Article
Reference
Theoretical Models on the Cu2O2 Torture Track: Mechanistic
Implications for Oxytyrosinase and Small-Molecule Analogues
CRAMER, Christopher J., et al.
Abstract
Accurately describing the relative energetics of alternative bis(μ-oxo) and μ-η2:η2 peroxo
isomers of Cu2O2 cores supported by 0, 2, 4, and 6 ammonia ligands is remarkably
challenging for a wide variety of theoretical models, primarily owing to the difficulty of
maintaining a balanced description of rapidly changing dynamical and nondynamical electron
correlation effects and a varying degree of biradical character along the isomerization
coordinate. The completely renormalized coupled-cluster level of theory including triple
excitations and extremely efficient pure density functional levels of theory quantitatively agree
with one another and also agree qualitatively with experimental results for Cu2O2 cores
supported by analogous but larger ligands. Standard coupled-cluster methods, such as
CCSD(T), are in most cases considerably less accurate and exhibit poor convergence in
predicted relative energies. Hybrid density functionals significantly underestimate the stability
of the bis(μ-oxo) form, with the magnitude of the error being directly proportional to the
percentage Hartree−Fock exchange in the functional. [...]
CRAMER, Christopher J., et al. Theoretical Models on the Cu2O2 Torture Track: Mechanistic
Implications for Oxytyrosinase and Small-Molecule Analogues. The journal of physical
chemistry. A, 2006, vol. 11, no. 5, p. 1991-2004
DOI : 10.1021/jp056791e
Available at:
http://archive-ouverte.unige.ch/unige:3631
Disclaimer: layout of this document may differ from the published version.
1 / 1
Theoretical Models on the Cu
2
O
2
Torture Track: Mechanistic Implications for
Oxytyrosinase and Small-Molecule Analogues
Christopher J. Cramer,*
,†
Marta Włoch,
‡
Piotr Piecuch,
‡,§
Cristina Puzzarini,
|
and
Laura Gagliardi
⊥
Department of Chemistry and Supercomputer Institute, UniVersity of Minnesota, 207 Pleasant St. SE,
Minneapolis, Minnesota 55455, Department of Chemistry, Michigan State UniVersity, East Lansing, Michigan
48824, Department of Physics and Astronomy, Michigan State UniVersity, East Lansing, Michigan 48824,
Dipartimento di Chimica “G. Ciamician”, UniVersita´ di Bologna, Via F. Selmi 2, I-40126 Bologna, Italy,
and Department of Physical Chemistry, Sciences II UniVersity of GeneVa, 30 Quai Ernest Ansermet,
CH-1211 GeneVa 4, Switzerland
ReceiVed: NoVember 23, 2005
Accurately describing the relative energetics of alternative bis(µ-oxo) and µ-η
2
:η
2
peroxo isomers of Cu
2
O
2
cores supported by 0, 2, 4, and 6 ammonia ligands is remarkably challenging for a wide variety of theoretical
models, primarily owing to the difficulty of maintaining a balanced description of rapidly changing dynamical
and nondynamical electron correlation effects and a varying degree of biradical character along the isomerization
coordinate. The completely renormalized coupled-cluster level of theory including triple excitations and
extremely efficient pure density functional levels of theory quantitatively agree with one another and also
agree qualitatively with experimental results for Cu
2
O
2
cores supported by analogous but larger ligands. Standard
coupled-cluster methods, such as CCSD(T), are in most cases considerably less accurate and exhibit poor
convergence in predicted relative energies. Hybrid density functionals significantly underestimate the stability
of the bis(µ-oxo) form, with the magnitude of the error being directly proportional to the percentage Hartree-
Fock exchange in the functional. Single-root CASPT2 multireference second-order perturbation theory, by
contrast, significantly oVerestimates the stability of bis(µ-oxo) isomers. Implications of these results for modeling
the mechanism of C-H bond activation by supported Cu
2
O
2
cores, like that found in the active site of
oxytyrosinase, are discussed.
