to the CR-CC(2,3),A approximation). Because of some small simplifications in the present computer implementation of the CR-CC(2,3),D method, the CR- CC(2,3),D energies may slightly depend on the choice of molecular coordinate system if there are orbital degeneracies. Although changes in the most accurate CR- CC(2,3),D energies for systems with orbital degeneracies due to changes of the coordinate system are minimal (0.1 millihartree or less), it is safer to calculate numerical CR-CC(2,3) derivatives for systems with orbital degeneracies using the CR-CC(2,3),A or CR-CC(2,3),B approximations. For this reason, the CR-CC(2,3),A energy is automatically passed to the numerical derivative calculations with GAMESS if they are requested by the user, with the most complete CR-CC(2,3),D approach providing the most accurate energetics. We should emphasize, however, that the above technical issues are only limited to systems with orbital degeneracies. When there are no orbital degeneracies (which is the case when the highest molecular symmetry group is an Abelian group), the present implementation of the CR-CC(2,3),D approach in GAMESS leads to perfectly invariant energies. The issue of a slight (0.1 millihartree or less) dependence of the CR-CC(2,3),D (also CR-CC(2,3),C) energies on the choice of molecular coordinate system when orbital degeneracies are present is only temporary and will be eliminated in the future releases of GAMESS via a suitable modification of the CR- CC(2,3) code. Since CR-CC methods can find use in applications involving bond breaking and reaction pathways, one has to make sure that the underlying solution of the CCSD equations, on which the completely renormalized [T], (T), (2,3), and (TQ) corrections are based, represents the same physical solution as those defining other regions of a given molecular potential energy surface. This remark is quite important, since, for example, diradical regions of potential energy surface are characterized by larger cluster amplitudes and one has to make sure that the properly converged values of these amplitudes are obtained. GAMESS is equipped with a good algorithm for converging CCSD equations and a restart option discussed in a later part of this document that facilitate converging larger cluster amplitudes in difficult cases. The user is encouraged to examine various interesting elements of the CC input and output. In addition to CC energies, GAMESS prints the largest T1 and T2 cluster amplitudes obtained in the CCSD calculations, the T1 diagnostic, norms of T1 and T2 vectors, and the R-CCSD[T], R-CCSD(T), and R-CCSD(TQ) denominators that define the renormalized and completely renormalized triples and quadruples corrections. For example, bond breaking and diradical cases are characterized by larger cluster amplitudes (particularly, T2) and a significant increase in the values of the R-CCSD[T], R-CCSD(T), and CR-CCSD(TQ) denominators, which damp unphysical triples and quadruples corrections of the standard CCSD[T], CCSD(T), and CCSD(TQ) approximations, compared to closed-shell regions of potential energy surface. As already mentioned, the CR- CC(2,3) calculations provide user with one-particle reduced density matrices, natural occupation numbers, and a number of one-electron properties, calculated at the CCSD level, in addition to the highly accurate CR-CC(2,3) and some other CR-CC energies. available computations (excited states) The equation of motion coupled cluster (EOMCC) method and the closely related response CC and symmetry-adapted cluster configuration interaction (SAC-CI) approaches provide very useful extensions of the ground-state CC theory to excited states. In the EOMCC theory, the excited states |PsiK> are obtained by applying the excitation operator R = R0 + R1 + R2 + ..., where R0, R1, R2, etc. are the reference, singly excited (1-particle-1-hole), doubly excited (2-particle-2-hole), etc. components of R, to the CC ground state |Psi0>. Thus, the EOMCC expression for the excited state |PsiK> is |PsiK> = R |Psi0> = R exp(T) |Phi> = (R0+R1+R2+...) exp(T1+T2+...) |Phi> . In practice, the standard EOMCC calculations are performed by diagonalizing the CC similarity transformed Hamiltonian H-bar = exp(-T) H exp(T) in the space of excited determinants included in the cluster operator T and the excitation operator R. For example, the basic EOMCCSD calculations defined by the truncation schemes T=T1+T2 and R=R0+R1+R2 are performed by diagonalizing exp(-T1-T2) H exp(T1+T2) in the space of singly and doubly excited determinants defining the CCSD (T=T1+T2) approximation. The direct result of such diagonalization are the vertical excitation energies omegaK = EK - E0 (EK and E0 and the excited- and ground- state energies, respectively). The EOMCC methods have several advantages. The most expensive steps of the basic EOMCCSD calculations scale only as No**2 * Nu**4 and yet the accuracy of the EOMCCSD results for excited states dominated by one-electron transitions (single excitations or singles or 1-particle-1- hole excitations) is very good. The errors in the EOMCCSD calculations for such states are often on the order of 0.1- 0.3 eV, which is acceptable in many applications. The EOMCCSD approximation and other standard EOMCC methods have an ease of application that is not matched by the multi- reference techniques, since formally the EOMCC theory is a single-reference formalism. Thus, the EOMCC methods are particularly well suited for calculations where active orbital spaces required in CASSCF-related calculations become very large or difficult to identify. Given sufficient computational resources, the EOMCCSD calculations for systems involving up to 10-20 light or a few heavy atoms are nowadays (meaning year 2004 and on) routine. The EOMCCSD method works reasonably well for excited states dominated by singles, but it fails to describe states dominated by two-electron transitions (doubles) and potential energy surfaces along bond breaking coordinates. These failures can be remedied by the CR- EOMCCSD(T) approximations described below. The EOMCC programs incorporated in GAMESS enable user to perform standard EOMCCSD calculations employing the RHF reference determinant. They also enable to improve the EOMCCSD results by adding the state-selective noniterative corrections due to triples to the ground and excited-state CCSD/EOMCCSD energies via the completely renormalized EOMCCSD(T) (CR-EOMCCSD(T)) approaches developed by the Piecuch group. The CR-EOMCCSD(T) approaches represent extensions of the ground-state CR-CCSD(T) method to excited states. In particular, in analogy to the CR-CCSD(T) approximation, the excited-state CR-EOMCCSD(T) approaches are based on the formalism of the method of moments of coupled-cluster equations (MMCC). Moreover, the CR- EOMCCSD(T) methods preserve the relatively low computer costs and ease of use of the ground-state CCSD(T) calculations. The most expensive noniterative