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. Schlegel, J.S. Binkley,
Int. J. Quantum Chem. Symp. 14, 545 (1978).
R.J. Bartlett and G.D. Purvis,
Int. J. Quantum Chem. Symp. 14, 561 (1978).
J. Paldus, J. Chem. Phys. 67, 303 (1977)
[orthogonally spin-adapted formulation].
Linearized Coupled-Cluster Method with Doubles (LCCD)
J. Cizek, J. Chem. Phys. 45, 4256 (1966).
J. Cizek, Adv. Chem. Phys. 14, 35 (1969).
R.J. Bartlett, I. Shavitt, Chem.Phys.Lett.50, 190 (1977)
57, 157 (1978) [Erratum].
R. Ahlrichs, Comp. Phys. Commun. 17, 31 (1979).
Coupled-Cluster Method with Singles and Doubles (CCSD) -
G.D.Purvis III, R.J.Bartlett, J.Chem.Phys. 76, 1910 (1982)
[spin-orbital formulation].
P. Piecuch, J. Paldus, Int.J.Quantum Chem. 36, 429 (1989).
[orthogonally spin-adapted formulation].
G.E.Scuseria, A.C.Scheiner, T.J.Lee, J.E.Rice,
H.F.Schaefer III, J. Chem. Phys. 86, 2881 (1987)
[non-orthogonally spin-adapted formulation].
G.E. Scuseria, C.L. Janssen, H.F.Schaefer III
J. Chem. Phys. 89, 7382 (1988)
[non-orthogonally spin-adapted formulation].
T.J. Lee and J.E. Rice, Chem. Phys. Lett. 150, 406 (1988)
[non-orthogonally spin-adapted formulation].
Coupled-Cluster Method with Singles and Doubles and
Noniterative Triples, CCSD[T] = CCSD+T(CCSD) -
M. Urban, J. Noga, S. J. Cole, and R. J. Bartlett,
J. Chem. Phys. 83, 4041 (1985).
P. Piecuch and J. Paldus, Theor. Chim. Acta 78, 65 (1990)
[orthogonally spin-adapted formulation].
P. Piecuch, S. Zarrabian, J. Paldus, and J. Cizek,
Phys. Rev. B 42, 3351-3379 (1990)
[orthogonally spin-adapted formulation].
P. Piecuch, R. Tobola, and J. Paldus,
Int. J. Quantum Chem. 55, 133-146 (1995)
[orthogonally spin-adapted formulation].
Coupled-Cluster Method with Singles and Doubles and
Noniterative Triples, CCSD(T) -
K. Raghavachari, G. W. Trucks, J. A. Pople, M. Head-Gordon,
Chem. Phys. Lett. 157, 479 (1989).
Equation of Motion Coupled-Cluster Method, Response
CC/Time Dependent CC Approaches, SAC-CI (Original Ideas), -
H. 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. Kowalski,
in A.K. Wilson (Ed.), Recent Progress in Electron
Correlation Methodology, ACS Symposium Series, Vol.
XXX, Am. Chem. Society, Washington, D.C., 2006, p. XX
[in press; ground states].
P.-D. Fan and P. Piecuch, Adv. Quantum Chem., in press
(2006).
Renormalized and Completely Renormalized Coupled-Cluster
Methods, Method of Moments of Coupled-Cluster Equations
(Initial Original Papers, Ground States) -
P. Piecuch, K. Kowalski,
in J. Leszczynski (Ed.), Computational Chemistry:
Reviews of Current Trends, Vol. 5, World Scientific,
Singapore, 2000, p. 1.
K. Kowalski, P. Piecuch, J. Chem. Phys. 113, 18 (2000).
K. Kowalski, P. Piecuch, J. Chem. Phys. 113, 5644 (2000).
Biorthogonal Method of Moments of Coupled-Cluster Equations
and Size Extensive Completely Renormalized Coupled-Cluster
Singles, Doubles, and Non-iterative Triples Approach (CR-
CC(2,3)=CR-CCSD(T)L; Initial Original Papers) ?
P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105(2005).
P. Piecuch, M. Wloch, J.R. Gour, and A. Kinal, Chem. Phys.
Lett. 418, 467-474 (2006).
Renormalized and Completely Renormalized Coupled-Cluster
Methods, Method of Moments of Coupled-Cluster Equations
(Other Original Papers, Higher-Order Methods, Ground-State
Benchmarks) -
K. Kowalski, P. Piecuch, Chem. Phys. Lett. 344, 165 (2001).
P. Piecuch, S.A. Kucharski, K. Kowalski,
Chem. Phys. Lett. 344, 176 (2001).
P. Piecuch, S.A. Kucharski, V. Spirko, K. Kowalski,
J.Chem.Phys. 115, 5796 (2001).
P. Piecuch, K. Kowalski, and I.S.O. Pimienta,
Int. J. Mol. Sci. 3, 475 (2002).
