Chemical Physics 328 (2006) 132–138
www.elsevier.com/locate/chemphys
Computational studies of one-electron properties of lithium hydride
in confinement
J.M.H. Lo, M. Klobukowski
*
Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G 2G2
Received 14 March 2006; accepted 22 June 2006
Available online 28 June 2006
Abstract
Dipole moments, dipole polarizabilities, and electric field gradients of the ground and the first 1R+ excited electronic state as well as
the dissociative state a 3R+ of a LiH molecule subjected to a cylindrical harmonic confining potential were calculated by means of the
second-order configuration interaction method. The response of these quantities to the confining potential was discussed in terms of the
bonding characteristics of the electronic states of a LiH molecule.
Ó 2006 Elsevier B.V. All rights reserved.
PACS: 31.15.Ar; 31.15.Pf; 31.25.Nj; 31.50.Bc; 31.50.Df; 33.15.Kr
Keywords: Dipole moments; Dipole polarizabilities; Electric field gradients; Confining potential; Lithium hydride
1. Introduction
Studies of systems confined by various forms of external potentials have commenced at the beginning of quantum mechanics. Following the early work of Fock [1] on
the confinement by magnetic fields, the concept of confinement has been utilized in several branches of science
such as nuclear physics [2], condensed-matter physics [3],
and surface chemistry [4]. The breakthrough in semi-conductors and nanotechnology in the past two decades,
where artificial atoms and molecules [5] can be constructed in experiments, provides a platform for verifying
the validity of the developed confinement models and
stimulates the advancement of the theory of confinement.
Confined many electron systems were reviewed by Jaskólski a decade ago [6]. More recent reviews were focused on
magnetic properties [7] and structure and ionization [8] of
confined atoms.
*
Corresponding author. Tel.: +1 780 492 2568; fax: +1 780 492 8231.
E-mail address: [email protected] (M. Klobukowski).
0301-0104/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemphys.2006.06.019
Recently, properties of quantum dots and confined
atoms were studied by using three-dimensional harmonic
oscillator potentials of spherical, elliptical, prolate, and
oblate symmetries [9–11]. Similar studies have been performed on molecular systems such as H2 [12,13], Li2 [14],
Be2 [15], Beþ
2 [16], and NeH [17]. These studies demonstrated that the spatial restrictions imposed on the electronic wavefunction by symmetry and strength of the
confining potential change geometries and spectral properties of the confined systems. These changes are seen in the
removed orbital degeneracy, modification of equilibrium
internuclear distances, and appearance of new points of
avoided crossing.
Distortion of the electron density in a molecule is
reflected in the variation of electric and magnetic properties
of the molecule. Measurement of these quantities and comparison with the values found in free space can provide
structural information about the molecular environment.
An excellent example demonstrating the usefulness of this
concept is provided by the studies of the auride ion done
by Sadlej et al. [18] who successfully accounted for the lack
of colour of the tetramethylammonium auride crystal by
using the helium cluster confinement model.
J.M.H. Lo, M. Klobukowski / Chemical Physics 328 (2006) 132–138
As the simplest neutral heterodiatomic molecule, lithium
hydride has been the subject of numerous benchmarking
theoretical studies due to the small number of electrons
allowing for the use of highly sophisticated methods and
extended basis sets. In the present study the electric dipole,
dipole polarizability, and electric field gradient of the LiH
molecule embedded in an axially symmetric harmonic
oscillator potential were calculated. For several values of
the confinement parameter x the electric dipole moment
and dipole polarizability were computed for three electronic states: the X 1R+ ground state, the bound excited
A 1R+ state, and the dissociative a 3R+ state. The electric
field gradient (EFG) components were calculated only for
the X and A 1R+ states. The dependence of the computed
quantities on the internuclear distance was also
investigated.
