A theoretical Investigation of hyperpolarizability for small GanAsm clusters

A theoretical investigation of hyperpolarizability for small GanAsm (n+m =
4–10) clusters
Y.-Z. Lan, W.-D. Cheng, D.-S. Wu, J. Shen, S.-P. Huang et al.
Citation: J. Chem. Phys. 124, 094302 (2006); doi: 10.1063/1.2173993
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A theoretical investigation of hyperpolarizability for small GanAsm
„n+m=4–10… clusters
Y.-Z. Lan, W.-D. Cheng,a͒
D.-S. Wu, J. Shen, S.-P. Huang, H. Zhang,
Y.-J. Gong, and F.-F. Li
Fujian Institute of Research on the Structure of Matter, State Key Laboratory of Structural Chemistry,
The Graduate School of the Chinese Academy of Sciences, Fuzhou, Fujian 350002,
People’s Republic of China
͑Received 14 November 2005; accepted 19 January 2006; published online 1 March 2006͒
In this paper, the second and third order polarizabilities of small GanAsm ͑n+m=4–10͒ clusters are
systematically investigated using the time dependent density functional theory ͑TDDFT͒/6-311
+G*
combined with the sum-over-states method ͑SOS/ /TDDFT/6-311+G*
͒. For the static second
order polarizabilities, the two-level term ͑␤vec.2͒ makes a significant contribution to the ␤vec for all
considered GanAsm clusters except for the Ga3As4 cluster. And, for the static third order
polarizabilities, the positive channel ͑͗␥͘II͒ makes a larger contribution to ͗␥͘tot than the negative
channel ͑͗␥͘I͒. Similar to the cubic GaAs bulk materials, the small GanAsm cluster assembled
materials exhibit large second order ͑1ϫ10−6
esu͒ and third order susceptibilities ͑5ϫ10−11
esu͒.
The dynamic behavior of ␤͑−2␻;␻,␻͒ and ␥͑−3␻;␻,␻,␻͒ show that the small GanAsm cluster
will be a good candidate of nonlinear optical materials due to the avoidance of linear resonance
photoabsorption. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2173993͔
I. INTRODUCTION
Clusters exhibit properties remarkably different from
those of the bulk materials,1,2
and the cluster-assembled ma-
terials have novel mechanical, electrical, and optical
properties.3–6
The energy band structure and the phonon dis-
tribution can be completely different for clusters as com-
pared with the crystalline bulk. Clusters have localized states
that are not present in the bulk. The transition from a higher
to a lower energy state may proceed via one or more local-
ized intermediate states. In recent years, there has been con-
siderable interest in the theoretical7–15
and experimental16,17
investigations of the semiconductor clusters such as Si, Ge,
and group III-V elements. As an example of group III-V
cluster, the optical properties of the gallium arsenide ͑GaAs͒
cluster are very different from those of the bulk. The exis-
tence of free surfaces in the GanAsm clusters results in the
absorption spectra7
exhibiting long absorption tails in the
low energy region. The static polarizabilities8
exceed the
bulk limit and tend to decrease with increasing cluster size.
The variable tendencies of static polarizabilities also imply
the “metalliclike” nature of the clusters. The polarizabilities
for the GanAsm ͑m+n=4–30͒ clusters reside above and be-
low the bulk limit in the experimental investigation.16
On the basis of the special behavior of the absorption
spectra and static polarizabilities in GaAs clusters, we be-
lieve that the investigation of nonlinear optical properties
will be interesting and important. For the nonlinear optical
properties of GaAs bulk materials, people have performed
many theoretical and experimental investigations.18–34
The
results of these studies suggest that GaAs should have large
␹͑2͒
͑0͒͑10−7
esu͒ and ␹͑3͒
͑0͒͑10−11
esu͒ values. However,
there are only few reports of theoretical studies13,15
for the
hyperpolarizabilities of the GaAs cluster. The present work
will provide the systematical insight into the nonlinear opti-
cal properties of the GanAsm ͑n+m=4–10͒ clusters based on
the calculated results by ab initio time-dependent density
functional theory ͑TDDFT͒ method combined with the sum-
over-states ͑SOS͒ formalism.