steps of the CR-EOMCCSD(T) approach scale as No**3 * Nu**4. The CR- EOMCCSD(T) option (CCTYP=CR-EOM) is a unique feature of GAMESS. At this time, the applicability of the EOMCCSD and CR-EOMCCSD(T) codes in GAMESS is limited to singlet states. The main advantage of the MMCC-based CR-EOMCCSD(T) approximations, in addition to their "black-box" character and relatively low computer costs, is their high (0.1 eV or so) accuracy in the calculations of excited states dominated by double excitations and excited-state potential energy surfaces along bond breaking coordinates, for which the standard EOMCCSD method fails (producing errors on the order of 1 eV or even bigger). In this regard, the CR- EOMCCSD(T) methods are quite similar to the CR-CCSD(T) approach, which is capable of describing ground-state potential energy surfaces involving single bond breaking. As a matter of fact, when limited to the ground-state problem, the CR-EOMCCSD(T) approximations become essentially identical to the CR-CCSD(T) method. There are, however, small differences and the CR-EOMCCSD(T) energies of the ground state are slightly different than the CR- CCSD(T) energies discussed in the earlier section. This is due to the fact that the original CR-CCSD(T) approximation has been designed for the ground states only, whereas the CR-EOMCCSD(T) approaches apply to ground and excited states and this required small modifications in the ground-state energy equations. A few different variants of the CR-EOMCCSD(T) method, termed the CR-EOMCCSD(T),IX, CR-EOMCCSD(T),IIX, and CR- EOMCCSD(T),III approaches (X=A,B,C,D) have been proposed and included in GAMESS. Types I, II, and III refer to three different ways of defining the approximate wave functions |PsiK> that are used to construct the CR- EOMCCSD(T) triples corrections to EOMCCSD energies in the underlying MMCC formalism. Types I and II use perturbative expressions for |PsiK> in terms of cluster components T1 and T2 and excitation components R0, R1, and R2. Type III uses additional CISD (CI singles and doubles) calculations in designing the wave functions |PsiK> that enter the CR- EOMCCSD(T) triples corrections. Thus, user should be aware of the fact that CR-EOMCCSD(T),III calculations involve the single-reference CISD calculations, in addition to the CCSD, EOMCCSD, and (T) steps common to all CR-EOMCCSD(T) methods. This increases the CPU timings of the CR- EOMCCSD(T),III calculations, when compared to CR- EOMCCSD(T),IX and CR-EOMCCSD(T),IIX (X=A-D) approaches. Additional letters A-D that label the CR-EOMCCSD(T),I and CR-EOMCCSD(T),II approximations refer to different ways of treating perturbative denominators in evaluating the (T) triples corrections (D means full treatment of these denominators, based on the diagonal matrix elements of the triples-triples block of the CCSD similarity transformed Hamiltonian, A means the crudest treatment through bare orbital energies). The user interested in further details is referred to a 2004 paper by Kowalski and Piecuch (J. Chem. Phys. 120, 1715-1738 (2004)). Our experience to date indicates that the CR- EOMCCSD(T),ID and CR-EOMCCSD(T),III methods are the most accurate ones when it comes to the calculations of excited states dominated by double excitations and excited-state potential energy surfaces along bond breaking coordinates, at least for moderate bond stretches. The CR-EOMCCSD(T),ID and CR-EOMCCSD(T),III methods are particularly good when examining the total energies of excited states (for example, as functions of nuclear geometries). If the user is only interested in vertical excitation energies rather than total energies, the good balance between ground and excited states, particularly when excited states are dominated by doubles, can be achieved by considering mixed approximations, such as CR-EOMCCSD(T),ID/IB. The ID/IB acronym means that the excitation energy is obtained by subtracting the CR-EOMCCSD(T),IB ground-state energy from the CR-EOMCCSD(T),ID energy of excited state. Other mixed approaches (IID/IB, etc.) are obtained in a similar way. The ID/IB results are particularly good when the excited states have significant doubly excited character. The fact that the CR-EOMCCSD(T),ID results for excited states are usually better than the CR-EOMCCSD(T),IA,IB,IC results is related to a better treatment of perturbative denominators in evaluating the (T) triples corrections in the CR- EOMCCSD(T),ID approximation. In addition to the total CR-EOMCCSD(T),IX, CR- EOMCCSD(T),IIX (X=A-D), and CR-EOMCCSD(T),III energies and vertical excitation energies based on the idea of mixing different approximations for excited and ground states (the ID/IA, IID/IA, ID/IB, and IID/IB excitation energies), GAMESS prints the so-called DELTA-CR-EOMCCSD(T) values (the del(IA), del(IB), del(IC), del(ID), del(IIA), del(IIB), del(IIC), del(IID), and del(III) energies). These are the vertical excitation energies obtained by directly correcting the EOMCCSD excitation energies rather than the total CCSD/EOMCCSD energies by triples corrections. For example, del(ID) refers to the vertical excitation energy obtained by subtracting the CCSD ground-state energy from the excited-state CR-EOMCCSD(T),ID energy. The DELTA-CR- EOMCCSD(T) values may be somewhat worse than the pure CR- EOMCCSD(T) (e.g., CR-EOMCCSD(T),ID) or CR-EOMCCSD(T),III) or mixed CR-EOMCCSD(T) (e.g., CR-EOMCCSD(T),ID/IB)) values of vertical excitation energies for states dominated by doubles, but they may provide a reasonable balance between ground and excited states and somewhat bigger improvements for vertical excitation energies corresponding to states dominated by singles. The DELTA-CR-EOMCCSD(T) methods provide a reasonably good balance between improvements in the results for excited states dominated by singles and improvements in the results for excited states dominated by doubles, but one should treat this remark with caution. In addition to the above CR-EOMCCSD(T) results, GAMESS also prints the so-called (T)/R excitation energies. These are the analogs of the EOMCCSD(T~) excitation energies proposed by Watts and Bartlett, obtained by using the right eigenvectors of the CCSD similarity transformed and right- hand moments of EOMCCSD equations rather than the left eigenstates of EOMCCSD and left-hand analogs of the EOMCCSD moments (see K. Kowalski and P. Piecuch, J. Chem. Phys. 120, 1715-1738 (2004) for details). Just like the EOMCCSD(T~) method of Watts and Bartlett, the (T)/R approach is based on the idea of directly correcting the EOMCCSD vertical excitation energies by triples. In analogy to the EOMCCSD(T~) method, the (T)/R corrections improve the EOMCCSD results for states dominated by singles, but they may fail to produce reasonable results for states dominated by doubles and for excited-state potential energy surfaces along bond breaking coordinates. The CR-EOMCCSD(T) methods are considerably more robust in this regard. In performing the