2. Computational details
The confining potential W(ri) was taken as an isotropic
two-dimensional harmonic oscillator potential centered at
the origin of the coordinate system
1
1
W ðri Þ ¼ x2 r2i ¼ x2 ðx2i þ y 2i Þ;
2
2
ð1Þ
where x defines the strength of the applied harmonic oscillator potential. The potential of Eq. (1) is a special case of
the general power-series confining potential [9]:
1 X nt þ1
2n
W ðri Þ ¼
x ðt bt Þ t ;
ð2Þ
2 t¼x;y;x t
where bt defines the origin of the potential. The harmonic
oscillator form of Eq. (2), with nt = 1, was employed because of the ease of programming the relevant integrals
in computer codes that use Gaussian basis functions.
The confining potentials defined in Eqs. (1) and (2) act
only on electrons. In all calculations it was assumed that
the principal axis of the cylindrical harmonic oscillator
potential overlaps with the molecular axis of LiH, taken
to be the z-axis. The cylindrical symmetry of the potential
ensures that there is no net interaction between the confining potential and the nuclei.
The wavefunctions of the ground and the first two
excited electronic states of LiH were calculated using the
full second-order configuration interaction (FSOCI)
method implemented in the program GAMESS-US
[19,20]. The zeroth-order configurations were generated
by the CASSCF method with the active space composed
of the 1s, 2s, and 2p orbitals of Li and the 1s orbital of
H. All electrons were correlated in the CASSCF calculations. In the subsequent second-order configuration interaction (SOCI) calculations all possible configurations
were generated by the single and double excitations from
the CASSCF reference configurations. No excitations were
allowed from the 1r MO. The renormalized Davidson correction [21] was used to estimate the contributions of quadruple excitations. This CASSCF/SOCI technique has
133
been extensively used in the molecular calculations involving heavy metal atoms [22–25], and very satisfactory performance was observed when compared to the multireference CI method.
Accuracy of the calculated dipole moments and polarizabilities strongly depends on the basis set quality in terms
of the active space size and the diffuseness of the orbital
exponents [26]. Therefore, two fairly large basis sets were
employed in the present calculations. The basis set of Jaszuński and Roos [27] for H, derived from Huzinaga’s 10s
basis set [28], was augmented by an additional 2s6p3d set
and contracted in the (411111111/111111/111) scheme. A
diffuse p function with the exponent of 0.01 was added to
this basis set, leading to the resulting 9s7p3d set. Its performance has been verified in the calculations of the static
polarizability and hyperpolarizability of H2 molecule.
Excellent agreement was found with the accurate results
of Kołos and Wolniewicz [29] who used explicitly correlated wavefunctions. The basis set for Li was the
(10s6p4d)/[5s3p2d] set of Sadlej et al. [30] optimized for
the calculations of molecular electric properties. Additional
set of 1s1p1d functions with exponents as = 0.03, ap = 0.03,
and ad = 0.02 was used giving rise to a 6s4p3d basis. At the
midpoint position between the Li and H nuclei a set of
4s4p4d Gaussian functions was included in order to properly describe the distorted electron density at this region
due to the harmonic oscillator potential [9]. Their exponents a were the same in each symmetry, and their values
of 0.025, 0.050, 0.075, and 0.100 were defined by the
expression a = x/2 where x was the confinement
parameter.
Unlike electric dipole moments and electric field gradients which were evaluated directly from the electron densities derived from the CASSCF/SOCI wavefunctions, the
dipole polarizability components were calculated using
the energy-based finite-field perturbation theory [31]. Kurtz
and coworkers have shown that numerical accuracy of the
finite field method is very sensitive to the precision of the
energy calculations [32]. Therefore, all thresholds were
tightened in GAMESS and with the convergence criterion
of 1020 a.u. selected in all the energy evaluations. Following the suggestion of Kurtz et al. [32], a small electric field
of 0.001 a.u. was used so as to avoid the unnecessary significant configurational changes of LiH.
3. Results and discussion
3.1. Dipole moments
Initially, potential energy curves for several low-lying
states of lithium hydride were calculated to calibrate the
performance of the method and basis sets. Fig. 1 shows
the potential energy curves for LiH molecule in both free
space and in a cylindrical confining potential.
