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J Phys Chem A. Author manuscript; available in PMC 2010 July 14.

Published in final edited form as:

PMCID: PMC2903681

NIHMSID: NIHMS212637

Department of Chemistry, Quantum Theory Project, 2328 New Physics Building, University of Florida, Gainesville, FL 32611-8435, 352-392-6973

Kenneth M. Merz, Jr.: ude.lfu.ptq@zrem

The publisher's final edited version of this article is available at J Phys Chem A

See other articles in PMC that cite the published article.

Heats of formation were calculated using coupled-cluster methods for a
series of zinc complexes. The calculated values were evaluated against
previously conducted computational studies using density functional methods as
well as experimental values. Heats of formation for nine neutral ZnX_{n}
complexes [X = -Zn, -H, -O, -F_{2}, -S, -Cl,
-Cl_{2}, -CH_{3}, (-CH_{3})_{2}]
were determined at the CCSD and CCSD(T) levels using the
6–31G** and TZVP basis sets, as well as the
LANL2DZ-6–31G** (LACVP**) and
LANL2DZ-TZVP hybrid basis sets. The CCSD(T)/6–31G**
level of theory was found to predict the heat of formation for the non-alkyl Zn
complexes most accurately. The alkyl Zn species were problematic in that none of
the methods that were tested accurately predicted the heat of formation for
these complexes. For the seven non-alkyl species, the
CCSD(T)/6–31G** level of theory was shown to predict the
most accurate heat of formation values. In instances where experimental
geometric parameters were available, these were most accurately predicted by the
CCSD/6–31G** level of theory; going to CCSD(T) did not
improve agreement with the experimental values.

Zinc complexes are critically important in biological systems, serving in
both a structural and a catalytic capacity.^{1}^{,}^{2} Indeed, zinc trails only iron
as the most ubiquitous transition metal in biological systems. Key complexes
include, but are certainly not limited to, Human Carbonic Anhydrase,^{3}^{–}^{5}
Carboxypeptidase,^{6} Alcohol
Dehydrogenase^{7} and so-called “zinc
fingers”^{8}^{,}^{9} which play structural roles in DNA recognition. Many of
these systems have been studied extensively via X-ray crystallography and
spectroscopic methods, including NMR.

The literature to date contains many computational studies on systems that
contain zinc. In a 1991 study, Kaupp *et al*. probed the structures
of ZnR_{2} complexes with 1,4-diaza-1,3-butadienes using pseudopotential
calculations.^{10} A subsequent study by
Kaupp and von Schnering probed the structures and binding energies of
(ZnX_{2})_{2} dimers at the MP2 and HF levels of theory using
pseudopotentials.^{11} Kabelac and Hobza
examined the binding of Zn^{2+} with the nucleic acid bases adenine,
guanine, cytosine and thymine at the MP2/TZVP level of theory.^{12} In recent work, Rayon and coworkers have probed
binding energies and geometries of several Zn^{II} complexes using MP2 and
density functional methods for optimizations, along with single point calculations
at the CCSD(T)/aug-cc-pVTZ level of theory.^{13}

We have recently employed a variety of popular density functional methods
spanning the GGA, meta-GGA, hybrid-GGA and meta-hybrid-GGA functional classes, a
total of 12, in the calculation of both heat of formation
(Δ*H _{f}*) and ionization potential for
transition metal complexes.

Although not as abundant as work with lower level methods, some literature
precedent for the application of coupled-cluster methods to
Δ*H _{f}* calculations in small organic
systems does exist. Dixon and coworkers have investigated iodine fluorides using
CCSD(T) methods

There is also some precedent for the use of these methods in
Δ*H _{f}* calculations on metal-containing
systems. A popular approach is extrapolation towards the complete basis set (CBS)
limit and has been applied to transition metal compounds from Sc-Zn.