CR-EOMCCSD(T) calculations, user should realize that the EOMCCSD method can provide a wrong state ordering if low-lying doubly excited states are mixed up with singly excited states in the electronic spectrum. This may require calculating a larger number of EOMCCSD states before correcting them for triples. An example of this situation has been described in K. Kowalski and P. Piecuch, J. Chem. Phys. 120, 1715-1738 (2004). The EOMCCSD method provides an incorrect ordering of the singlet A1 states of ozone, so that one must use the third excited EOMCCSD state of the singlet A1 (1A1) symmetry (the fourth 1A1 state total, using the CCSD/EOMCCSD energy ordering of ground and excited states) to calculate the noniterative CR-EOMCCSD(T) triples correction that describes the first excited singlet A1 (the second 1A1) state. Without calculating several states of each symmetry at the EOMCCSD level prior to CR-EOMCCSD(T) calculations, one would risk losing information about some important low- lying doubly excited states. Because of the inherent limitations of the EOMCCSD approximation, complicated doubly excited states resulting from the EOMCCSD calculations may be shifted to high energies, mixing with the singly excited states that are accurately described by the EOMCCSD method. After correcting the EOMCCSD energies for the effect of triples, these doubly excited states may become low-lying states. This is exactly what we observe in the case of ozone and other cases of severe quasi- degeneracies. The issues of size extensivity in the EOMCCSD and CR- EOMCCSD(T) calculations are highly complex and much beyond the scope of this writing. Briefly, none of the EOMCC methods are rigorously size extensive and yet all EOMCC methods are very useful in great many applications. The EOMCCSD approach is size intensive for excited states dominated by singles and the EOMCCSD energies correctly separate when the one-electron charge-transfer excitations are considered. Thus, the EOMCCSD approach correctly describes the dissociation of a singly excited system (AB)* into the A* + B, A + B*, (A+) + (B-), and (A-) + (B+) fragments (* designates a one-electron excitation). We must remember, however, that the above separability properties of the EOMCCSD energies are no longer true if the reference determinant |Phi> does not separate correctly (for example, the RHF determinant does not correctly separate if the AB -> A+B fragmentation involves the dissociation of the closed-shell system AB into open-shell fragments A and B). As in the case of the ground-state CR-CCSD(T) approach, the CR-EOMCCSD(T) methods slightly violate the rigorous size extensivity/intensivity (at the level of 1-2 millihartree for systems with up to 30-50 correlated electrons), but at the same time the CR-EOMCCSD(T) approaches significantly improve a poor description of excited states with significant double excitation components by the EOMCCSD method. As a result, lack of strict size extensivity of the CR-EOMCCSD(T) theories is of relatively minor significance in applications for systems with up to at least 50 correlated electrons [see M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch, J. Chem. Phys. 122, 214107-1 - 214107-15 (2005) for a thorough discussion of the complicated extensivity issues in EOMCCSD and CR-EOMCCSD(T) calculations]. The user is encouraged to examine various interesting elements of the EOMCC input and output. In addition to EOMCC energies, GAMESS prints the largest R1 and R2 excitation amplitudes and the so-called reduced excitation level (REL) diagnostic, which provides information about the character of a given excited state (REL close to 1 means singly excited, REL close to 2 means doubly excited). GAMESS also prints the R0 value (the coefficient at the reference in the EOMCCSD wave function). If a molecule has symmetry and R0 equals 0, user immediately learns the excited state has a different symmetry than the ground state. GAMESS provides full information about irreps of the calculated excited states. density matrices and properties One of the major advantages of EOMCC methods, including EOMCCSD, is a relatively straightforward access to reduced density matrices and molecular properties that these methods offer. This is done by considering the left eigenstates of the similarity transformed Hamiltonian H-bar = exp(-T) H exp(T) mentioned in the earlier sections. The similarity transformed Hamiltonian H-bar is not hermitian, so that, in addition to the right eigenstates R|Phi>, which define the "ket" CC or EOMCC wave functions discussed in the previous section, we can also define the left eigenstates of H-bar,form a biorthonormal set. We can use these eigenstates to calculate expectation values and transition matrix elements of quantum-mechanical operators (observables), involving the CC and EOMCC ground and excited states, as follows: = , where W-bar = exp(-T) W exp(T) is a similarity transformed form of the observable W we are interested in and where we added labels K and M to operators L and R to indicate the CC/EOMCC electronic states they are associated with. The operator W could be, for example, a dipole or quadrupole moment. It could also be a product of creation and annihilation operators, which we could use to calculate the reduced density matrices. For example, if the operator W = (ap-dagger) aq, where ap-dagger and aq are the creation and annihilation operators associated with the spin- orbitals p and q, respectively, we can calculate the CC or EOMCC one-body reduced density matrix in the electronic state K, Gamma(qp,K), as Gamma(qp,K) = . For the corresponding transition density matrix involving two different states K and M, say ground and excited states or some other combination, we can write Gamma(qp,KM) = . By having access to reduced density matrices, we can calculate various properties analytically. For example, by calculating the one-body reduced density matrices of ground and excited states and the corresponding transition density matrices, we can determine all one-electron properties and the corresponding transition matrix elements involving one- electron properties using a single mathematical expression: = Sum_pq Gamma(qp,KM), where