As can be seen from Table 1, the calculated binding
energies and excitation energies of free LiH are in very
good agreement with the experimental values. In spite of
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J.M.H. Lo, M. Klobukowski / Chemical Physics 328 (2006) 132–138
0.3
0.3
(b) ω = 0.10
0.25
0.25
B 1Π
(a) ω = 0.00
Energy / a.u.
0.2
B Π
3
b Π
Li(2p)+H(1s)
0.15
Li(2px,y)+H(1s)
0.2
1
1 +
A Σ
0.15
b 3Π
1 +
A Σ
3 +
a Σ
0.1
Li(2pz)+H(1s)
a 3Σ+
0.1
Li(2s)+H(1s)
Li(2s)+H(1s)
1 +
1 +
X Σ
0.05
X Σ
0.05
0
0
1
2
3
4 5 6 7
RLi-H / Angstrom
8
1
2
3 4 5 6
RLi-H / Angstrom
7
8
Fig. 1. CASSCF/SOCI potential energy curves for several low-lying electronic states of LiH (a) for the zero field (x = 0.00 a.u.) and (b) for x = 0.10 a.u.
Energies are plotted with respect to the potential minimum of the ground X 1R+ state.
Table 1
Calculated excitation energies Te (in cm1) and binding energies De (in eV)
of LiH in free space
Parameter
State
Calc.
Expt. [33]
De
X 1R+
A 1R+
B 1P
2.4692
1.0619
0.0314
2.5154
1.0765
0.035
Te
X!A
X!B
A!B
26031
34371
8339
26510
34912
8402
the fact that the calculated equilibrium distance in the
ground electronic state (1.6102 Å) is slightly larger than
the measured value of 1.5949 Å [34], the CASSCF/SOCI
method is able to provide reliable wavefunctions and energies for the subsequent calculations of various one-electron
properties of LiH.
Several remarkable changes appear on the potential
curves when the confining potential is applied. Compared
to the field-free case (Fig. 1(a)), the binding energy for
the X 1R+ state increases while that for the A 1R+ state
decreases. The dissociation channel that leads to the
asymptotic products of Li(2P) + H(2S) splits into two
branches. The lower branch is of r symmetry and connects
with the 2pz configuration of Li while the upper branch is
of p symmetry, giving rise to the Li(2px) and Li(2py) products. These changes may be attributed to the geometric
constraint imposed by the confining potential as the axially
symmetric potential lifts the triple degeneracy of the p subshell and creates two subsets: {px, py} and {pz}, the latter
one being less destabilized and lower in energy [15]. The
second dissociation channel shifts the two P states towards
higher energy and greatly increases their Te values. In addition, the relative stability of 2pz orbital with respect to 2px
and 2py orbitals leads to variation of the electronic configuration of the A 1R+ state which is manifested in its electric
properties.
Electric dipole moments for the first three electronic
states of LiH were computed as expectation values using
the SOCI wavefunctions for a range of internuclear distances. The calculated dipole moments for the X 1R+ state
for several values of x are shown in Fig. 2.
Since the Li atom in lithium hydride was located at the
positive end of z-axis, a positive value of the dipole
moment indicates accumulation of the electron density at
the more electronegative H atom. The large magnitude of
Fig. 2. Electric dipole moments for the X 1R+ state of LiH. The bottom
line corresponds to x = 0.0 a.u. and the successive lines correspond to
increments in x of 0.05 a.u.
J.M.H. Lo, M. Klobukowski / Chemical Physics 328 (2006) 132–138
the electric dipole moment reveals strong ionic character of
the ground electronic state of the LiH molecule which
results from the partial charge transfer from Li to H. The
CASSCF/SOCI dipole moment at the experimental equilibrium distance of 1.5957 Å is 5.897 Debye which agrees
very well (within 0.3%) with the experimental value of
5.882 Debye [35]. The dipole moment reaches its maximum
at R 2.7 Å where the ground state potential energy curve
crosses the Li+H ionic potential curve [36]. At larger values of the internuclear distance R the electric dipole
moment gradually vanishes because of the increasing covalent bond character that leads the X 1R+ state to the correct
dissociation limit of the Li and H atoms in their ground
states.