II. COMPUTATIONAL PROCEDURES
The calculation method we used in this paper is based on
the perturbation expansion of the Stark energy of the mol-
ecule as a function of electric field. The total energy ͑Stark
energy͒ of a molecular system in an external field E may be
written in a power series by
Utot = Utot − uiEi − 1
2! ␣ijEiEj − 1
3! ␤ijkEiEjEk
− 1
4! ␥ijklEiEjEkEl − ... . ͑1͒
Here the repeated indices represent the summation of coor-
dinate components. Ei is a component of the external field, ui
is a component of the permanent dipole moment, and ␣ij,
␤ijk, and ␥ijkl are the polarizability and the second and third
order polarizabilities, respectively.
The tensor component of the frequency-dependent polar-
izability ␤ijk͑−␻p;␻1,␻2͒ and ␥ijkl͑−␻p;␻1,␻2,␻3͒ can be
written as35,36
␤ijk͑− ␻p;␻1,␻2͒
= ͑2␲/h͒2
͚p
ͭ ͚mn͑m n͒
Ј ͗g͉ri͉n͗͘n͉⌬rj͉m͗͘m͉rk͉g͘
͑␻ng − ␻p͒͑␻mg − ␻2͒
+ ͚m
Ј ͗g͉ri͉m͗͘m͉⌬rj͉m͗͘m͉rk͉g͘
͑␻mg − ␻p͒͑␻mg − ␻2͒
ͮ, ͑2͒
a͒
Electronic mail:cwd@ms.fjirsm.ac.cn
THE JOURNAL OF CHEMICAL PHYSICS 124, 094302 ͑2006͒
0021-9606/2006/124͑9͒/094302/8/$23.00 © 2006 American Institute of Physics124, 094302-1

␥ijkl͑− ␻p;␻1,␻2,␻3͒
= ͑2␲/h͒3
K͑− ␻p,␻1,␻2,␻3͒e4
ϫ͚ͭp
͚ͫm,n,o
Ј ͗g͉ri͉m͗͘m͉⌬rj͉n͗͘n͉⌬rk͉o͗͘o͉ri͉g͘
͑␻mg − ␻p͒͑␻ng − ␻2 − ␻3͒͑␻og − ␻2͒
ͬ
− ͚p
͚ͫm,n
Ј ͗g͉ri͉m͗͘m͉rj͉g͗͘g͉rk͉n͗͘n͉ri͉g͘
͑␻mg − ␻p͒͑␻ng − ␻3͒͑␻ng + ␻2͒
ͬͮ, ͑3͒
where ͚Ј indicates a summation over all states except for the
ground state; ͗g͉ri͉n͘ is an electronic transition moment cor-
responding to the i͑i=x,y,z͒ component between the refer-
ence state ͗g͉ and excited state ͗n͉; ͗n͉⌬rj͉m͘ denotes the
dipole difference operator equal to ͗n͉rj͉m͘−͗g͉rj͉g͘␦nm; ␻mg
is the transition frequency between excited state m and ref-
erence state g; ␻1, ␻2, ␻3 are the frequencies of the applied
radiation field; ␻p is the polarization response frequency; and
the prefactor K͑−␻p;␻1,␻2,␻3͒ is a numeric coefficient.36
In
Eq. ͑2͒, the terms include three-level-type terms ͑␤vec.3͒ in
addition to the two-level contributions ͑␤vec.3͒, and in Eq.
͑3͒, the first summation involves the two-photon allowed
states, referred to as the ͗␥͘II terms, and the second summa-
tion involves four-photon volleys between the reference state
and one-photon allowed states, referred to as the ͗␥͘I terms.
The sum-over-states perturbation theory expression for the
hyperpolarizability ͓Eqs. ͑2͒ and ͑3͔͒ indicates that one re-
quires transition dipole moments between the ground and
excited states and between the excited states, excitation en-
ergies, and excited state dipole moments to compute the ␤
and ␥ responses. These data will be obtained by the
TDDFT/6-311+G*
method in this work.