are matrix elements of the one-body property operator W in a basis set of molecular spin-orbitals used in the calculations. The calculation of reduced density matrices provides the most convenient way of calculating CC and EOMCC properties of ground and excited states. In addition, by having reduced density matrices, one can calculate CC and EOMCC electron densities, rhoK(x) = Sum_pq Gamma(qp,K) (phi_q(x))* phi_p(x), where phi_p(x) and phi_q(x) are molecular spin-orbitals and x represents the electronic (spatial and spin) coordinates. By diagonalizing Gamma(qp,K), one can determine the natural occupation numbers and natural orbitals for the CC or EOMCC state |PsiK>. The above strategy of handling molecular properties analytically by determining one-body reduced density matrices was implemented in the CC/EOMCC programs incorporated in GAMESS. At this time, the calculations of reduced density matrices and selected properties are possible at the CCSD (ground states) and EOMCCSD (ground and excited states) levels of theory (T=T1+T2, R=R1+R2, L=L0+L1+L2). Currently, in the main output the program prints the CCSD and EOMCCSD electric multipole (dipole, quadrupole, etc.) moments and several other one-electron properties that one can extract from the CCSD/EOMCCSD density matrices, the EOMCCSD transition dipole moments and the corresponding dipole and oscillator strengths, and the natural occupation numbers characterizing the CCSD/EOMCCSD wave functions. In addition, the complete CCSD/EOMCCSD one- body reduced density matrices and transition density matrices in the RHF molecular orbital basis and the CCSD and EOMCCSD natural orbital occupation numbers are printed in the PUNCH output file. The eigenvalues of the density matrix (natural occupation numbers) are ordered such that the corresponding eigenvectors (CCSD or EOMCCSD natural orbitals) have the largest overlaps with the consecutive ground-state RHF MOs. Thus, the first eigenvalue of the density matrix corresponds to the CCSD or EOMCCSD natural orbital that has the largest overlap with the RHF MO 1, the second with RHF MO 2, etc. This ordering is particularly useful for analyzing excited states, since in this way one can easily recognize orbital excitations that define a given excited state. One has to keep in mind that the reduced density matrices originating from CC and EOMCC calculations are not symmetric. Thus, if we, for example, want to calculate the dipole strength between states K and M for the x component of the dipole mu_x, |

|**2, we must write | |**2 = , where each matrix element in the above expression is evaluated using the expression for shown above. A similar remark applies to the corresponding component of the oscillator strength, (2/3)*|EK-EM|*| |**2, which we have to write as (2/3)*|EK-EM|* . In other words, both matrix elements and have to be evaluated, since they are not identical. This is reflected in the GAMESS output, where the user can see quantities such as the left and right transition dipole moments. From the above description, it follows that in order to calculate reduced density matrices and properties using CC and EOMCC methods, one has to determine the left as well as the right eigenstates of the similarity transformed Hamiltonian H-bar. For the ground state, this is done by solving the linear system of equations for the deexcitation operator Lambda (in the CCSD case, the one- and two-body components Lambda1 and Lambda2). For excited states, we can proceed in several different ways. We can solve the linear system of equations for the amplitudes defining the EOMCC deexcitation operator L, after determining the corresponding EOMCC excitation operator R and excitation energy omega (recommended option, default in GAMESS), or we can solve for the L and R amplitudes simultaneously in the process of diagonalizing the similarity transformed Hamiltonian. These different ways of solving the EOMCC problem are discussed in section "Eigensolvers for excited- state calculations." As already mentioned, the left eigenstates of the similarity transformed Hamiltonian of the CCSD approach are also used to construct the triples corrections to CCSD energies defining the rigorously size extensive completely renormalized CR-CC(2,3) approximation. This is why the user gets an immediate access to electrostatic multipole moments and other one-electron properties calculated at the CCSD level, when running the CR-CC(2,3) calculations. excited state example ! excited states of methylidyne cation...CH+ ! Basis set and geometry come from a FCI study by ! J.Olsen, A.M.Sanchez de Meras, H.J.Aa.Jensen, ! P.Jorgensen Chem. Phys. Lett. 154, 380-386(1989). ! ! EOMCC methods give: ! STATE EOMCCSD ID/IA IID/IA ID/IB IID/IB FCI ! B1 (1Pi) 3.261 3.226 3.226 3.225 3.224 3.230 ! A1 (1Delta) 7.888 6.988 6.963 6.987 6.962 6.964 ! A1 (1Sigma+) 9.109 8.656 8.638 8.654 8.637 8.549 ! A1 (1Sigma+) 13.580 13.525 13.526 13.524 13.525 13.525 ! B1 (1Pi) 14.454 14.229 14.221 14.228 14.219 14.127 ! A1 (1Sigma+) 17.316 17.232 17.220 17.231 17.219 17.217 ! A2 (1Delta) 17.689 16.820 16.790 16.819 16.789 16.833 ! Note the improvements in the EOMCCSD results by the ! CR-EOMCCSD(T) appproaches (e.g., ID/IB) for the Sigma+ ! state at 8.549 eV and both Delta states. ! ! The ground state CCSD dipole is z=-0.645, and the ! right/left transition moment to the first pi state ! is x=0.297 and 0.320, with oscillator strength 0.0076 ! $contrl scftyp=rhf cctyp=cr-eom runtyp=energy icharg=1 units=bohr $end $system mwords=5 $end $ccinp ncore=0 $end $eominp nstate(1)=4,2,2,0 minit=1 noact=3 nuact=7 ccprpe=.true. $end $data CH+ at R=2.13713...basis set from CPL 154, 380 (1989) Cnv 2 Carbon 6.0 0.0 0.0 0.16558134 S 6 1 4231.610 0.002029 2 634.882 0.015535 3 146.097 0.075411 4 42.4974 0.257121 5 14.1892 0.596555 6 1.9666 0.242517 S 1 ; 1 5.1477 1.0 S 1 ; 1 0.4962 1.0 S 1 ; 1 0.1533 1.0 S 1 ; 1 0.0150 1.0 P 4 1 18.1557 0.018534 2 3.9864 0.115442 3 1.1429 0.386206 4 0.3594 0.640089 P 1 ; 1 0.1146 1.0 P 1 ; 1 0.011 1.0 D 1 ; 1 0.75 1.0 Hydrogen 1.0 0.0 0.0 -1.97154866 S 3 1 1.924060D+01 3.282800D-02 2 2.899200D+00 2.312080D-01 3 6.534000D-01 8.172380D-01 S 1 ; 1 1.776D-01 1.0 S 1 ; 1 2.5D-02 1.0 P 1 ; 1 1.0 1.0 $end resource requirements User can perform LCCD, CCD, and CCSD calculations, that is without calculating the [T], (T), (2,3), and (TQ) corrections, or calculate the entire set of the standard and renormalized [T], (T), (2,3), and (TQ) ground-state corrections, in addition to the CCSD energies. User can also perform the EOMCCSD calculations of excited states and stop at EOMCCSD or continue to obtain some or all CR- EOMCCSD(T) triples corrections (cf. the values of input variable CCTYP in $CONTRL and $EOMINP group). Finally, user can perform the calculations of ground-state properties at the CCSD level or calculate ground- and excited-state properties. It is also possible to combine some of the above calculations. For example, one can calculate the CCSD and EOMCCSD properties and obtain triples corrections to the calculated CCSD and EOMCCSD energies from a single input (see the example above). The CR-CC(2,3) calculation produces the MBPT(2) and CCSD energies, and CCSD one- electron properties and density matrices, in addition to the CR-CC(2,3) and some other CR-CC triples corrections to the CCSD energies, again all from a single input (CCTYP=CR- CCL). The most expensive steps in CC/EOMCC calculations scale as follows: LCCD, CCD, CCSD, EOMCCSD No**2 times Nu**4 (iterative) CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T], CR-CCSD(T), CR-CC(2,3) (#1), CR-EOMCCSD(T) (#2) No**3 times Nu**4 (non-iterative) plus No**2 times Nu**4 (iterative) CCSD(TQ), R-CCSD(TQ), CR-CCSD(TQ) No**2 