We desired to expand our efforts towards the calculation of heat of formation
values to include more powerful computational methods. Our current efforts are
focused on the application of coupled-cluster methods toward this end. Higher level
calculations such as CASPT2 were not employed, as these are not viable for larger
Zn-containing systems and one of our primary reasons for these investigations is the
uncovering of computational methodologies suitable for use in QM and QM/MM type
calculations on biological systems incorporating one or more Zn centers. With this
in mind we have focused on using relatively modest basis sets (e.g.,
6–31G** and TZVP) combined with the CCSD (coupled-cluster
with single and double excitations) and CCSD(T) (coupled-cluster with single and
double and perturbative triple excitations) methods for the purpose of this study.
CCSD(T) provides an excellent compromise between accuracy and computational cost and
we wish to evaluate its performance in determining
Δ*H _{f}* values for ZnX

We have chosen a series of zinc complexes to be the focus of our initial work
with coupled cluster methods for Δ*H _{f}*
calculations. The nine chosen ZnX

All calculations were carried out on a SUN cluster featuring dual 2.5GHz
Opteron nodes using the Gaussian 03^{34} suite
of programs. All geometry optimizations incorporated standard gradient methods. For
all single point calculations, the SCF=TIGHT keyword was used. The
SCF=XQC keyword was applied in all instances, as SCF convergence was often
problematic, especially for higher energy multiplets. CCSD and CCSD(T) calculations
were run as implemented in Gaussian 03.^{35}^{–}^{40} Frequency
calculations were conducted on all geometries (at the minimum energy multiplicity)
to insure all calculated lowest energy structures resided at local minima on the
potential energy surface. Where applicable, calculations were done at the UCCSD at
UCCSD(T) levels. All other calculations are closed shell. The Pople type
6–31G** and triple-ζ quality TZVP basis sets were
used as implemented in Gaussian 03.^{41}^{,}^{42} LACVP** calculations were
run using the GEN keyword for the basis set. In these calculations, the LANL2DZ
basis/pseudopotential was used for Zn and the 6–31G** basis
set for the nonmetal atoms. A second set of calculations was run which applied the
TZVP basis set to the nonmetals while retaining LANL2DZ on the Zn atom. T1
diagnostics were computed with Gaussian 03 at the
CCSD/6–31G** and CCSD/TZVP levels of theory.^{43}^{,}^{44} This is a
measure that identifies instances where multi-reference effects may be important.
While multi-reference methods are beyond the intent of this investigation, but we
have placed these values in the supporting information, as the results identify several compounds for
which multi-reference approaches should be considered (ZnO, ZnS and
ZnF_{2}).

For all Zn species considered, we initially desired to optimize the 1, 3, 5
and 7 multiplicities for even electron species and the 2, 4, 6 and 8 multiplicities
for odd electron species as done in our previous DFT work.^{14} This worked well for most CCSD calculations, although
it was sometimes difficult to achieve SCF convergence for high energy
multiplicities. CCSD(T) calculations failed for a large number of high energy
multiplicities, although the CCSD ground state could always be converged for smaller
complexes using CCSD(T) calculations. Zn(CH_{3})_{2} proved to have
serious convergence problems in the geometry optimization procedure for the CCSD(T)
calculations, which uses the numerical eigenvector following algorithm in Gaussian,
and attempts at the CCSD(T) level were abandoned for this system at all
multiplicities. At C_{1} symmetry, the lowest for this system,
Zn(CH_{3})_{2} experienced difficulties with the number of
variables as well. Enforcing D_{3h} symmetry to decrease the variables
considered did not provide any relief for convergence related problems.

Heats of formation (Δ*H _{f}*) for all
complexes were computed using the method outlined in the Gaussian white paper on
Thermochemistry in the Gaussian 03 online manual.

$$\mathrm{\Delta}{H}_{f}(\text{M},298\text{K})=\mathrm{\Delta}{H}_{f}(\text{M},0\text{K})+(({H}_{M}(298\text{K})-{H}_{M}(0\text{K}))-\mathrm{\sum}\text{x}({H}_{x}(298\text{K})-{H}_{x}(0\text{K}))$$

(1)

$$\mathrm{\Delta}{H}_{f}(298\text{K})=627.5095({E}_{\text{CORR}})+31.17-627.5095({E}_{\text{Zn}})+\mathrm{\Delta}{H}_{f}(\text{atom},298\text{K})-627.5095({E}_{\text{atom}})$$

(2)

*E*_{CORR} is identified as the sum of electronic and
thermal enthalpies from the output of the Gaussian frequency calculation (which
includes thermal and ZPE corrections to the energy). *E*_{Zn}
and *E*_{atom} are the energies of the Zn and nonmetal atoms
at a given level of theory. The constant 31.17 (kcal/mol) in Equation 2 is the
Δ*H _{f}* (Zn, 298K) taken from the NIST chemistry
WebBook

$$\sqrt{\frac{1}{n}\sum _{i}^{n}{({x}_{i}-\overline{x})}^{2}}$$

(3)