In order to systematically investigate the hyperpolariz-
ability of small GaAs cluster, we select a series of GanAsm
͑n+m=4–10͒ clusters as study target. The electronic struc-
ture and stability of small GaAs clusters were investigated in
detail by several theoretical studies.7,9
Based on these theo-
retical results, we configure 12 small GanAsm clusters
͑shown in Fig. 1͒ and optimize the geometries of all consid-
ered GanAsm clusters with symmetry constraint ͑shown in
Fig. 1͒ in the DFT/6-311+G*
level before calculating the
optical properties. The geometry optimizations and calcula-
tions of excited state properties are performed in the GAUSS-
IAN 98 program.37
III. RESULTS AND DISCUSSIONS
A. The second order nonlinear properties
1. The static second order polarizability ␤„0…
The static second order polarizabilities ␤͑0;0,0͒ of all
considered GanAsm clusters are listed in Table I, where ␤vec
is the vector component along the ground state dipole mo-
ment direction and can be defined as
TABLE I. Relative contribution of two-level ͑␤vec.2͒ and three-level terms ͑␤vec.3͒ to the second order polariz-
ability ͑␤vec͒ for small GanAsm ͑n+m=4–10͒ clusters based on the SOS/ /TDDFT/6-311+G*
method in the
static case. The ␤ and ␹͑2͒
͑0͒ are in units of 10−30
cm5
/esu ͑i.e., 10−30
esu͒ and 10−8
esu, respectively. Note that,
if we use the SI unit, the conversion factors between SI and esu are ␤͑esu͒=3.7114ϫ10−19
C3
m3
J−2
=4.1917ϫ10−8
m4
V−1
.
GanAsm Symmetry ␤vec ␤vec.2 ␤vec.3 ␤vec.2 /␤vec.3 ␤tot ␹͑2͒
͑0͒
Ga2As2 D2h 0.0 0.0 0.0 0.0 0.0
Ga2As3 D3h 0.0 0.0 0.0 0.0 0.0
Ga3As2 Cs 5.51 4.15 1.36 3.05 5.63 706.06
Ga3As3 C2v 7.77 9.21 −1.44 −6.40 7.95 824.88
Ga3As4 Cs 2.49 0.78 1.71 0.46 2.56 226.51
Ga4As3 C2v 0.60 0.49 0.11 4.45 0.61 54.53
Ga4As4͑1͒ C2h 0.0 0.0 0.0 0.0 0.0
Ga4As4͑2͒ Td 0.0 0.0 0.0 0.0 0.0
Ga4As5 C2v 0.91 0.99 −0.08 −12.38 0.94 64.76
Ga5As4 C2v 1.89 3.77 −1.88 −2.00 1.89 131.26
Ga5As5͑1͒ Cs 0.13 0.63 −0.50 −1.26 1.30 80.93
Ga5As5͑2͒ C1 2.18 3.03 −0.85 −3.56 2.23 138.83
FIG. 1. The geometrical structures of the clusters GanAsm ͑n+m=4–10͒.
094302-2 Lan et al. J. Chem. Phys. 124, 094302 ͑2006͒

␤vec = ͑␮1␤1 + ␮2␤2 + ␮3␤3͒/͉␮͉,
and ␤tot is obtained by
␤tot = ͑␤x
2
+ ␤y
2
+ ␤z
2
͒1/2
where ␤i =
1
3
͚i,j
͑␤ijj + ␤jij + ␤jji͒ ͑i,j = x,y,z͒.
͑4͒
␤vec will be equal to ␤tot when the charge transfer is unidi-
rectional and parallel to the molecular dipole moment. The
definition of the molecular Cartesian direction x, y, and z is
shown in Fig. 1. Note that the ␤ values of the Ga2As2,
Ga4As4͑1͒, and Ga4As4͑2͒ clusters equal zero because the
geometry of these clusters possesses the inversion symmetry.