times Nu**5 or Nu**6 (#3) (non-iterative) plus No**3 times Nu**4 (non-iterative) plus No**2 times Nu**4 (iterative) ---- (#1) In addition to the usual No**2 times Nu**4 iterative CCSD steps and No**3 times Nu**4 non-iterative steps needed to determine the (2,3) triples correction, the CR-CC(2,3) calculations require extra No**2 times Nu**4 iterative steps needed to obtain the left CCSD state, which enters the CR-CC(2,3) triples correction formula. (#2) In addition to the No**2 times Nu**4 iterative CCSD and EOMCCSD steps and No**3 times Nu**4 non-iterative (T) steps that are common to all CR-EOMCCSD(T) models, the CR-EOMCCSD(T),III method requires the iterative No**2 times Nu**4 steps of CISD. The CR-EOMCCSD(T),IX and CR-EOMCCSD(T),IIX (X=A-D) methods do not require these additional CISD calculations. (#3) To reduce the cost, the program will automatically choose between the No**2 times Nu**5 and Nu**6 algorithms in the (Q) part, depending on the ratio of Nu to No. ---- The cost of calculating the standard CCSD[T] and CCSD(T) energies and the cost of calculating the R-CCSD[T] and R- CCSD(T) energies are essentially the same. The cost of calculating the triples corrections of the CR-CCSD[T] and CR-CCSD(T) approaches is essentially twice the cost of calculating the standard CCSD[T] and CCSD(T) corrections. Similar relationships hold between the costs of the CCSD(TQ), R-CCSD(TQ), and CR-CCSD(TQ) calculations. The cost of calculating the triples corrections of the CR- CC(2,3),X (X=A-D) approaches is also twice the cost of calculating the CCSD[T] and CCSD(T) triples corrections, but additional No**2 times Nu**4 iterative steps are required to generate the left CCSD state after converging the CCSD equations in order to calculate the final CR- CC(2,3) energies. Although the noniterative triples corrections may be seen to grow as the seventh power of the system size, they often require less time than the sixth power iterations of the CCSD step, while providing a great increase in accuracy. Similar remarks apply to the CR- EOMCCSD(T) calculations: The cost of the CR-EOMCCSD(T) calculation for a single electronic state, in its noniterative triples part, is twice the cost of computing the standard (T) corrections of CCSD(T). The total CPU time of the CR-EOMCCSD(T) calculations scales linearly with the number of calculated states. In spite of the formal N**6 scaling, the calculations of the CCSD/EOMCCSD properties per single electronic state are considerably less expensive than the CCSD calculations for two reasons. First of all, the process of obtaining the left eigenstates of the similarity transformed Hamiltonian H-bar can reuse the intermediates (matrix elements of H-bar) which are obtained in the prior CCSD calculations. Second, converging left eigenstates of H-bar is usually much quicker than converging the CCSD equations when one obtains the left eigenstates of H-bar by solving the linear system of equations for the L deexcitation amplitudes after determining the R excitation amplitudes and excitation energies. This means that computing properties at the CCSD/EOMCCSD level is not very expensive once the CCSD and EOMCCSD right eigenvectors are obtained. Similar remarks apply to the CR-CC(2,3) calculations, which require the left CCSD eigenstates in addition to the CCSD T1 and T2 amplitudes: The determination of the left CCSD states that are needed to determine the non-iterative triples corrections of the CR-CC(2,3) approach makes the entire CCSD part of the CR-CC(2,3) calculation only somewhat more expensive than the regular CCSD iterations needed to obtain T1 and T2 clusters. The CCSD(TQ), R-CCSD(TQ), and CR- CCSD(TQ) calculations are more expensive than the CCSD(T) calculations, in spite of the fact that all of these methods use non-iterative N**7 steps. This is related to the fact that the No**2 times Nu**5 steps of the (TQ) methods are more expensive than the No**3 times Nu**4 steps of the (T) approaches. On the other hand, the CCSD(TQ), R- CCSD(TQ), and CR-CCSD(TQ) methods are much less expensive than the iterative ways of obtaining the information about quadruply excited clusters. This is a result of an efficient use of diagram factorization in coding the CCSD(TQ), R-CCSD(TQ), and CR-CCSD(TQ) methods, which leads to a reduction of the N**9-type steps in the original (Q) expressions to N**7 steps. Rough estimates of the memory required are: CCSD 4 No**2 times Nu**2 + No times Nu**3 CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T) 4 No**2 times Nu**2 + No times Nu**3 CR-CCSD[T], CR-CCSD(T) No**2 times Nu**2 + 2 * No times Nu**3 (faster algorithm) 4 No**2 times Nu**2 + No times Nu**3 (slower, less memory) CR-CC(2,3) The most expensive routine requires 3 * No * Nu**3 + 3 * Nu**3 + 5 * No**2 *Nu**2 words CCSD(TQ),b, R-CCSD(TQ)-n,x (n=1,2;x=a,b), CR-CCSD(TQ),x (x=a,b) 2 * No times Nu**3 + No**2 times Nu**2 + Nu**3, preceded and followed by steps that require memories, such as, for example, 3 * Nu**3 + 5 * No**2 * Nu**2 EOMCCSD No times Nu**3 + 4 No**2 times Nu**2 (MEOM=0,1) if MEOM=2, add to this (4 times number of roots + 2) times No**2 times Nu**2 CR-EOMCCSD(T),IX, 2 * No times Nu**3 + 3 No**2 times Nu**2 CR-EOMCCSD(T),IIX(X=A-D) [MTRIP=1 in $EOMINP] CR-EOMCCSD(T) 3 * No times Nu**3 + 5 No**2 times Nu**2 all variants (faster algorithm) [MTRIP=2 in $EOMINP] CR-EOMCCSD(T),III 2 * No times Nu**3 + 5 No**2 times Nu**2 [MTRIP=3 in $EOMINP] CR-EOMCCSD(T) 2 * No times Nu**3 + 5 No**2 times Nu**2 all variants (slower algorithm) [MTRIP=4 in $EOMINP] The program automatically selects the algorithm for the CR- CCSD[T] and CR-CCSD(T) calculations, depending on the amount of available memory. A similar remark applies to the EOMCCSD calculations, where some additional reductions of memory requirements are possible if memory is low. The above estimates are rough. The time required for calculating the CR-CCSD[T] and CR- CCSD(T) triples corrections is only twice the time used to calculate the standard CCSD[T] and CCSD(T) corrections. Thus, by just doubling the CPU time for the noniterative triples corrections and by selecting CCTYP=CR-CC, we gain access to all six noniterative triples corrections (the CCSD[T], CCSD(T), R-CCSD[T], R-CCSD(T), CR-CCSD[T], and CR- CCSD(T) energies) plus, of course, to the MBPT(2) and CCSD energies. At the same time, the CR-CCSD[T] and CR-CCSD(T) results for stretched nuclear geometries and diradicals are better than the results of the conventional CCSD[T] and CCSD(T) calculations. In some cases, choosing CCTYP=R-CC might be reasonable, too. The choice CCTYP=R-CC gives five different energies (CCSD, CCSD[T], CCSD(T), R-CCSD[T], and R-CCSD(T)) for the price of three (CCSD, CCSD[T], and CCSD(T)) as the there is no extra time needed for the R- theories compared to the standard ones. If we ignore the iterative CCSD steps and additional iterative steps needed to determine the left CCSD state, the time required