Summarized in Table 2 are calculated
heats of formation for Zn complexes at the CCSD and CCSD(T) levels for all
applications of the 6–31G** basis set (stand-alone, and as
part of LACVP**). A plot of all calculated values versus the
experimental Δ*H _{f}* for an illustrative comparison
is given in Figure 1. For each metal entry,
most data points are grouped rather closely together. Significant deviations from
the experimental values were found in calculated

The calculated Δ*H _{f}* values for

Overall, calculations using the 6–31G** basis set
always perform better than their LACVP** counterparts. In 6 of 8
cases, CCSD(T)/6–31G** outperforms
CCSD/6–31G** with the two exceptions being ^{2}ZnH
(by 0.1 kcal/mol) and ^{1}ZnF_{2} (by 1.6 kcal/mol).
CCSD(T)/LACVP** outperforms CCSD/LACVP** in all
cases except for ^{2}ZnH. The average error associated with
CCSD(T)/6–31G** is -10.1 kcal/mol, a 2.5 kcal/mol
improvement over the average for CCSD/6–31G**. The average
error increases nearly twofold for both CCSD and CCSD(T) calculations using the
LACVP** basis set.

Next we will focus on the geometries of species for which experimental data
are available: ^{2}ZnH, ^{1}ZnF_{2},
^{1}ZnCl_{2} and ^{1}Zn(CH_{3})_{2}.
Table 3 contains a summary of Zn-X bond
lengths for all complexes, with the aforementioned available literature values. For
^{2}ZnH, the Zn-H bond length is calculated to within 0.001 A at the
CCSD/6–31G** and CCSD(T)/6–31G**
levels of theory. The deviation from experimental is significantly larger for both
coupled-cluster methods using the LACVP** basis set; 0.057 A for
CCSD and 0.058 A for CCSD(T). For ^{1}ZnF_{2}, the reverse trend is
observed in that coupled-cluster methods incorporating the LACVP**
basis set calculate Zn-F bond lengths to within 0.01 A, whereas methods utilizing
strictly the Pople-style 6–31G** basis set arrive at
equilibrium bond lengths 0.033 A lower than the experimental value. For
^{1}ZnCl_{2}, CCSD/6–31G** predicts the
most accurate bond length, within 0.007 A of the experimental value
(CCSD(T)/6–31G** is nearly as accurate, off by 0.008 A).
Both methods with the LACVP** basis set are off by at least 0.04 A.
Finally, the Zn-C bond distance in ^{1}Zn(CH_{3})_{2} is
calculated to within 0.002 A at the CCSD/6–31G** level of
theory and within 0.063 A at CCSD/LACVP**. The Zn-X distances for
which experimental data are not available are scattered at best amongst the levels
of theory, but with no literature values available it is difficult to assess which
methods are “correct” in their predictions. Three points are
immediately obvious: (1) the 6–31G** basis set is preferred
over LACVP** for these heat of formation predictions, (2) no obvious
trend with respect to over- or underestimation of bond lengths and (3) applying the
CCSD(T) level offers no significant improvement on calculated equilibrium bond
lengths with a constant basis set; indeed, the geometries typically *deviate
more from the experimental value* using the higher-level method.

In order to further probe the effect of basis set on these calculations, we
ran all calculations using the TZVP basis set. Calculated heats of formation for
ZnX_{n} complexes at the CCSD and CCSD(T) levels for all applications of
the TZVP basis set (stand-alone, and in conjunction with LANL2DZ) are summarized in
Table 4 and Figure 3. All calculated Δ*H _{f}* values
were overestimated using these two basis sets. As with the
6–31G** basis sets, the two methyl Zn species had poorly
predicted Δ

As was observed using the Pople-style basis set, TZVP and LANL2DZ-TZVP
results for ^{2}ZnCH_{3} were quite poor. All errors well surpassed
100% of the experimental value, ranging from a low of 162.7% for
CCSD(T)/LANL2DZ-TZVP to a high of 175.4% at the CCSD/TZVP level of theory.
Of eight methods utilized during the course of this study, not one does an adequate
job of predicting the Δ*H _{f}* value for