For the Ga2As3 cluster with D3h symmetry, there are only
four nonvanishing tensor components: xxx, yxx, xxy, and
xyx, and these components satisfy the relation xxx=−yxx=
−xxy=−xyx, which leads to ␤i equal to zero. Here, we will
show interest in the relative magnitudes of the two-level ver-
sus three-level term contributions ͑␤vec.2/␤vec.3͒. This con-
cept had been firstly introduced by Kanis et al.38
in the in-
vestigation of ␤ for a variety of molecular chromophoric
units, and a question had been given as to whether the fea-
ture of ␤vec.2/␤vec.3 is generally correct or merely fortunate.
In this work, the feature of our results is very similar to
that of the investigation of ␤ for a variety of molecular chro-
mophoric units.38
As shown in Table I, both ␤vec.2 and ␤vec.3
are not negligible in these ␤vec results, and the ␤vec.2 makes a
significant contribution to the ␤vec for all considered GanAsm
clusters except for the Ga3As4 cluster, whose ␤vec.2/␤vec.3
equals 0.46. Therefore, the question given by Kanis et al.
also exists in the feature of ␤vec.2/␤vec.3 for inorganic com-
pound. However, a general conclusion still cannot be drawn
due to the lack of enough calculated results. More investiga-
tions of a series of inorganic compounds such as the silicon
cluster will be performed in the future. The exception of the
Ga3As4 cluster, that is, the three-level term provides a sig-
nificant contribution to ␤vec, can be further understood by
investigating the electronic origin of ␤vec. Figure 2 shows the
plot of ␤vec.2 and ␤vec.3 versus the number of excited states
for the Ga3As4 cluster. It is found that the 79th and 80th
excited states make larger contributions to ␤vec.3 than ␤vec.2.
The corresponding three-level term involves the ground state
and two excitation states, and it can be described by the Sg
→S79→S80→Sg channel. The excitations are as as follows:
the system leaves the ground state for the excited state S79,
then leaves for the excited state S80 before returning to the
ground state. ͑The relationships in the following sections
have similar meaning.͒ The significant excited states can be
related with the orbital-to-orbital transitions by configuration
TABLE II. The transition dipole moments and transition energies between
ground state and excited states, and between excited states for the Ga3As4
cluster.
States
Transition dipole moment ͑a.u.͒
Transition
energy
͑eV͒
Oscillator
strengthx y z
0–79 0.2376 −0.1040 0.0 4.545 0.0075
0–80 0.2599 0.1771 0.0 4.554 0.011
79–80 −0.7744 −1.0075 0.0 0.009 0.0004
FIG. 2. Plot of ␤vec.2͑0͒ and ␤vec.3͑0͒ vs the number of excited states for the
Ga3As4 cluster.
FIG. 3. The molecular orbitals contributing to the 79th and 80th excited
states for the Ga3As4 cluster.
FIG. 4. Plot comparing the ␹͑2͒
͑0͒ between the GaAs assembled materials
and the cubic bulk materials. “32” indicates the Ga3As2 cluster. The letters
a–l indicate the ␹͑2͒
͑0͒ of the GaAs bulk materials based on different cal-
culation methods ͑a–g͒ and experiments ͑h–l͒.
094302-3 Hyperpolarizability for GanAsm clusters J. Chem. Phys. 124, 094302 ͑2006͒

interaction mixing coefficients. The 79th and 80th excited
states are primarily composed of −0.541 20MO110→120
+0.628 42MO113→124 and 0.563 81MO110→120
+0.679 13MO113→124, respectively. The corresponding mo-
lecular orbitals ͑MOs͒ are shown in Fig. 3. Having an insight
into the geometry structure of the Ga3As4 cluster and the
charge distribution of molecular orbitals, we find that the
charge transferring from the As atom to the Ga atom mainly
contributes to these two excited states. Furthermore, we
present the transition dipole moments and transition energies
between the ground state and these two excited states, and
between the excited states in Table II. The large contribution
of ␤vec.3 to ␤vec significantly arises from the large transition
dipole moments.