Introduction
Various metalloenzymes containing one, two, or more copper
atoms activate molecular oxygen to oxidize organic substrates.
1-8
Tyrosinase is one such enzyme: a so-called type 3 copper
protein, it oxidizes tyrosine residues to their corresponding
o-quinones within an active site containing two copper atoms,
both of which are involved in the oxidation reaction.
1,5,6,9
At
least in part to better understand the mechanism of tyrosinase,
an enormous amount of effort has gone into the study of the
structures, spectroscopy, and reactivities of 2:1 Cu/O
2
model
systems.
2,10-14
The variety of ways in which two supported copper(I) atoms
can bind O
2
turns out to be surprisingly rich (Figure 1).
12,15
The three dominant structural motifs that have been character-
ized experimentally are the trans end-on µ-η
1
:η
1
peroxo, the
side-on µ-η
2
:η
2
peroxo, and the bis(µ-oxo), although select
examples of the other three cases shown in Figure 1 have been
identified or inferred. In the first two cases, the O
2
fragment is
reduced to a peroxide dianion by oxidation of the two copper
atoms to Cu(II). In the third case, the O-O bond is broken and
the formal oxidation states of the core atoms are Cu(III) and
O
2-
. In general, the particular Cu
2
O
2
motif observed is strongly
dependent on the nature of the ligands. Thus, for example,
tetradentate ligands tend to favor end-on peroxo binding,
bidentate ligands favor bis(µ-oxo) as they ideally accommodate
the preferred square-planar ligand arrangement for Cu(III), and
binding motifs associated with tridentate ligands tend to vary
depending on the rigidity and/or steric requirements of the ligand
itself.
12,15
Indeed, for selected sets of ligands, it has been
demonstrated that more than one motif can be accessed. For
example, Jung et al.
16
have demonstrated that the binding of
oxygen to a dicopper complex supported by a bridging
N,N,N′,N′-bis{2-(2-pyridyl)ethyl}-1,4-butanediamine ligand shows
a kinetic preference for end-on peroxo but a thermodynamic
preference for side-on peroxo.
Another example that has received special attention involves
ligands that permit access to both the side-on peroxo and bis-
* To whom correspondence should be addressed. E-mail: cramer@
pollux.chem.umn.edu.
†
University of Minnesota.
‡
Department of Chemistry, Michigan State University.
§
Department of Physics and Astronomy, Michigan State University.
|
Universita´ di Bologna.
⊥
Sciences II University of Geneva.
Figure 1. Binding motifs for Cu
2
O
2
.
1991J. Phys. Chem. A 2006, 110, 1991-2004
10.1021/jp056791e CCC: $33.50 © 2006 American Chemical Society
Published on Web 01/06/2006
(µ-oxo) forms.
11,13,17
The former geometry has been shown by
X-ray crystallography to be present in oxyhemocyanin, an
oxygen-transporting protein with the imidazole rings of six
histidine residues serving as ligands.
18,19
Although no X-ray
crystal structure has yet been obtained for oxytyrosinase, spectral
data for oxyhemocyanin and oxytyrosinase are sufficiently
similar that there is little doubt that the resting state of
oxytyrosinase also involves a side-on µ-η
2
:η
2
peroxo with five
or six histidine ligands.
6
However, Tolman and co-work-
ers,
11,13,17,20,21
and later others,
22-26
have demonstrated that in
select instances the relative energies of isomeric µ-η
2
:η
2
peroxo
and bis(µ-oxo) complexes can be sufficiently close to one
another, and the barrier to their interconversion sufficiently low,
that the two species may be in rapid equilibrium with one
another; theoretical studies have rationalized many of the
electronic structural details governing this phenomenon.
27-34
This observation potentially complicates mechanistic analysis
in model systems and, by extension, oxytyrosinase, because the
kinetics for systems characterized by a rapid preequilibrium can
be indistinguishable from systems involving only a single
reactant.
