Is the Hyperpolarizability of Cu
2
Negative? A Study of Basis Set and Electron Correlation
Effects
George Maroulis*
Department of Chemistry, UniVersity of Patras, GR-26500 Patras, Greece
ReceiVed: May 5, 2003; In Final Form: June 18, 2003
The dipole (hyper)polarizability of the copper dimer has been obtained from conventional ab initio and density
functional theory calculations. A very large (23s16p12d6f) basis set consisting of 346 Gaussian-type functions
is thought to provide reference results of near-Hartree-Fock quality for all properties. We obtain R j) 102.54
and ΔR) 41.89 for the mean and the anisotropy of the dipole polarizability (R
R
/e
2
a
0
2
E
h
-1
). For the Cartesian
components and the mean of the hyperpolarizability (10
-3
γ
Rγδ
/e
4
a
0
4
E
h
-3
) we obtain γ
zzzz
) 309, γ
xxxx
) 209,
γ
xxzz
) 87, and γ j) 244. Electron correlation lowers R j but increases considerably ΔR. The effect on the
hyperpolarizability is enormous, as the longitudinal component γ
zzzz
is drastically reduced, while γ
xxxx
and
γ
xxzz
are nearly halved. At the CCSD(T) level of theory with a [7s6p6d2f] basis set we obtain R j) 93.82, ΔR
) 67.09 and γ
zzzz
) 18, γ
xxxx
) 101, γ
xxzz
) 35, and γ j) 86. The dipole polarizability varies as [R j (R) -
R j (R
e
)]/e
2
a
0
2
E
h
-1
) 28.09(R - R
e
) + 4.69(R - R
e
)
2
- 0.52(R - R
e
)
3
- 0.36(R - R
e
)
4
and [ΔR(R) - ΔR-
(R
e
)]/e
2
a
0
2
E
h
-1
) 49.58(R - R
e
) + 11.92(R - R
e
)
2
- 1.94(R - R
e
)
3
-1.32(R - R
e
)
4
around the experimental
bond length R
e
) 2.2197 Å. B3LYP density functional theory calculations with a [8s7p7d5f] basis set yield
R j) 77.62, ΔR) 44.73e
2
a
0
2
E
h
-1
, and γ j) (95.9 × 10
3
)e
4
a
0
4
E
h
-3
. These values differ from the conventional
ab initio results. The present investigation shows that the longitudinal component and the mean of the
hyperpolarizability are positive around R
e
, in conflict with previous findings. The extension of (hyper)-
polarizability calculations to higher copper clusters is highly nontrivial and will require the development of
new computational strategies.
The structure of copper clusters has been the object of
numerous theoretical and experimental studies.
1-7
Of particular
interest are efforts focusing on the general physicochemical
behavior of these systems. Such work includes the bonding of
acetylene to copper clusters,
8
the reaction of Cu
2
with ethylene,
9
the bonding of ammonia, carbon monoxide, and ethylene to
copper atom, dimer, and trimer,
10
the optical spectra of copper
dimer and trimer in superfluid helium,
11
the bonding of CO and
NO to Cu
2
,
12
the simulation of copper cluster deposition on
copper,
13
the physisorption of copper microclusters on MgO-
(100),
14
the bonding of ammonia to small copper clusters,
15
the
collision between Cu
2
and an Ar film,
16
the identification of
the Cu
2
(N
2
)
n
complexes,
17
and the optical properties and redox
behavior of copper clusters.
18
More, in-depth experimental
19-22
or theoretical
23-26
studies have been reported for the copper
dimer. Remarkably few papers have been published on the
dipole polarizability of copper clusters. Recent work by Ca-
laminici et al.,
27
Jaque ´ and Toro-Labbe ´,
28
and Cao et al.
29
included density functional theory (DFT) calculations of the
static dipole polarizability of copper clusters Cu
n
, n e 13. Lastly,
Shigemoto et al.
30
reported a study on the axial component of
the dipole hyperpolarizability (γ
zzzz
) of the copper dimer. Their
findings brought forth the possibility of a negative dipole
hyperpolarizability for this important diatomic molecule.
In this paper we report conventional ab initio and DFT
calculations of the static (hyper)polarizability of Cu
2
. We rely
on a finite-field approach,
31
presented in some detail in previous
work.
32-34
Our study includes an investigation of electric
correlation effects on the dipole properties and their bond-length-
or R-dependence around the experimental equilibrium bond
length R
e
. Electron correlation correction effects were obtained
via Møller-Plesset perturbation theory (MP) and coupled-cluster
techniques (CC).
35-41
Thus, the conventional ab initio methods
adopted in this work are self-consistent field (SCF), second-
(MP2) and fourth-order (MP4) Møller-Plesset perturbation
theory, single and double excitation coupled cluster theory
(CCSD), and its extension CCSD(T) which includes an estimate
of connected triple excitations by a perturbational treatment. In
addition to the above methods, we have added calculations
performed with a widely used DFT method, B3LYP.
42,43
We
expect our B3LYP results to provide valuable information on
the performance of DFT methods on copper clusters. Special
attention has been paid to the design of suitable basis sets of
Gaussian-type functions (GTF). This is a matter of basic im-
portance to molecular property calculations.
44-46
We have de-
signed basis sets for Cu
2
relying on a variety of substrates. Thus,
we eschew as much as possible the appearance of systematic
errors linked to their composition. We rely mostly on a rich,
(17s10p6d) primitive basis set contracted to A0 ≡ [6s3p3d] as
[842111;631;411].
47
We considered a sequence of basis sets built
upon this substrate. Their compositions are as follows:
* Electronic address: maroulis@upatras.gr.
A1 ≡ [7s6p5d1f], 114 contracted GTF, from A0 +
s(0.013896), p(0.099537, 0.036540), d(0.0847, 0.0252),
and f(0.0252)
A2 ≡ [7s6p6d], 110 CGTF, from A0 + s(0.013896),
p(0.099537, 0.036540), and d(0.5206, 0.0847, 0.0252)
6495 J. Phys. Chem. A 2003, 107, 6495-6499
10.1021/jp0352128 CCC: $25.00 © 2003 American Chemical Society
Published on Web 07/29/2003