Variation of the dipole moment with the internuclear
distance for the a 3R+ and A 1R+ differs from that found
in the ground electronic state. At small internuclear distances, both the a 3R+ and A 1R+ states possess negative
dipole moments, in contrast to the positive dipole moment
for the X 1R+ state. The one-electron promotion 2r ! 3r
leads to the excited 1r22r3r configuration that brings
about a shift of an electron towards the Li atom and
reverses the dipole orientation. The dipole moment for
the a 3R+ state behaves as the dipole moment in the ground
state and converges to zero for very large internuclear distance due to the increasing covalent character.
The dependence of the dipole moment on the internuclear distance R for the A 1R+ state is more complex. As
shown in Fig. 3 the dipole moment increases rapidly when
R increases from 1.0 Å until its maximum appears at 4.8 Å.
This observation is consistent with the MCSCF studies of
Docken and Hinze who found that the ionic potential
energy curve intersects the A 1R+ state potential energy
curve at 4.2 Å [36]. Similarly to the X 1R+ and a 3R+ states,
the configuration mixing with the covalent Li(2pz) + H(1s)
a
135
configuration diminishes the dipole moment for the A 1R+
state at larger values of R and yields the expected zero
dipole moment at the asymptotic limit.
Confinement effects on the electric dipole moments of
these states are fairly distinct, as illustrated in Figs. 2 and
3. The effect is more profound in the X 1R+ state in which
the magnitude of the dipole moment increased by 50% for
x = 0.20 a.u. The internuclear distance at which the dipole
moment reaches its maximum is shifted to larger values of
R. On the other hand, the extremal values of the dipole
moments for the a 3R+ and A 1R+ states are not significantly affected by the presence of confining potential. In
both cases the values are only slightly increased by 0.3 a.u.
For the X 1R+ and A 1R+ states under confinement the
maximum of the dipole moment is shifted towards larger
values of the internuclear distance R. In confinement, both
the ionic and covalent potential energy curves are shifted to
higher energies, with the ionic potential experiencing smaller shift. As a result, the potential energy curves of the X
1 +
R and A 1R+ states cross the ionic potential curve at larger internuclear distances and lead to the shifts of maxima
on the dipole moment curve toward larger R. The stability
of the ionic character can also explain the larger dipole
moments for the confined LiH molecule.
3.2. Dipole polarizabilities
Dependence of the static dipole polarizabilities of LiH
on the internuclear distance in the three R+ states, as computed by the finite field method at the CASSCF/SOCI
level, is shown in Fig. 4.
Recently, these values have been calculated by Mérawa
et al. [37] using the time-dependent gauge invariant (TDGI)
approach and the CCSD(T) method. Their values,
although slightly smaller, are in good agreement with the
b
Fig. 3. Electric dipole moments for LiH in the a 3R+ state (a) and the A 1R+ state (b). The lines correspond to the following values of confinement strength:
x = 0.0 a.u. —–, x = 0.05 a.u. – – – , x = 0.10 a.u. - - - , x = 0.15 a.u. , and x = 0.20 a.u. – –.
136
J.M.H. Lo, M. Klobukowski / Chemical Physics 328 (2006) 132–138
Fig. 4. Static dipole polarizability components of the X 1R+, a 3R+, and A
1 +
R states of the LiH molecule in free space. The lines correspond to the
following polarizabilities: axx (X 1R+) ——, azz (X 1R+) – – –, axx (a 3R+) - -, azz (a 3R+) , axx (A 1R+) – –, and azz (A 1R+) - -.
results obtained in the present work. The differences may
be attributed to the differences between the TDGI and
the energy-based finite-field method which is very sensitive
to the accuracy of the computed energies of LiH in the static electric fields. The smaller polarization and diffuse sets
in Sadlej’s 5s3p2d basis compared to the 7s5p3d1f basis
set [38] and the auxiliary functions at the mid-bond position may also contribute to larger dipole polarizability,
although the actual dependence of dipole polarizability
on basis set is not certain, especially in correlated calculations [39].