M.J. McGuire, K. Kowalski, P. Piecuch,
J. Chem. Phys. 117, 3617 (2002).
P. Piecuch, S.A. Kucharski, K. Kowalski, M. Musial,
Comput. Phys. Comm., 149, 71 (2002).
I.S.O. Pimienta, K. Kowalski, and P. Piecuch,
J. Chem. Phys. 119, 2951 (2003).
S. Hirata, P.-D. Fan, A.A. Auer, M. Nooijen, P. Piecuch,
J. Chem. Phys. 121, 12197 (2004).
K. Kowalski and P. Piecuch, J. Chem. Phys. 122, 074107
(2005).
P.-D. Fan, K. Kowalski, and P. Piecuch, Mol. Phys. 103,
2191 (2005).
Completely Renormalized Coupled-Cluster Methods, Examples
of Large-Scale Applications to Ground-State Properties -
I. Ozkan, A. Kinal, M. Balci,
J.Phys.Chem. A 108, 507 (2004).
R.L. DeKock, M.J. McGuire, P. Piecuch, W.D. Allen,
H.F. Schaefer III, K. Kowalski, S.A. Kucharski, M. Musial,
A.R. Bonner, S.A. Spronk, D.B Lawson, S.L. Laursen,
J. Phys. Chem. A 108, 2893 (2004).
M.J. McGuire, P. Piecuch, K. Kowalski, S.A. Kucharski,
M. Musial, J. Phys. Chem. A 108, 8878 (2004).
M.J. McGuire, P. Piecuch
J. Am. Chem. Soc. 127, 2608 (2005).
A. Kinal, P. Piecuch, J. Phys. Chem. A 110, 367 (2006).
C.J. Cramer, M. Wloch, P. Piecuch, C. Puzzarini, and L.
Gagliardi, J. Phys. Chem. A 110, 1991 (2006).
Completely Renormalized Equation of Motion Coupled-Cluster
Methods, Method of Moments of Coupled-Cluster Equations
for Ground and Excited States (Original Papers) -
K. Kowalski P. Piecuch, J. Chem. Phys. 115, 2966 (2001).
K. Kowalski P. Piecuch, J. Chem. Phys. 116, 7411 (2002).
K. Kowalski P. Piecuch, J. Chem. Phys. 120, 1715 (2004).
M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch, J. Chem.
Phys. 122, 214107 (2005).
Also, multi-reference and other externally corrected MMCC
methods including ground and excited states,
K. Kowalski and P. Piecuch,
J. Molec. Struct.: THEOCHEM 547, 191 (2001).
K. Kowalski and P. Piecuch, Mol. Phys. 102}, 2425 (2004).
M.D. Lodriguito, K. Kowalski, M. Wloch, and P. Piecuch
J. Mol. Struct: THEOCHEM, in press (2006).
Completely Renormalized Equation of Motion Coupled-Cluster
Methods, Method of Moments of Coupled-Cluster Equations
for Ground and Excited States (Selected Benchmarks and
Applications)
C.D. Sherrill, P. Piecuch, J.Chem.Phys. 122, 124104 (2005)
R.K. Chaudhuri, K.F. Freed, G. Hose, P. Piecuch,
K. Kowalski, M. Wloch, S. Chattopadhyay, D. Mukherjee,
Z. Rolik, A. Szabados, G. Toth, and P.R. Surjan,
J. Chem. Phys. 122, 134105-1 (2005).
K. Kowalski, S. Hirata, M. Wloch, P. Piecuch, and T.L.
Windus, J. Chem. Phys. 123, 074319 (2005).
S. Nangia, D.G. Truhlar, M.J. McGuire, and P. Piecuch
J. Phys. Chem. A 109, 11643 (2005).
P. Piecuch, S. Hirata, K. Kowalski, P.-D. Fan, and T.L.
Windus, Int. J. Quantum Chem. 106, 79 (2006).
M. Wloch, M.D. Lodriguito, P. Piecuch, and J.R. Gour
Mol. Phys., in press (2006).
S. Coussan, Y. Ferro, A. Trivella, P. Roubin, R. Wieczorek,
C. Manca, P. Piecuch, K. Kowalski, M. Wloch, S.A.
Kucharski, and M. Musial, J. Phys. Chem. A, in press
(2006).
Completely Renormalized Coupled-Cluster and Equation of
Motion Coupled-Cluster Methods, GAMESS Implementations -
P. Piecuch, S.A. Kucharski, K. Kowalski, M. Musial,
Comput. Phys. Comm., 149, 71 (2002).
K. Kowalski and P. Piecuch
J. Chem. Phys. 120, 1715 (2004).
M. Wloch, J.R. Gour, K. Kowalski, and P. Piecuch,
J. Chem. Phys. 122, 214107 (2005).
P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105 (2005).
K. Kowalski, P. Piecuch, M. Wloch,S.A. Kucharski,
M. Musial, and M.W. Schmidt, in preparation.
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)