35
Thus, even in the absence of observing a spectral
signature for a bis(µ-oxo) intermediate, because it is present at,
say, part per million concentration, one cannot a priori discount
the possibility that this same isomer is so much more reactive
than the µ-η
2
:η
2
peroxo that product derives predominantly from
prior isomerization and subsequent reaction.
Notwithstanding this complication, a sufficient number of
model compounds presenting the various motifs have now been
examined such that general reactivity trends have been associ-
ated with each one.
12,13,17,36
In particular, end-on peroxo species
have been characterized as basic (subject to protonation to
generate hydrogen peroxide) and nucleophilic (reactive with
acylating agents) but not particularly electrophilic (no oxidation
of phosphines to phosphine oxides). Side-on peroxo compounds,
by contrast, fail to exhibit basic character and enhance the
electrophilicity of the bound O
2
fragment, as judged by their
abilities to abstract H atoms from good donors, oxidize
phosphines to phosphine oxides, and hydroxylate aromatic rings
both intra- and intermolecularly. Finally, bis(µ-oxo) compounds
have been demonstrated also to be nonbasic and to be strongly
electrophilic, oxidizing both activated and aliphatic C-H bonds,
hydroxylating aromatic rings, and serving as oxidizing agents
in electron-transfer reactions.
Given the complications alluded to above with respect to
distinguishing between direct reactivity and prior equilibration
(and, of course, the possibility that both these pathways may
be kinetically competitive), computational protocols have also
been brought to bear to characterize energetic and mechanistic
details for particular reactions. Studies of C-H bond activation
by bis(µ-oxo) compounds have documented that this motif
engages directly in H-atom transfer/rebound type mecha-
nisms.
37-39
A very recent density functional study also found a
transition-state structure for hydroxylation of a coordinated
phenolate by a bis(µ-oxo) species.
40
With respect to oxytyro-
sinase itself, Siegbahn has examined in considerable detail
possible reactions of a side-on peroxo species coordinated by
six imidazole ligands with either phenol or phenolate.
41-43
Siegbahn found that the coordination of phenolate to copper
causes the side-on µ-η
2
:η
2
peroxo species to distort to a µ-η
2
:
η
1
geometry, which may either react directly itself or may further
decoordinate to a µ-η
2
:η
0
peroxyl radical species that acts as
the electrophilic species (the relative energies of the two species
were within the estimated error range of the calculations, hence
the qualification as to which, if either, is more mechanistically
relevant).
43
Implicit in several of the theoretical studies described above
is the assumption that the energy difference between the bis-
(µ-oxo) and µ-η
2
:η
2
peroxo motifs is accurately predicted by
the particular level of theory employed. However, no prior
calculation has tested its employed theory in a quantitatiVe
fashion against an experimentally characterized bis(µ-oxo)/µ-
η
2
:η
2
peroxo equilibrium, nor has it been possible to demonstrate
convergence of a suitably complete level of electronic structure
theory for a model system more simply ligated than an
experimentally characterized system to establish a benchmark
against which to compare other more practical levels of theory.
In this work, we compare several different density functional
(DFT) models, the single-root multireference second-order
perturbation theory (CASPT2), and standard and completely
renormalized (CR) coupled-cluster (CC) methods with respect
to their abilities to predict accurate energetics for the conversion
of the bis(µ-oxo) Cu
2
O
2
2+
core to the µ-η
2
:η
2
peroxo motif.