Despite systematically larger magnitude of the present
values, the trends in the dipole polarizability components
for the X 1R+ and A 1R+ states are the same as the ones
predicted by Mérawa et al. [37]. The axx components for
both states are small compared to the azz counterparts
because of the dominant r-type bonding character in these
states. The maximum that appears on the azz curve in the X
1 +
R state at 4.0 Å was attributed by Kołos and Wolniewicz
[29] to the dipole–induced dipole interaction. A similar feature is also found in the A 1R+ state where the maximum of
azz occurs at about 5.8 Å. Interestingly, due to the largest
negative contribution of the X 1R+ state to the polarizability for the A 1R+ state [37], azz in the A 1R+ state exhibits a
minimum at the same value of R where the maximum azz
for the X 1R+ state is present.
At very large internuclear distances the polarizability
components axx and azz for these three R+ states converge
to two distinct values corresponding to the sum of the
atomic polarizabilities of H(2S) and Li(2S), for the X 1R+
and a 3R+ states, and H(2S) and Li(2P), for the A state.
The finite field calculations yielded the limiting values of
177.5 a.u. for the X 1R+ and a 3R+ states and 140.5 a.u.
for the A 1R+ state. These values are larger, in particular
the latter one, than the TDGI and CCSD(T) values
deduced by Mérawa et al. [37].
Due to cylindrical symmetry of the confining potential
the xx and zz components of the dipole polarizabilities will
be affected in different degree and the anisotropy
Da = azz axx would increase. These expectations are confirmed by the plots shown in Fig. 5 where the components
axx (the top panel) and azz (the bottom panel) for the three
states exhibit completely different responses to the confining potential. As expected, the axx components are suppressed because of the radial compression of the electron
density. On the other hand, the azz components dramatically increase in the presence of confining potential as the
z-axis is the only unconfined degree of freedom that allows
for the distortion of electron density when a weak electric
field is applied. The enhancement of azz is more pronounced for the X 1R+ state due to the larger contribution
from the A 1R+ state by the induced configuration mixing
of the 2s and 2pz orbitals of Li. Furthermore, the maximum
of azz is shifted toward larger R, in contrast to the case of
the A 1R+ state, where the maximum appears at approximately the same internuclear distance regardless of the
strength of the confining potential. An intriguing feature
concerning the evolution of the azz for the a 3R+ state is
noteworthy. An isosbestic point appears at 2.60 Å indicating a switch of the electron configuration. The reduction of
azz at R < 2.60 Å is caused by the strong electrostatic interaction of the ionic configuration. For the internuclear distances greater than 2.60 Å the inclusion of covalent 2s and
2pz character reduces the Coulomb attraction along the
molecular axis and thus increases the dipole polarizability
azz.
3.3. Electric field gradients
Electric field gradients (EFGs) at the nuclei of a molecule are a measure of the second derivatives of the external
electric potential arising from the surrounding nuclei and
electrons. They may be expected to be vulnerable to the
applied confining potential because of the induced distortion of the electron density of the entire molecule. The
EFGs at the Li and H nuclei for the internuclear distance
R = 1.60 Å were computed for several values of x are listed
in Table 2.
Due to the linear symmetry of the molecule and the
traceless property of the EFG tensor, Vxx + Vyy + Vzz = 0
and Vxx = Vyy. The details of the charge distribution
around the nucleus of interest can therefore be determined
solely by Vzz. As seen in Table 2 the magnitudes of Vzz for
both Li and H are small and indicate ionic character of the
ground electronic state of LiH as inferred from the large
electric dipole moment (Section 3.1). The isolated closedshell Li+(1s2) and H(1s2) ions do not contribute to the
EFG because of their spherical symmetry [40]. The nonzero EFGs at the Li and H nuclei reveal that the electron
distribution of the bonding electrons in the 2r molecular
J.M.H. Lo, M. Klobukowski / Chemical Physics 328 (2006) 132–138
137
a
c
e
b
d
f
Fig. 5. Responses of the static dipole polarizability a of the LiH molecule to the confining potential. The panels show axx of X 1R+ state (a), azz of X 1R+
state (b), axx of a 3R+ state (c), azz of a 3R+ state (d), axx of A 1R+ state (e), azz of A 1R+ state (f). The lines correspond to the following values of
confinement strength: x = 0.0 a.u. —–, x = 0.05 a.u. – – – , x = 0.10 a.u. - - - , x = 0.15 a.u. , and x = 0.20 a.u. Æ - Æ -.