2. The static second order susceptibility ␹„2…
„0…
In order to compare with the second order susceptibili-
ties of the GaAs bulk materials, we estimate the second order
susceptibilities ␹͑2͒
͑0͒ of the GanAsm cluster assembled ma-
terials. The ␹͑2͒
͑0͒ can be obtained in terms of the following
formula:
␹ijk
͑2͒
͑− ␻p;␻1,␻2͒ = Nfi͑␻p͒fj͑␻1͒fk͑␻2͒
ϫ␤ijk͑− ␻p;␻1,␻2͒, ͑5͒
where N is the cluster number density of the GanAsm cluster
assembled materials and defined as the product of the mass
density ͑d͒ of the cluster assembled materials and
Avogadro’s constant ͑NA͒ divided by the molar mass ͑M͒ of
the cluster, that is, N=dϫNA/M; f͑␻i͒ is the local field fac-
tor at the radiation frequency ␻i and can be obtained by the
following formula: f͑␻i͒=͓␧͑I͒
͑␻i͒+2͔/3, where ␧͑I͒
͑␻i͒ is
TABLE III. Relative contributions of negative ͑͗␥͘I͒ and positive channels ͑͗␥͘II͒ to the third order polarizabil-
ity ͑͗␥͘tot͒ for small GanAsm ͑n+m=4–10͒ clusters based on the SOS/ /TDDFT/6-311+G*
method in the static
case. The ␥ and ␹͑3͒
͑0͒ are in units of 10−36
and 10−11
esu, respectively. Note that, if we use the SI unit, the
conversion factors between SI and esu are ␥͑esu͒=1.2380ϫ10−25
C4
m4
J−3
=1.3982ϫ10−14
m5
V−2
.
GanAsm Symmetry ͗␥͘I ͗␥͘II ͗␥͘I /͗␥͘II ͗␥͘tot ␹͑3͒
͑0͒
Ga2As2 D2h −37.78 23.18 −1.63 −14.6 −11.51
Ga2As3 D3h −5.67 12.37 −0.46 6.7 4.19
Ga3As2 Cs −4.46 15.21 −0.29 10.75 6.83
Ga3As3 C2v −10.32 42.44 −0.24 32.12 16.88
Ga3As4 Cs −0.92 8.14 −0.11 7.22 3.24
Ga4As3 C2v −5.07 7.28 −0.69 2.21 1.00
Ga4As4͑2͒ C2h −11.35 47.6 −0.24 36.25 14.29
Ga4As4͑2͒ Td −9.9 41.67 −0.24 31.77 12.53
Ga4As5 C2v −2.33 3.92 −0.59 1.59 0.55
Ga5As4 C2v −1.47 6.94 −0.21 5.47 1.92
Ga5As5͑1͒ Cs −3.78 8.44 −0.40 4.66 1.47
Ga5As5͑2͒ C1 −3.63 14.25 −0.25 10.62 3.34
FIG. 5. Dynamic behavior of second order polarizability for the small
GanAsm cluster.
FIG. 6. Plot of ␤vec.2, ␤vec.3, ␤vec, and ␤tot vs the number of excited states for
the Ga3As2 cluster in the 0.65 eV case.

the dielectric constant of the cluster assembled materials at
frequency ␻i. Here, the mass density ͑d=5.316 g/cm3
͒ and
static dielectric constant ͓␧͑I͒
͑0͒=13.2͔ are approximated to
the value of the GaAs bulk materials.39
The ␹͑2͒
͑0͒ of all considered GanAsm cluster assembled
materials are listed in Table I. Comparisons of the ␹͑2͒
͑0͒
values of the GanAsm cluster assembled materials with the
cubic GaAs bulk materials based on different theories and
experiments19–29
are shown in Fig. 4. The calculated results
͑marked a–g in Fig. 4͒ and experimental data ͑marked h–l
in Fig. 4͒ of ␹͑2͒
by different groups localize at the range
from 100ϫ10−8
to 200ϫ10−8
esu for the cubic GaAs bulk
materials. It is found that our calculated ␹͑2͒
͑0͒ of the
GanAsm cluster assembled materials are close to those of the
GaAs cubic bulk materials when the GanAsm clusters includ-
ing more than seven atoms ͑i.e., Ga3As4 cluster͒ are as-
sembled, and the ␹͑2͒
͑0͒ values ﬂuctuate around 100
ϫ10−8
esu.