for calculating the size extensive CR-CC(2,3) triples corrections is also only twice the time of calculating the CCSD[T] and CCSD(T) corrections. There is an additional bonus though: The CR-CC(2,3) calculations automatically produce a variety of CCSD one-electron properties at no extra cost. Similar remarks apply to quadruples and excited state calculations, although in the latter case a lot depends on user's expectations. If user is only interested in excited states dominated by singles and if accuracies on the order of 0.1-0.3 eV (sometimes better, sometimes worse) are acceptable, EOMCCSD is a good choice. However, it may be worth improving the EOMCCSD results by performing the CR-EOMCCSD(T) calculations, which often lower the errors in calculated excited states to 0.1 eV or less without making the calculations a lot more expensive (the CR-EOMCCSD(T) corrections are noniterative, so that the CPU time needed to calculate them may be comparable to the time spent in all EOMCCSD iterations). If there is a risk of encountering low-lying states having significant doubly excited contributions or multi-reference character, choosing CR- EOMCCSD(T) is a necessity, since errors obtained in EOMCCSD calculations for states dominated by doubles can easily be on the order of 1 eV. The CCSD(T) approach is often fine for closed-shell molecules, but there are cases, such as the vibrational frequencies of ozone and properties of other multiply bonded systems, where inclusion of quadruples is necessary. The CR-CCSD(T) approach is very useful in cases involving single bond breaking and diradicals, but CR-CC(2,3) and CR-CCSD(TQ) should be better. In addition, the CR-CC(2,3) method provides rigorously size extensive results. In cases of multiple bond dissociations, CR-CCSD(TQ) is a better alternative. The program is organized such that choosing a CR-CCSD(TQ) option (CCTYP=CR-CC(Q)) produces all energies obtained with CCTYP=CR-CCSD(T) and all CCSD(TQ), R-CCSD(TQ), and CR- CCSD(TQ) energies. By selecting CCTYP=CCSD(TQ), the user can obtain the CCSD(TQ) and R-CCSD(TQ) energies, in addition to the CCSD, CCSD[T], CCSD(T), R-CCSD[T], and R- CCSD(T) energies. We encourage the user to read papers, such as P.Piecuch, S.A.Kucharski, K.Kowalski, M.Musial Comput. Phys. Comm., 149, 71-96(2002); K. Kowalski and P. Piecuch, J. Chem. Phys., 120, 1715-1738 (2004); M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch, J. Chem. Phys. 122, 214107 (2005); K. Kowalski, P. Piecuch, M. Wloch, S.A. Kucharski, M. Musial, and M.W. Schmidt, in preparation, where time and memory requirements for various types of CC and EOMCC calculations are described in considerable detail. restarts in ground-state calculations The CC code incorporated in GAMESS is quite good in converging the CCSD equations with the default guess for cluster amplitudes. The code is designed to converge in relatively few iterations for significantly stretched nuclear geometries, where it is not unusual to obtain large cluster amplitudes whose absolute values are close to 1. This is accomplished by combining the standard Jacobi algorithm with the DIIS extrapolation method of Pulay. The maximum number of amplitude vectors used in the DIIS extrapolation procedure is defined by the input variable MXDIIS. The default for MXDIIS is as follows: MXDIIS = 5, for 5 < No*Nu, MXDIIS = 3, for 2 < No*Nu < 6, MXDIIS = 0, for No*Nu < 3. Thus, in the vast majority of cases, the default value of MXDIIS is 5. However, for very small problems, when the DIIS expansion subspace leads to singular systems of linear equations, it is necessary to reduce the value of MXDIIS to 2-4 (we chose 3) or switch off DIIS altogether (which is the case when MXDIIS = 0). It may, of course, happen that the solver for the CCSD equations does not converge, in spite of increasing the maximum number of iterations (input variable MAXCC; the default value is 30) and in spite of changing the default value of MXDIIS. In order to facilitate the calculations in all such cases, we included the restart option in the CC codes incorporated in GAMESS. Thus, user can restart a CCSD (or (L)CCD) calculation from the restart file created by an earlier CC calculation. In order to use the restart option, user must save the disk file CCREST (unit 70) from the previous CC run (cf. the GAMESS script rungms) and make sure that this file is copied to scratch directory where the restarted calculation is carried out. A restart is invoked by entering a nonzero value for IREST, which should be the number of the last iteration completed, and must be some value greater than or equal 3. Examples of using the restart option include the following situations: o The CCSD program did not converge in MAXCC iterations, but there is a chance to converge it if the value of MAXCC is increased. User restarts the calculation with the increased value of MAXCC. o User ran a CCSD calculation, obtaining the converged CCSD energy, but later decided to run CR-CCSD(T) or CR-CC(2,3) calculation. Instead of running the entire CCSD --> CR- CCSD(T) or CCSD --> CR-CC(2,3) task again, user restarts the calculation after changing the value of input variable CCTYP to CR-CC (the CR-CCSD(T) case) or CR-CCL (the CR-CC(2,3) case) and entering IREST to reuse the previous CCSD amplitudes, proceeding at once to the non- iterative triples corrections (left CCSD calculations and triples corrections in the CR-CC(2,3) case). o The CCSD program diverged for some geometry with a significantly stretched bond. User performs an extra calculation for a different nuclear geometry, for which it is easier to converge the CCSD equations, and restarts the calculation from the restart file generated by an extra calculation. This technique of restarting the CC calculations from the cluster amplitudes obtained for a neighboring nuclear geometry is particularly useful for scanning PESs and for calculating energy derivatives by numerical differentiation. There also are situations where restart of the ground- state CCSD calculations is useful for excited-state and property calculations: o User ran a CCSD, CCSD(T), or CR-CCSD(T) calculation, obtaining the converged CC energies for the ground state, but later decided to run an excited-state EOMCCSD or CR-EOMCCSD(T) calculations. Instead of running the entire CCSD --> EOMCCSD or CCSD --> CR-EOMCCSD(T) task, user restarts the calculation after changing the value of input variable CCTYP to EOM-CCSD or CR-EOM, selecting excited-state options in $EOMINP, and entering IREST greater or equal to 3 to reuse the previously converged CCSD amplitudes, proceeding at once to the excited-state (EOMCCSD or CR-EOMCCSD(T)) calculations. o User ran an EOMCCSD excited-state calculation, obtaining the converged CCSD amplitudes, but later discovered (by analyzing R1 and R2 amplitudes and REL values) that some states are dominated by doubles, so that the EOMCCSD results need to be improved by the CR-EOMCCSD(T) triples