Table 2
Electric field gradients (in au) of the ground electronic state of LiH at R = 1.60 Å
x
Li
H
Vxx
Vyy
Vzz
Vxx
Vyy
Vzz
0.00
0.05
0.10
0.15
0.20
0.021841
0.023369
0.027167
0.032142
0.037648
0.021841
0.023369
0.027167
0.032142
0.037648
0.043681
0.046739
0.054333
0.064284
0.075296
0.026709
0.025898
0.023594
0.020051
0.015537
0.026709
0.025898
0.023594
0.020051
0.015537
0.053417
0.051796
0.047188
0.040102
0.031073
orbital located is polarized in the region between the two
nuclei and that the ions are not perfectly spherical in the
molecular environment.
Although not very remarkable in magnitude, the confinement-induced variations of EFGs at the Li and H
nuclei are very informative as they reveal changes in the
electron distribution caused by the external harmonic
potential. Two opposite trends are observed for the EFGs
at Li and H. The Vzz at the H nucleus is reduced by 60%
when the parameter x is increased from 0 to 0.20 a.u.
Simultaneously, the Vzz of at the Li nucleus grows by
70%. The vanishing EFG at the H nucleus is consistent
with the conclusion from the wavefunction analysis that
the ionic Li+H character becomes more dominant with
increasing x in the ground state of the confined LiH mol-
ecule. The larger extent of charge transfer to the H atom
results in the closed s-shell 1s2 configuration which has zero
contribution to the EFG.
The situation at the Li nucleus is more complicated as
two effects play role in determining the Vzz. On one hand,
transfer of the electron density towards the H atom turns
the electronic configuration of the Li atom to be more
1s2-like and decreases the EFG. On the other hand, the
confining potential enhances the hybridization of the 2r
molecular orbital by increasing the contribution from the
Li(2pz) atomic orbital. That effect modifies the valence electron density of Li and provides a positive contribution to
the EFG. Because of the 1/r3 dependence of Vzz the second
effect is less important and the deformation of the valence
electron distribution dominates the change of EFG and
138
J.M.H. Lo, M. Klobukowski / Chemical Physics 328 (2006) 132–138
leads to the net increase of Vzz in the presence of the confining potential.
4. Conclusions
In the present work, the effects of a cylindrical harmonic
confining potential in one-electron properties in the ground
and first two excited states of LiH molecule were studied. It
was found that the dipole moment in the ground electronic
state is significantly increased by the confining potential
due to enhanced ionic character of the wavefunction. The
effects were less profound in the excited states, but the configuration interactions induced by the external potential
resulted in subtle changes of the dependence of the dipole
moment on the internuclear distance R.
The influence of confining potential on the components
of the static dipole polarizabilities of LiH is more drastic
than on the dipole moment. The spatial restriction imposed
by the cylindrical potential induces anisotropic response of
the axial and equatorial components of the polarizability
tensor of the confined LiH molecule. Compression of the
electron density in the equatorial direction leads to reduction of the axx and ayy components. The azz component is
enhanced by the polarization of the 2r molecular orbital.
The electric field gradients of LiH are also affected by
the application of the cylindrical repulsive potential. Opposite trends were observed for the Li and H nuclei. The
decreasing EFGs at the H nucleus are attributed to the
enhanced ionic closed-shell H character induced by the
confining potential. At the Li nucleus the EFG is increased
due to increased mixing of the Li(2pz) and 2r orbitals that
leads to polarization of the electron density.
Acknowledgement
This work was supported in part by the Natural Science
and Engineering Research Council of Canada (NSERC)
(PGS-B Scholarship to JMHL and NSERC Research
Grant to MK).
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