3. The dynamic second order polarizability
␤„−2␻;␻,␻…
In experiment, one often measures the ␤͑−2␻;␻,␻͒ us-
ing some special laser wavelengths such as 1064 nm
͑1.16 eV͒ and 1907 nm ͑0.65 eV͒. Therefore, it is important
and interesting to investigate the dynamic behavior of the
second order polarizability. We can obtain the dynamic be-
havior of ␤ using the SOS method, in which the laser fre-
quency ͑␻͒ is an input parameter in the SOS formulation
͓Eq. ͑2͔͒. Figure 5 shows the plots of the ␤tot value versus the
input photon energy from 0.0 to 1.3 eV. From Fig. 5, we ﬁnd
that the ␤tot value varies slowly until the input photon energy
approaches about 0.7 eV, and then exhibits different disper-
sions or resonances. For instance, in the case of the Ga3As2
cluster, there is a resonance peak at 0.65 eV, and the corre-
sponding ␤tot value equals 14.66ϫ10−30
cm5
/esu. In view of
the denominator of Eq. ͑2͒, the large ␤tot can be obtained at
an input photon energy close to one-photon ͑␻͒ or two-
photon ͑␻p=2␻͒ resonance energies. To identify the contri-
bution to the second order polarizability, we present the plot
of ␤vec.2, ␤vec.3, ␤vec, and ␤tot versus the number of excited
states for the Ga3As2 cluster at an input energy of 0.65 eV in
Fig. 6. It is found that the contribution of the third excited
state leads to the large ␤vec or ␤tot. The transition energy of
third excited state is 1.32 eV ͑Ϸ0.65ϫ2͒, which means that
the two-photon near-resonance peak occurs at the 2␻ fre-
quency.
Near resonance leads to a large ␤; however, the linear
absorption possibly occurs at near resonance of input energy,
which leads to high optical damage, thermal effects, and lim-
ited response times.40
For all considered GanAsm clusters, the
calculated7
and experimental17
effective photoabsorption
gaps are up to 5.0 eV, which means that the strong absorp-
tion should appear at about half of this energy. Therefore,
one can obtain the dynamic second order polaraziability at
the low-frequency region with the input photon energy rang-
ing from 0.0 to 2.4 ͑Ϸ5.0/2͒ eV in order to avoid the reso-
nance absorption.
B. The third order nonlinear properties
1. The static third order polarizability ␥„0…
The static third order polarizabilities ␥͑0͒ of all consid-
ered GanAsm clusters are listed in Table III, where ͗␥͘I and
͗␥͘II are the negative ͓the second set of summations in
TABLE IV. Nine components ␥iiii͑i=x,y,z͒͑ϫ10−36
esu͒ of the Ga2As2 and Ga2As3 clusters.
GanAsm Symmetry ␥xxxx ␥yyyy ␥zzzz ␥xxyy, ␥yyxx ␥xxzz, ␥zzxx ␥yyzz, ␥zzyy
Ga2As2 D2h −24.37 5.31 −12.95 0.01 −19.24 −1.28
Ga2As3 D3h 0.52 0.61 18.12 0.24 3.50 3.38
FIG. 7. The ␥xxxx and ␥zzzz values vs the number of excited states for the
Ga2As2 and Ga2As3 clusters, respectively.
FIG. 8. Plot comparing the ␹͑3͒
͑0͒ between the GanAsm cluster assembled
materials and the cubic GaAs bulk materials. “22” indicates the Ga2As2
cluster. The letters a– f indicate the ␹͑3͒
͑0͒ of the GaAs bulk materials based
on different calculation methods ͑a–d͒ and experiments ͑e and f͒.