corrections. Instead of running the entire CCSD --> CR-EOMCCSD(T) task, user restarts the calculation after changing the value of input variable CCTYP from EOM-CCSD to CR-EOM, and entering IREST greater or equal to 3 to reuse the previously converged CCSD amplitudes, proceeding at once to the EOMCCSD and CR-EOMCCSD(T) calculations. o User ran a CR-CCSD(T) calculation, obtaining the converged ground-state energies, but later decided to run CCSD and EOMCCSD properties. Instead of running the CCSD --> EOMCCSD task again, user restarts the calculation after changing the value of input variable CCTYP to EOM-CCSD, adding CCPRPE=.TRUE. and the desired values of NSTATE in $EOMINP, and entering IREST to reuse the previously converged CCSD amplitudes, proceeding at once to CCSD and EOMCCSD properties. initial guesses in excited-state calculations The EOMCCSD calculation is an iterative procedure which needs initial guesses for the excited states of interest. The popular initial guess for the EOMCCSD calculations is obtained by performing the CIS calculations (diagonalizing the Hamiltonian in a space of singles only). This is acceptable for states dominated by singles, but user may encounter severe convergence difficulties or even miss some states entirely if the calculated states have significant doubly excited character. One possible philosophy is not to worry about it and use the CIS initial guess only, since EOMCCSD fails to describe states with large doubly excited components. This is not the philosophy of the EOMCC programs in GAMESS. GAMESS is equipped with the CR- EOMCCSD(T) triples corrections to EOMCCSD energies, which are capable of reducing the large errors in the EOMCCSD results for states dominated by two-electron transitions, on the order of 1 eV, to 0.1 eV or even less. Thus, the ability to capture states with significant doubly excited contributions is an important element of the EOMCC GAMESS codes. Excited states with significant contributions from double excitations can easily be found by using the EOMCCSd (little d) initial guesses provided by GAMESS. In the EOMCCSd calculations (and analogous CISd calculations used to initiate the CISD calculations for the CR-EOMCCSD(T),III method), the initial guesses for the calculated excited states are defined using all single excitations (letter S in EOMCCSd and CISd) and a small subset of double excitations (the little d in EOMCCSd and CISd) defined by active orbitals or orbital range specified by the user. The inclusion of a small set of active double excitations in addition to all singles in the initial guess greatly facilitates finding excited states characterized by relatively large doubly excited amplitudes. GAMESS input offers a choice between the CIS and EOMCCSd/CISd initial guesses. The use of EOMCCSd/CISd initial guesses is highly recommended. This is accomplished by setting the input variable MINIT at 1 and by selecting the orbital range (active orbitals to define "little doubles" d) through the numbers of active occupied and active unoccupied orbitals (variables NOACT and NUACT, respectively) or an array of active orbitals called MOACT. eigensolvers for excited-state calculations The basic eigensolver for the EOMCCSD calculations is the Hirao and Nakatsuji's generalization of the Davidson diagonalization algorithm to non-Hermitian problems (the similarity transformed Hamiltonian H-bar is non-Hermitian). GAMESS offers the following three choices of EOMCCSD eigensolvers for the right eigenvalue problem (R amplitudes and energies only): o the true multi-root eigensolver based on the Hirao and Nakatsuji's algorithm, in which all states are calculated at once using a united iterative space (variable MEOM=2). o the single-root eigensolver, in which one calculates one state at a time, but the iterative subspace corresponding to all calculated roots remains united (variable MEOM=0). o the single-root eigensolver, in which one calculates one state at a time and each calculated root has a separate iterative subspace (variable MEOM=1). The latter option (MEOM=1) leads to the fastest algorithm, but there is a risk (often worth taking) that some states will be converged more than once. The true multi-root eigensolver (MEOM=2) is probably the safest, but it is also the most expensive solver and there are some risks associated with using it too. When MEOM=2, there is a risk that one root, which is difficult to converge, may cause the entire multi-root procedure fail in spite of the fact that all other roots participating in the calculation converged. The EOMCCSD program in GAMESS is prepared to handle this problem by saving individual roots that converged during multi-root iterations in case the entire procedure fails because of one or more roots which are difficult to converge. In this way, at least some roots are saved for the subsequent CR-EOMCCSD(T) calculations. The middle option (MEOM=0) seems to offer the best compromise. MEOM=0 is a single-root eigensolver, so there are no risks associated with loosing some states during multi-root calculations. At the same time, the use of the united iterative subspace for all calculated roots helps to eliminate the problem of MEOM=1 of obtaining the same root more than once. The single-root eigensolver with a united iterative subspace (MEOM=0) is recommended (and used as a default), although other ways of converging the right EOMCCSD equations (MEOM=1,2) are very useful too. As pointed out earlier, in order to calculate reduced density matrices and properties using CCSD and EOMCCSD methods, one has to determine the left as well as the right eigenstates of the non-Hermitian similarity transformed Hamiltonian H-bar. For the ground state, this is done by solving the linear system of equations for the deexcitation operator Lambda (in the CCSD case, the one- and two-body components Lambda1 and Lambda2). For the amplitudes defining the L1 and L2 components of the excited-state operator L, one can proceed in several different ways and these different ways are reflected in the EOMCCSD algorithm incorporated in GAMESS. One can, for example, solve the linear system of equations for the amplitudes defining the EOMCCSD deexcitation operator L=L1+L2, after determining the corresponding excitation operator R=R1+R2 and excitation energy omega. This is a highly recommended option, which is also a default in GAMESS. This option is executed with any choice of MEOM=0,1,2 and when the user selects CPRPE=.TRUE. In case of unlikely difficulties with obtaining the L1 and L2 components, one can solve for the EOMCCSD values of the L1,L2 and R1,R2 amplitudes and excitation energies simultaneously in the process of diagonalizing the similarity transformed Hamiltonian H-bar completely in a single sequence of iterations. This approach is reflected by the following two additional choices of the input variable MEOM: o MEOM=3, one root at a time, separate iterative space for each computed root, left and right eigenvectors of the similarity transformed Hamiltonian and energies (like MEOM=1, but both left and right eigenvectors are iterated). o MEOM=4, one root at a time, united iterative spaces for all calculated roots, left and right eigenvectors of the similarity transformed Hamiltonian and energies (like MEOM=0, but both left and right eigenvectors are iterated). In both cases, the user has to select CCPRPE=.TRUE. in order for these two choices of MEOM to work. references and citations required in publications Any publication describing the results of CC calculations obtained using GAMESS should give reference to the relevant papers. Depending on the specific CCTYP value, these are: CCTYP = LCCD, CCD, CCSD, CCSD(T) P. Piecuch, S.A. Kucharski, K. Kowalski, and M. Musial Comput. Phys. Commun. 