The predicted bond lengths were compared against experimental values for
^{2}ZnH, ^{1} ZnF_{2}, ^{1}ZnCl_{2} and
^{1}Zn(CH_{3})_{2} (Table 5). The CCSD(T)/TZVP geometry for ^{2}ZnH was closest to
the experimental r_{Zn–H}, overestimating by 0.025 Å. The
CCSD/TZVP level of theory best described the Zn-F bond length in
^{1}ZnF_{2}, off by only 0.006 A. Both the CCSD and CCSD(T)
methods in conjunction with the TZVP basis set gave a Zn-Cl bond distance of 2.109,
0.037 A higher than the experimental r_{ZnCl} value in
^{1}ZnCl_{2}. ^{1}Zn(CH_{3})_{2} could
only be optimized at the CCSD/LANL2DZ-TZVP level of theory and r_{Zn-C} was
overestimated by 0.056 A. For the three complexes that could be optimized at all
levels of theory incorporating TZVP or LANL2DZ-TZVP, the TZVP basis set performed
better although there was very little variation in the results for
^{1}ZnF_{2}. There was no clear separation between the CCSD and
CCSD(T) methods, with CCSD better for ^{1}ZnF_{2}, CCSD(T) better
for ^{2}ZnH (both by slim margins) and both methods producing identical
geometries for ^{1}ZnCl_{2}.

Overall, CCSD(T)/6–31G** and CCSD(T)/TZVP
calculations best predict the heat of formation for the 7 non-alkyl Zn complexes.
These methods are compared in Figure 3. The
CCSD(T)/6–31G** level of theory is shown to slightly
outperform the CCSD(T)/TZVP level in these Δ*H _{f}*
predictions. The difference in average errors between these two methods is 5.0
kcal/mol, with CCSD(T)/631G** averaging a −9.0 kcal/mol
deviation and CCSD(T)/TZVP differing from the experimental by an average of
−14.0 kcal/mol for the 7 non-alkyl complexes.

Generally, all levels of theory correctly predicted the appropriate spin
ground states for the Zn species investigated. Open shell species were found to
be ground state doublets, and closed shell species ground state singlets. There
are two notable exceptions, and these are the cases of ZnO and ZnS. The expected
ground state multiplicity, in so far as what species the heat of formation is
determined for, is ambiguous. Experimental work indicates that the available
heat of formation and Zn-X bond lengths are for the triplet.^{47}^{,}^{48} However,
the most accurate calculations to date on these complexes predict a ground state
singlet multiplicity.^{49}^{,}^{50} The triplet ground state was predicted for ZnO at
the CCSD/LACVP**, CCSD(T)/TZVP and CCSD/LANL2DZ-TZVP levels of
theory. The triplet was found to be the ground state of ZnS at only the
CCSD/LACVP** and CCSD/LANL2DZ-TZVP levels of theory. The
pseudopotential incorporating basis sets with the CCSD level of coupled-cluster
theory most often arrive at the triplet ground state for both ZnO and ZnS. The
lowest errors in calculated Δ*H _{f}* values were
observed using the TZVP and 6–31G** basis sets, which
most frequently predict the singlet ground state multiplicities. For ZnO,
theoretical studies support a triplet ground state for bond lengths in excess of
1.85A and a singlet for distances closer to the reported internuclear distance
of 1.69 Å.

We decided to further our investigation by applying a Douglas-Kroll-Hess
2^{nd} order relativistic correction (DKH) to calculations at the
CCSD/6–31G** level of theory as implemented in Gaussian
03.^{51}^{–}^{55} This correction was applied during the course of
both the geometry optimizations and frequency analyses. A comparison of
CCSD/6–31G** Δ*H _{f}*
values with and without this correction is provided in table 6. Addition of the DKH correction actually
results in a slight increase in the average heat of formation error. The error
is decreased with this correction for

CCSD/6–31G** Δ*H*_{f}
values with and without 2^{nd} order DKH relativistic correction for
non-alkyl Zn complexes.