Eq. ͑3͔͒ and positive ͓the first set of summations in Eq. ͑3͔͒
channel contributions to ͗␥͘, respectively. ͗␥͘I and ͗␥͘II can
be described by the virtual-excitation channel40
Sr→Sm
→Sr→Sn→Sr and Sr→Sm→Sn→Sp→Sr, respectively,
where Sr is a reference state, which is usually, but not nec-
essarily, the ground state; and Sm, Sn, and Sp are the excited
states. ͗␥͘tot is the sum of ͗␥͘I and ͗␥͘II. The ͗␥͘ is approxi-
mately obtained by
͗␥͘ = 1/5͑␥xxxx + ␥yyyy + ␥zzzz + ␥xxyy + ␥xxzz + ␥yyxx
+ ␥yyzz + ␥zzxx + ␥zzyy͒. ͑6͒
The values of ͗␥͘I/͗␥͘II ͑in Table III͒ and ͗␥͘tot mean that,
for all considered GanAsm clusters except for the Ga2As2
cluster, the ͗␥͘II makes a significant contribution to the ͗␥͘tot;
that is, the positive channel ͑Sr→Sm→Sn→Sp→Sr͒ which
involves three excited states makes a significant contribution
to the third order polarizability ͗␥͘tot. For the Ga2As2 cluster,
the negative channel ͑Sr→Sm→Sr→Sn→Sr͒ which in-
volves two excited states makes a significant contribution to
the ␥ because the ͗␥͘I/͗␥͘II equals −1.63.
In order to illustrate the significant contributions of these
excited states to the static third order polarizabilities of the
GanAsm cluster, as examples, we present nine main compo-
nents ␥iiii͑i=x,y,z͒ which contribute to ͗␥͘ of the Ga2As2 and
Ga2As3 clusters in Table IV. It is shown that the ␥xxxx and
␥zzzz are the largest components among the nine components
␥iiii͑i=x,y,z͒ for the Ga2As2 and Ga2As3 clusters, respectively.
Figure 7 gives the plot of ␥iiii͑i=x,y,z͒ value versus the number
of excited states. It is found that, for the Ga2As2 cluster, the
57th and 58th excited states make significant contributions to
the ␥xxxx. The virtual-excitation channel composed of these
two excited states can be described by the Sg→S57→Sg
→S58→Sg channel; that is, the negative channel gives the
most contribution to ␥xxxx and determines the sign of the
␥xxxx. Contrarily, for the Ga2As3 cluster, the large ͗␥͘II
mainly arises from the contribution of the positive Sg→S76
→S78→S83→Sg channel.
2. The static third order susceptibility ␹„3…
„0…
In order to compare with the static third order suscepti-
bility of the GaAs bulk materials, we estimate the static third
order susceptibilities ␹͑3͒
͑0͒ of the GanAsm cluster assembled
materials. The ␹͑3͒
͑0͒ can be obtained by
FIG. 9. Dynamic behavior of third order polarizability for the small GanAsm
cluster.
FIG. 10. The ͉␥zzyy͉ and ␥xxyy values vs the number of excited states for the
Ga4As3 and Ga4As4͑2͒ clusters, respectively.

␹ijkl
͑3͒
͑− ␻p;␻1,␻2,␻3͒ = Nfi͑␻p͒fj͑␻1͒fk͑␻2͒fl͑␻3͒
ϫ␥ijkl͑− ␻p;␻1,␻2,␻3͒, ͑7͒
where N and f are defined by Eq. ͑5͒. The calculated
␹͑3͒
͑0͒ for all considered GanAsm cluster assembled materials
are listed in Table III. And in Fig. 8, we show the calculated
␹͑3͒
͑0͒ of the GanAsm cluster assembled materials and the
cubic GaAs bulk materials based on different theories and
experiments.30–34
The calculated values ͑marked a–d in Fig.