We consider various ligand sets (Chart 1), including no ligands
at all (0), one (1), two (2), and three (3) ammonia ligands per
copper atom, and finally three imidazole ligands (4) per copper
atom; the last ligand set is directly relevant to the case of
oxytyrosinase. We find inter alia that (i) the CR-CC theory
makes predictions that are reasonably well converged with
respect to high-order correlation effects at the triple-excitation
level and considerably more accurate than the results of standard
CC calculations; (ii) single-root CASPT2 results are very
sensitive to active-space choice and, even with large active
spaces that provide relative isomer energies that are apparently
converged with respect to the orbitals included, this level
overestimates the stability of bis(µ-oxo) isomers relative to µ-η
2
:
η
2
peroxo isomers by 20-30 kcal mol
-1
; and (iii) DFT models
show extreme sensitivity to the incorporation of Hartree-Fock
(HF) exchange: pure functionals provide quantitatively useful
predictions but hybrid HF-DFT functionals, like the popular
B3LYP model, underestimate the stability of bis(µ-oxo) isomers
relative to µ-η
2
:η
2
peroxo isomers by 5-10 kcal mol
-1
for every
10% of HF exchange included in the functional. The consistency
between CR-CC predictions and those from pure DFT func-
CHART 1
1992 J. Phys. Chem. A, Vol. 110, No. 5, 2006 Cramer et al.
tionals for 0-3, together with their mutual good agreement in
the case of 3 with recently published
34
multireference config-
uration interaction (MRCI) data, strongly suggests that pure DFT
functionals will provide accurate predictions for Cu
2
O
2
systems
with larger and more realistic ligands, like the oxytyrosinase-
related case 4, for which calculations at such levels as CR-CC
or MRCI are not presently practical.
Theoretical Methods
Basis Sets. Four different basis sets were used in this work,
which we designate BS1, BS2, BS3, and BS4. In all four sets,
the Stuttgart pseudopotential and associated basis functions were
used for Cu.
44
In BS1, the largest basis set, which was employed
for most energy calculations, the atomic natural orbital (ANO)
basis set of Pierloot et al.
45
was used. For N and O we used a
[10s6p3d | 4s3p2d] contraction, whereas for H we used a [8s4p
| 2s1p] contraction. In BS2, the same primitive functions were
contracted as 3s2p1d and 1s, respectively. BS3 was used only
for geometry optimizations of 3 and employed the Pople basis
set 6-31G(d) for H, N, and O.
46
BS4 was used only for 4 and
consisted of the Pople basis sets
46
STO-3G and 6-311G(d) for
H and O, respectively, and the MIDI! basis set
47
for C and N.
Density Functionals. We assayed three different pure func-
tionals. The BLYP functional combines the generalized-gradient
approximation (GGA) exchange functional of Becke
48
with the
GGA correlation functional of Lee, Yang, and Parr.
49
The
mPWPW91 functional combines the GGA exchange
50
and
correlation
51
functionals of Perdew and Wang as modified by
Adamo and Barone.
52
The TPSS functional is a meta GGA
functional.
53
We also considered hybrid HF-DFT functionals.
54
The
B3LYP,
55
mPW1PW91,
52
MPW1K,
56
and TPSSh
57
functionals
incorporate 20%, 25%, 48.2%, and 10% HF exchange, respec-
tively, into their corresponding functionals.
For singlet-state calculations, unstable restricted (R) self-
consistent-field (SCF) solutions were reoptimized at the unre-
stricted (U) SCF level.
58
All restricted solutions were checked
for instability. Singlet energies from unrestricted calculations
were computed in two ways. First, the raw broken-spin-
symmetry (BS) SCF energy was used without modification.
Second, the sum method
59
was employed to eliminate spin
contamination from the triplet state in the SCF solution. In this
approach, the singlet energy is computed as
60-62
where the triplet energy is computed for the single-determinantal
high-spin configuration S
z
) 1 (at the BS geometry) and 〈S
2
〉 is
the expectation value of the total-spin operator applied to the
Kohn-Sham (KS) determinant for the unrestricted S
z
) 0
calculation. Gra¨fenstein and Cremer
63
have shown that values
of 〈S
2
〉 computed at the DFT level have diagnostic value in
assessing spin contamination; thus, the sumBS approach is more
physically realistic than using the raw BS energy.