149, 71-96 (2002). CCTYP = R-CC, CR-CC, CCSD(TQ), CR-CC(Q) P. Piecuch, S.A. Kucharski, K. Kowalski, and M. Musial Comput. Phys. Commun. 149, 71-96 (2002); K. Kowalski and P. Piecuch J. Chem. Phys. 113, 18-35 (2000); K. Kowalski and P. Piecuch J. Chem. Phys. 113, 5644-5652 (2000). CCTYP = CR-CCL P. Piecuch, S.A. Kucharski, K. Kowalski, and M. Musial Comput. Phys. Commun. 149, 71-96 (2002); P. Piecuch and M. Wloch J. Chem. Phys. 123, 224105/1-10 (2005). CCTYP = EOM-CCSD, CR-EOM P. Piecuch, S.A. Kucharski, K. Kowalski, and M. Musial Comput. Phys. Commun. 149, 71-96 (2002); K. Kowalski and P. Piecuch, J. Chem. Phys. 120, 1715-1738 (2004); M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch, J. Chem. Phys. 122, 214107-1 - 214107-15 (2005). CCTYP = CR-EOML P. Piecuch, J. R. Gour, and M. Wloch Int. J. Quantum Chem. 109, 3268-3304(2009) and the first two papers cited for CR-EOM just above CCTYP = IP-EOM2, EA-EOM2 J. R. Gour, P. Piecuch, M. Wloch J. Chem. Phys. 123, 134113/1-14(2005) J. R. Gour, P. Piecuch J. Chem. Phys. 125, 234107/1-17(2006) In addition, the explicit use of CCPRP=.TRUE. in $CCINP and/or the use of CCPRPE=.TRUE. in $EOMINP should reference M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch, J. Chem. Phys. 122, 214107/1-15 (2005). --- The rest of this section is a list of references to the original formulation of various areas in Coupled-Cluster Theory relevant to methods available in GAMESS: Electronic structure: J. Cizek, J. Chem. Phys. 45, 4256 (1966). J. Cizek, Adv. Chem. Phys. 14, 35 (1969). J. Cizek, J. Paldus, Int.J.Quantum Chem. 5, 359 (1971). Nuclear theory (examples): F. Coester, Nucl. Phys. 7, 421 (1958). F. Coester, H. Kuemmel, Nucl. Phys. 17, 477 (1960). K. Kowalski, D.J. Dean, M. Hjorth-Jensen, T. Papenbrock, P. Piecuch, Phys. Rev. Lett. 92, 132501 (2004). D.J. Dean, J.R. Gour, G. Hagen, M. Hjorth-Jensen, K. Kowalski, T. Papenbrock, P. Piecuch, M. Wloch, Nucl. Phys. A. 752, 299 (2005). M. Wloch, D.J. Dean, J.R. Gour, P. Piecuch, M. Hjorth- Jensen, T. Papenbrock, K. Kowalski, Eur. Phys. J. A 25 (Suppl. 1), 485 (2005). M. Wloch, J.R. Gour, P. Piecuch, D.J. Dean, M. Hjorth- Jensen, T. Papenbrock, J. Phys. G: Nucl. Phys. 31, S1291 (2005). M. Wloch, D.J. Dean, J.R. Gour, M. Hjorth-Jensen, K. Kowalski, T. Papenbrock, P. Piecuch, Phys. Rev. Lett. 94, 212501 (2005). P. Piecuch, M. Wloch, J.R. Gour, D.J. Dean, M. Hjorth- Jensen, T. Papenbrock, in V. Zelevinsky (Ed.), Nuclei and Mesoscopic Physics, AIP Conference Proceedings, Vol. 777 (AIP Press, 2005), p. 28. D.J. Dean, M. Hjorth-Jensen, K. Kowalski, T. Papenbrock, M. Wloch, and P. Piecuch, in Key Topics in Nuclear Structure, Proceedings of the 8th International Spring Seminar on Nuclear Physics, edited by A. Covello (World Scientific, Singapore, 2005), p. 147. Coupled-Cluster Method with Doubles (CCD) - J. Cizek, J. Chem. Phys. 45, 4256 (1966). J. Cizek, Adv. Chem. Phys. 14, 35 (1969). J. Cizek, J. Paldus, Int.J.Quantum Chem. 5, 359 (1971). J.A. Pople, R. Krishnan, H.B. 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Monkhorst, Int. J. Quantum Chem. Symp. 11, 421 (1977). K. Emrich, Nucl. Phys. A 351, 379 (1981). H. Sekino and R.J. Bartlett, Int. J. Quantum Chem. Symp. 18, 255 (1984). E. Daalgard and H. Monkhorst, Phys. Rev. A 28, 1217 (1983). M. Takahashi and J. Paldus, J. Chem. Phys. 85, 1486 (1986). H. Koch and P. Jorgensen, J. Chem. Phys. 93, 3333 (1990). H. Nakatsuji, K. Hirao, Chem. Phys. Lett. 47, 569 (1977). H. Nakatsuji, K. Hirao, J.Chem.Phys. 68, 2053, 4279 (1978). Equation of Motion Coupled-Cluster Method with Singles and Doubles, EOMCCSD - J. Geertsen, M. Rittby, and R.J. Bartlett, Chem. Phys. Lett. 164, 57 (1989). J.F. Stanton and R.J. Bartlett, J. Chem. Phys. 98, 7029 (1993). Method of Moments of Coupled-Cluster Equations and Renormalized and Completely Renormalized Coupled-Cluster Methods (Overviews) - P. Piecuch, K. Kowalski, I.S.O. Pimienta, S.A. Kucharski, in M.R. Hoffmann, K.G. Dyall (Eds.), Low-Lying Potential Energy Surfaces, ACS Symposium Series, Vol. 828, Am. Chem. Society, Washington, D.C., 2002, p. 31 [ground and excited states]. P. Piecuch, K. Kowalski, I.S.O. Pimienta, M.J. McGuire, Int. Rev. Phys. Chem. 21, 527 (2002) [ground and excited states]. P. Piecuch, I.S.O. Pimienta, P.-F. Fan, K. Kowalski, in J. Maruani, R. Lefebvre, E. Brandas (Eds.), Progress in Theoretical Chemistry and Physics, Vol. 12, Advanced Topics in Theoretical Chemical Physics, Kluwer, Dordrecht, 2003, p. 119 [ground states]. P. Piecuch, K. Kowalski, I.S.O. Pimienta, P.-D. Fan, M. Lodriguito, M.J. McGuire, S.A. Kucharski, T. Kus, M. Musial, Theor. Chem. Acc. 112, 349 (2004) [ground and excited states]. P. Piecuch, M. Wloch, M. Lodriguito, and J.R. Gour, in S. Wilson, J.-P. Julien, J. Maruani, E. Brandas, and G. Delgado-Barrio (Eds.), Progress in Theoretical Chemistry and Physics, Vol. 15, Recent Advances in the Theory of Chemical and Physical Systems, Springer, Berlin, 2006, p. XX, in press [excited states]. P. Piecuch, I.S.O. Pimienta, P.-D. Fan, and K. 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T1 diagnostic: T.J.Lee, P.R.Taylor Int.J.Quantum Chem., S23, 199- 207(1989). It is often assumed that T1>0.02 indicates that CCSD may not be correct for a system which is not very single reference in nature. (T) corrections tolerate greater singles amplitudes. However, T1 diagnostic is in many cases misleading, since one can easily have small (or even vanishing) T1 cluster amplitudes due to symmetry and a significant configurational quasi-degeneracy and multi- reference character. In general, in typical multi-reference situations, such as bond stretching and diradicals, one observes a significant increase of T2 cluster amplitudes. The larger values of T2 amplitudes are a clear signature of a multi-reference character of the wave function. The CR- CCSD(T), CR-CCSD(TQ), and CR-CC(2,3) methods tolerate significant increases of T2 amplitudes in cases of single- bond breaking and diradicals. CCSD(T) and CCSD(TQ) approaches cannot do this, when the spin-adapted RHF references are employed. Written by Piotr Piecuch, Michigan State University (updated March 18, 2006)