8͒ and experimental data ͑marked e and f in Fig. 8͒ of
␹͑3͒
͑0͒ by different groups localize at about 5ϫ10−11
esu for
the cubic GaAs bulk materials. From Fig. 8, we find that our
calculated ␹͑3͒
͑0͒ of all considered GanAsm clusters fluctuate
around 5ϫ10−11
esu.
3. The dynamic third order polarizability
␥„−3␻;␻,␻,␻…
As the ␤͑−2␻;␻,␻͒, we can obtain the dynamic behav-
ior of ␥͑−3␻;␻,␻,␻͒. Figure 9 shows the plots of the
͗␥͑3␻͒͘ versus the input photon energy from 0.0 to 1.3 eV.
From Fig. 9, we find that all considered GanAsm assembled
clusters exhibit the wide nonresonance frequency region
͑Ͻ0.7 eV͒ except for the Ga4As3 and Ga4As4͑2͒ clusters.
There is separately a resonance peak at 0.372 and 0.496 eV
for the Ga4As3 ͓Fig. 9͑a͔͒ and Ga4As4͑2͒ ͓Fig. 9͑b͔͒ clusters.
The presence of a resonance peak means that the input pho-
ton energy is close to the transition energy of some excited
state. Obviously, the linear absorption mentioned above also
influences the third order nonlinear optical process. There-
fore, it is necessary to analyze the electron origin of contri-
bution to large ͗␥͘, similar to the investigation above of ␥͑0͒
of the Ga2As2 and Ga2As3 clusters. The component with the
largest contribution to ͗␥͘ is ␥zzyy and ␥xxyy for the Ga4As3
and Ga4As4͑2͒ clusters at the corresponding resonance input
photon energy ͑i.e., 0.372 and 0.496 eV͒, respectively. Fig-
ure 10 shows the plot of ␥zzzz and ␥xxyy components versus
the number of excited states. From Fig. 10, we find that the
contributions can be viewed as the negative channel ͑the ab-
solute value of ␥zzyy in Fig. 10͒ and positive channel contri-
butions for the Ga4As3 and Ga4As4͑2͒ clusters, respectively.
For the Ga4As3 cluster, the third ͑S3͒ and sixth ͑S6͒ excited
states make significant contributions to the ␥xxyy. The transi-
tion energies of S3 and S6 are 1.11 and 1.23 eV, which are
much lower than the photoabsorption gaps of 5.0 eV ͑Refs. 7
and 17͒ mentioned above. These two excited states ͑third and
sixth͒ form the negative virtual-excitation channel contribu-
tion Sr→S3→Sr→S6→Sr. For the Ga4As4͑2͒ cluster, three
excited states ͑fourth, fifth, and sixth͒ mainly contribute to
the ␥xxyy.
IV. CONCLUSIONS
We systematically investigate the second and third order
nonlinear optical properties of a series of small GanAsm ͑n
+m=4–10͒ clusters using the TDDFT/6-311+G*
combined
with the SOS method. The small GanAsm ͑n+m=4–10͒ clus-
ters as well as the GaAs bulk materials exhibit large nonlin-
ear susceptibilities ͑10−6
esu for ␹͑2͒
and 10−11
esu for ␹͑3͒
͒
compared with the general nonlinear susceptibility values41
͑10−8
esu for ␹͑2͒
and 10−15
esu for ␹͑3͒
͒. Due to the large
photoabsorption gaps ͑up to 5.0 eV͒, it is expected that the
resonance absorption can be avoided in the frequency-
dependent ͑Ͻ2.4 eV͒ nonlinear optical experiments. Our cal-
culated results also expect that the small GanAsm cluster as-
sembled materials will be a good candidate of nonlinear
materials.
ACKNOWLEDGMENTS
This investigation was based on the work supported by
the National Natural Science Foundation of China under
Project No. 20373073, the Key Foundation of the Fujian
Province ͑No. 2004HZ01-1͒, and the Foundation of State
Key Laboratory of Structural Chemistry ͑No. 030060͒.
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.

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