Single-Reference Post-SCF Levels. We performed a variety
of standard single-reference CC calculations
64,65
using the CC
method with singles and doubles (CCSD),
66-68
the CC method
with singles, doubles, and noniterative triples (CCSD(T)),
69
and,
to explore the role of higher-order correlations, the CC method
with singles, doubles, and noniterative triples and quadruples
(CCSD(TQ)), using variant “b” of the factorized CCSD(TQ)
approximation.
70
However, although the CCSD, CCSD(T), and
CCSD(TQ) methods offer a highly accurate treatment of
dynamic correlation effects (particularly the last two), they
usually fail, sometimes dramatically, in cases that require a well-
balanced description of dynamical and nondynamical correlation
effects, e.g., in systems that have a significant biradical
character
71-73
and reaction paths involving significant bond
stretching or breaking
74-84
(indeed, achieving such balance is
challenging for any ab initio methodsas shown below, even
multireference techniques can have difficulties when a multi-
dimensional reference space that is larger than present practical
limits permit is required to treat different types of correlation
on an equal footing).
To improve on the CC model, we performed calculations
using the recently developed
70,74,75,82-84
completely renormalized
CCSD(T) (CR-CCSD(T)) and CCSD(TQ) (CR-CCSD(TQ))
methods, which can accurately and effectively deal with reactive
potential-energy surfaces involving bond stretching,
74-84
biradicals,
71-73
and other cases of electronic near-degeneracies
within a single-reference description employing an RHF refer-
ence. We used variant “b” of the CR-CCSD(TQ) approach,
70
which enables us to determine if the CR-CCSD(T) results are
reasonably well converged with respect to higher-order cor-
relation effects. Although the CR-CCSD(T) and CR-CCSD(TQ)
approaches provide a robust description of biradical systems,
they slightly violate the rigorous size extensivity of the CC
theory (at the level of 0.5-1.0% of the changes in the correlation
energy along a reaction pathway). Also, certain types of high-
order dynamic correlations, which are represented, for example,
as disconnected products of singly, doubly, and triply excited
clusters and which may or may not play a role in the systems
examined in this study, are neglected at the CR-CCSD(T) level
(as they are also in CCSD(T)). To address the issues of
convergence with respect to higher-order correlations neglected
in CR-CCSD(T) and size-extensivity, we also performed
calculations using the recently proposed CR-CCSD(T)
L
ap-
proach, which is based on a new, biorthogonal formulation of
the method of moments of CC equations (a formalism used to
design all CR-CC methods
74,75,82,84
) employing the left eigen-
states of the similarity-transformed Hamiltonian of CC theory.
85-87
The CR-CCSD(T)
L
method, also referred to as the CR-CC(2,3)
approach,
86,87
is a rigorously size-extensive, improved variant
of the CR-CCSD(T) approach, which describes the aforemen-
tioned higher-order correlations neglected in the CR-CCSD(T)
calculations while eliminating the failures of standard CC
methods, such as CCSD(T), in calculations involving single-
bond breaking and biradicals. Additional details are available
in the Supporting Information.
In all single-reference correlated calculations, we used the
RHF determinant as a reference. The RHF, second-order many-
body perturbation theory (MBPT(2) or MP2), CCSD(T), CR-
CCSD(T), and CR-CCSD(T)
L
calculations were performed for
all systems up to 3. The smaller BS2 basis set was used for
CR-CCSD(T)
L
calculations on 3 (all other CC and CR-CC
calculations were carried out with BS1). Because of resource
demands, CR-CCSD(TQ)/BS1 calculations were carried out for
0 and 1 only. We explicitly correlated the 4s and 3d electrons
of the Cu atoms, the 2s and 2p electrons of the N and O atoms,
and the 1s electrons of the H atoms. For the bare system 0,we
also computed CR-CCSD(T)
L
results after adding the 12 3p
electrons of the Cu atoms to those being correlated.