Table 7 contains results for
CCSD(T)/6–31G** calculations including relativistic
effects for six ZnX_{n} complexes. The alkyl Zn species were omitted,
and efforts to include this correction for ^{1}Zn_{2} failed.
The average error increases by 1.4 kcal/mol with the inclusion of this
correction compared to the same set of six compounds using standard
CCSD(T)/6–31G** calculations. Only the error for
^{1}ZnF_{2} decrease with the 2^{nd} order
correction, while all other errors increase in magnitude. The most significant
increase in error is associated with ZnO. There is also a ground state
multiplicity change for this species, which is predicted to be a triplet with
inclusion of the DKH correction whereas at the
CCSD(T)/6–31G** level of theory it is predict to possess
a ground state singlet multiplicity.

There are two viable conformations for
^{1}Zn(CH_{3})_{2}, specifically conformations which
have pseudo-eclipsed hydrogen atoms, and pseudo-anti hydrogen atoms (Figure 4). Both conformations were examined
during the course of this study. The pseudo-eclipsed conformation was found to
be a minimum at both the CCSD/6–31G** and CCSD/TZVP
levels of theory, while the pseudo-anti conformer was calculated to be a
transition state as evidenced by the presence of one negative mode in the
vibrational analysis. Still, the energy difference between the two conformations
is quite small 0.03 kcal/mol at CCSD/6–31G** which
implies virtually no barrier to free rotation of the methyl groups as this value
is lower than *kT*. This is not surprising, as a bridging Zn atom
separates these substituents. A plot of the unrelaxed potential energy surface
for methyl rotation is displayed in Figure
5 (relative energy as a function of H-C-C-H twist angle).

Relative Conformational Energy vs. H-C-C-H Twist for
^{1}Zn(CH_{3})_{2}
(CCSD/6–31G**); unrelaxed scan in 5°
increments. Minima at −120, 0 and 120° correspond **...**

The eclipsed conformation is perhaps favored over the gauche due to
stabilizing hyperconjugative interactions with the *d*-orbitals
of the bridging Zn atom and the C-H antibonding orbitals; the eclipsed
conformation affords more of these interactions than does the gauche. Therefore
the potential energy surface is essentially a 60° shift of that seen in
ethane (where the pseudo-gauche conformation is most stable and the
pseudo-eclipsed is a rotational transition state). We are currently
investigating bulkier alkyl substituents (e.g. ZnEt_{2} and
Zn^{i}Pr_{2}) to see if this trend of eclipsed
conformational preference continues. Unfortunately, the size of these species
will preclude any investigations using higher level methods such as CCSD(T), but
we are confident that density functional methods and CCSD calculations will
prove adequate in these efforts.

For nine ZnX_{n} complexes, Δ*H _{f}*
values were calculated using the CCSD and CCSD(T) coupled cluster methods in
conjunction with the 6–31G**, TZVP,
LANL2DZ-6–31G** and LANL2DZ-TZVP basis sets. Generally, the
6–31G** basis set was found to outperform the other three
and the CCSD(T)/6–31G** level of theory provided
Δ

In general, CCSD/6–31G** calculations seem
appropriate for the prediction of heat of formation values for non alkylated
species, with the CCSD(T) method providing slight improvement. The
CCSD/6–31G** does a good job reproducing the experimental
bond distances in these systems. The addition of a 2^{nd} order
Douglas-Kroll-Hess relativistic correction does not provide overall improvement in
the calculated Δ*H _{f}* values. Rather, this term
offers modest improvement in some instances while actually resulting in an increased
error for other entries. The fact that CCSD values are comparable is important, as
this method is clearly more amenable to calculations on larger systems whereas the
CCSD(T) calculations would quickly become too resource-intensive to be viable. We
are currently extending our research to include complexes of other third row
transition metals to determine the extent of applicability of coupled-cluster
methods to these systems.

Click here to view.^{(52K, xls)}

Click here to view.^{(44K, doc)}

We thank the NIH (GM066859 and GM044974) for supporting this research. MNW wishes to thank the NIH for support in the form of an NRSA postdoctoral fellowship (F32GM079968). We thank the members of the Merz group for useful discussions during the course of this research. MNW specifically thanks Duane Williams for his numerous editorial comments.

Supporting Information Available: T1 diagnostics computed at the CCSD/6–31G** and CCSD/TZVP levels of theory and spreadsheets detailing all heat of formation calculations at all investigated levels of theory. This information is available free of charge on the internet at http://pubs.acs.org.

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