Multireference SCF and Post-SCF Levels. The complete
active-space (CAS) SCF method
88
was used to generate mo-
lecular orbitals (MOs) and reference functions for subsequent
multiconfigurational second-order perturbation calculations of
the dynamic correlation energy (CASPT2).
89
Unless stated
E
singlet
)
2E
〈S
z
〉)0
- 〈S
2
〉E
〈S
z
〉)1
2 - 〈S
2
〉
(1)
Theoretical Models on the Cu
2
O
2
Torture Track J. Phys. Chem. A, Vol. 110, No. 5, 2006 1993
otherwise, it is implicit in this paper that CASSCF refers to the
optimization of the ground-state MOs (i.e., there is no state
averaging including excited states) and CASPT2 refers also to
the application of the model only to the ground state without
permitting any sort of multistate mixing involving higher-energy
roots. Full details of orbital selections, energies, and configu-
ration and reference weights are provided in the Supporting
Informationswe report here only the most critical details.
The performance of the CASSCF/CASPT2 model, which has
been widely used for the quantitative modeling of various
aspects of transition-metal chemistry,
90-92
depends critically on
the choice of orbital active space for the CAS reference. To
better describe our active spaces, we will adhere to a convention
whereby the z axis of a Cartesian frame is defined by the Cu-
Cu vector, the y axis is defined by the O-O vector, and the x
axis is then orthogonal to the Cu
2
O
2
core. Although full details
for all spaces are available in the Supporting Information, it is
appropriate here to be explicit about our largest active space
for 3 because of the prior work on this system at the CASPT2
level by Flock and Pierloot.
30
These authors first considered an
(8,10) active space comprised of the σ and σ* orbitals dominated
by appropriate combinations of the O 2p
y
basis functions, the
bonding and antibonding combinations of the 3d
yz
basis func-
tions from each Cu, two sets of formally π and π* orbitals
dominated by appropriate combinations of O 2p
z
(occupied) and
3p
z
(correlating virtual) basis functions, an additional σ
OO
virtual
orbital dominated by O 3p
y
basis functions, and an unspecified
linear combination of Cu 4d
yz
basis functions. Flock and
Pierloot
30
further expanded this to a (12,14) active space by
adding two sets of formally π and π* orbitals dominated by
appropriate combinations of O 2p
x
(occupied) and 3p
x
(cor-
relating virtual) basis functions. In our case for 3, we found
after extensive analysis that two of the virtual orbitals in this
(12,14) space were not particularly helpful, namely the 4d
yz
combination and the π* orbital formed from 3p
z
basis functions.
In their place, we included 2 occupied orbitals (generating a
(16,14) active space) dominated by bonding and antibonding
combinations of O 2s orbitals. A graphical depiction of the (16,-
14) active space for 3 is also available as Supporting Informa-
tion. We also considered a still larger (14,15) active space
(generating 5179830 Slater determinants for the singlet ground
state) which led to similar results, suggesting that, in this case
(and analogously for 0-2), the CASSCF and CASPT2 results
are well converged with respect to further expansion of the
active space, at least within practical limits, which do not permit
further expansion to a full-valence active space.
Two sets of CASPT2 calculations followed each CASSCF
calculation. In the first, the O 1s and Cu 2s and 3s orbitals were
kept frozen. In the second, the Cu 3p orbitals were also kept
frozen. All CASPT2 calculations employed a real level shift
93
of 0.1 au in combination with a new technique
94
that shifts active
orbital energies to simulate ionization energies for orbitals
excited out of and electron affinities for orbitals excited into
the active space. In the cases examined here, however, intruder
states did not appear to pose any significant problems.
Geometries. For systems 0, 1, and 2, idealized bis(µ-oxo)
and µ-η
2
:η
2
peroxo core geometries were generated. For the
former motif, the Cu-Cu and O-O distances in the D
2h
core
were taken as 2.8 and 2.3 Å, respectively. For the latter motif,
these same distances were taken to be 3.6 and 1.4 Å,
respectively. For 1, ammonia ligands were added to each Cu
along the Cu-Cu axis with Cu-N distances of 2.0 Å, N-H
distances of 1.0 Å, CuNH angles of 110°, and OCuNH dihedral
angles consistent with the C
2h
point group. For 2, all internal
coordinates were the same as for 1 except that the N atoms
were placed so as to make NCuCu angles of 135° and each
ammonia ligand had one N-H bond eclipsing a Cu-O bond,
thereby generating a structure belonging to the D
2h
point group.
For 3, the geometries of both motifs were optimized at the
B3LYP/BS3 level (R and U for the bis(µ-oxo) and µ-η
2
:η
2
peroxo cases, respectively) within the C
2h
point group. For 4,
geometries were fully optimized within the C
2h
and C
i
point
groups at both the R and U BLYP/BS3 and B3LYP/BS3 levels.
Intermediate geometries along linear isomerization paths were
generated for 0-3 according to
where q
i
is a given atomic Cartesian coordinate and F is the
fraction of progress along the isomerization coordinate (so that
0 and 100 define the bis(µ-oxo) and µ-η
2
:η
2
peroxo geometries,
respectively). We present results for F values of 0, 20, 40, 60,
80, and 100. Reference to a particular structure along an
isomerization coordinate will henceforth be made by adding F
as a subscript to the structure cardinal; e.g., 2
20
refers to the
structure 20% converted from 2
0
to 2
100
.
In the case of 4, additional end-on µ-η
1
:η
1
peroxo geometries
in the C
i
point group were obtained from full optimization at
the R and U BLYP/BS4 and B3LYP/BS4 levels. The natures
of all stationary points for 4 were confirmed by the calculation
of analytic force constants.
Software. Full details on software are provided in the
Supporting Information.
Results
Cu
2
O
2
2+
. The energies for structures of 0 as a function of F
were computed at all levels of theory using the BS1 basis set.
At the CAS and CASPT2 levels, an (8,8) and a large (16,14)
active space were employed. At the DFT level, restricted
solutions to the Kohn-Sham SCF equations could not be
successfully converged for the pure functionals BLYP, mP-
WPW91, and TPSS, although they could be for the hybrid
functionals. However, even those RDFT solutions that could
be converged were all unstable to symmetry breaking (computed
〈S
2
〉 values for the KS determinants ranged from 0.4 to 1.0
depending on structure and functional). Electronic energies at
many levels relative to 0
100
are provided in Table 1. Absolute
electronic energies and a more complete list of relative energies,
BS-DFT 〈S
2
〉 values, and
3
B
2g
electronic energies for all
structures (the final two quantities are required to compute
sumBS-DFT energies) may be found in the Supporting Informa-
tion.
A considerable variation in relative energies as a function of
theoretical level is observed. For 0
0
, the predicted relative energy
ranges over more than 90 kcal mol
-1
! Figure 2 presents
predicted structural relative energies graphically for select levels
of theory. Table 1 and Figure 2 indicate that the CR-CC results
are reasonably well converged, both with respect to the effect
of higher-than-triple excitations (quadruples play a small role)
and additional correlations involving triply excited clusters,
brought by the size-extensive CR-CCSD(T)
L
approach. There
is a very good agreement between the CR-CC and pure DFT
results. CASSCF and CASPT2 produce energies which are
considerably below those obtained with the CR-CC and pure
DFT approaches, and there is little substantive difference
between the (8,8) and (16,14) active-space results. Unlike CR-
CC, the standard CC approaches do not produce the well-
converged results and seem to fail in the biradical F ) 60-
q
i
(F) ) q
i
(0) +
F
100
[q
i
(100) - q
i
(0)] (2)
1994 J. Phys. Chem. A, Vol. 110, No. 5, 2006 Cramer et al.
