THE JOURNAL OF CHEMICAL PHYSICS 125, 114315 共2006兲
Theoretical spectroscopy of the calcium dimer in the A 1⌺+u, c 3⌸u,
and a 3⌺+u manifolds: An ab initio nonadiabatic treatment
Béatrice Bussery-Honvault and Jean-Michel Launay
Laboratoire PALMS, UMR 6627 du CNRS, Université de Rennes 1, Campus de Beaulieu,
35042 Rennes Cedex, France
Tatiana Korona and Robert Moszynskia兲
Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland
共Received 15 November 2005; accepted 19 June 2006; published online 20 September 2006兲
Nonadiabatic theory of molecular spectra of diatomic molecules is presented. It is shown that in the
fully nonadiabatic framework, the rovibrational wave functions describing the nuclear motions in
diatomic molecules can be obtained from a system of coupled differential equations. The
rovibrational wave functions corresponding to various electronic states are coupled through the
relativistic spin-orbit coupling interaction and through different radial and angular coupling terms,
while the transition intensities can be written in terms of the ground state rovibrational wave
function and bound rovibrational wave functions of all excited electronic states that are electric
dipole connected with the ground state. This theory was applied in the nearly exact nonadiabatic
calculations of energy levels, line positions, and intensities of the calcium dimer in the
A 1⌺+u 共1 1S + 1 1D兲, c 3⌸u共1 3 P + 1 1S兲, and a 3⌺+u 共1 3 P + 1 1S兲 manifolds of states. The excited state
potentials were computed using a combination of the linear response theory within the
coupled-cluster singles and doubles framework for the core-core and core-valence electronic
correlations and of the full configuration interaction for the valence-valence correlation, and
corrected for the one-electron relativistic terms resulting from the first-order many-electron Breit
theory. The electric transition dipole moment governing the A 1⌺+u ← X 1⌺+g transitions was obtained
as the first residue of the frequency-dependent polarization propagator computed with the
coupled-cluster method restricted to single and double excitations, while the spin-orbit and
nonadiabatic coupling matrix elements were computed with the multireference configuration
interaction wave functions restricted to single and double excitations. Our theoretical results explain
semiquantitatively all the features of the observed Ca2 spectrum in the A 1⌺+u 共1 1S + 1 1D兲,
c 3⌸u共1 3 P + 1 1S兲, and a 3⌺+u 共1 3 P + 1 1S兲 manifolds of states. © 2006 American Institute of Physics.
关DOI: 10.1063/1.2222348兴
I. INTRODUCTION
Over the past decades the calcium dimer attracted the
interest of theoreticians and experimentalists. Due to the
equal number of bonding and antibonding electrons, Ca2
does not form a typical chemical bond. However, the dissociation energy 共about ⬇1100 cm−1兲 is larger than in typical
van der Waals complexes bound by dispersion forces and
rather similar to hydrogen-bonded systems such as the water
dimer. Most of the ab initio calculations on the calcium
dimer reported in the literature thus far1–9 concerned with the
ground state potential energy curve1–4 and the van der Waals
constants governing the long-range behavior of the ground
state potential.5–9 To our knowledge only three theoretical
papers considered the excited states of the calcium
dimer.4,10,11
The calcium dimer was the subject of numerous highresolution spectroscopic studies in the gas phase12–21 and in
rare gas matrices.22–27 The dissociation energy of the ground
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2006/125共11兲/114315/15/$23.00
state of the Ca dimer was first estimated from the RydbergKlein-Rees 共RKR兲 inversion of the spectroscopic data for the
B 1⌺+u ← X 1⌺+g transitions.15 In two recent papers, Tiemann
and coworkers20,21 reported a much more elaborate study of
Ca2 by measuring the B 1⌺+u ← X 1⌺+g transitions covering
over 99.8% of the ground state well depth, and reported the
dissociation energy for the ground state of the calcium dimer
with an astonishing accuracy of 0.01 cm−1. Note parenthetically that the theoretical value reported in Ref. 4 is in a very
good agreement with the experiment. As a by-product of the
experimental work reported in Refs. 20 and 21 the spectroscopic constants and the dissociation energy for the excited
B 1⌺+u were obtained.
The spectrum corresponding to the B 1⌺+u ← X 1⌺+g transitions could easily be assigned using standard spectroscopic
techniques because the B state dissociating into 1S + 1 P atoms
is relatively well isolated, thus not perturbed by any other
electronic state. This is not the case for the A 1⌺+u state dissociating into 1S + 1D atoms. Here the potential energy curve
of the A state crosses the curve of the c 3⌸u state dissociating
into 1S + 3 P states, and the corresponding spectrum cannot
125, 114315-1
© 2006 American Institute of Physics
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114315-2
J. Chem. Phys. 125, 114315 共2006兲
Bussery-Honvault et al.
easily be assigned. The experimental investigations of the A
state was started by Andrews and co-workers.22–26 By using
various experimental techniques in the argon matrices Andrews and co-workers were able to assign the upper electronic state as the A state, but no reliable information about
the well depth and the spectroscopic constants could be obtained. The first gas phase observation of the A state was
reported by Hofmann and Harris.17 These authors measured
the spectrum corresponding to A 1⌺+u ← X 1⌺+g by filtered laser excitation, and showed that the rovibrational levels of the
A state are strongly perturbed by the c 3⌸u state. By applying
an approximate deperturbation procedure they could determine the dissociation energy and the spectroscopic constants
of the A state, and get some 共indirect兲 information on the c
state. It is worth noting here that the analysis of the spectrum
reported in Ref. 17 was not complete. In particular, the key
information on the spin-orbit coupling between the A and c
states could not be obtained from the deperturbation of the
spectra, while theoretical considerations of the spin-orbit
coupling matrix elements between the A and c states based
on a simple linear combination of atomic orbitals 共LCAO兲molecular orbitals 共MO兲 picture were not correct. Gondal et
al.18 observed the A state levels by collision-induced fluorescence, and obtained the spectroscopic constants for this state,
essentially in agreement with the results of Hofmann and
Harris.17
The coupling of the A and c states is due to the spin-orbit
interactions. However, another state spin-orbit coupled with
the c 3⌸u state is the lowest a 3⌺+u state dissociating into 1S
+ 3 P atoms. This state is very difficult to observe because it is
directly coupled only with the c state, which in turn only
perturbs the A state. Direct observation of the a state was
reported in Ref. 19. The authors of Ref. 19 claimed to populate the a state by laser excitation and collisions of free calcium atoms, and to observe collision-induced spectra corresponding to the a state.
On the theory side only two papers10,11 reported potential
energy curves for the A, c, and a states. Both calculations
showed the reversed 1 P- 1D state ordering already at the
atomic level, so the molecular results may not be very accurate. Indeed, the binding energies determined in Refs. 10 and
11 are 7% and 35% off the best experimental estimates.17
The spin-orbit coupling was considered only in Ref. 11.
However, due to the reversed order of the atomic 1 P and 1D
states, the matrix elements reported correspond to the coupling of the B 1⌺+u state with the c 3⌸u state, not important
from the spectroscopic point of view. The nonadiabatic couplings have not been considered in the literature thus far.
It was shown in Refs. 28 and 29 that it is possible to trap
and cool calcium atoms, and recently photoassociation spectra involving cold calcium atoms were reported.30 The observed photoassociation spectrum of the calcium dimer from
cold calcium atoms showed regular vibrational series from
the B 1⌺+u state, but the efficiency of the spontaneous emission to the bound vibrational levels of the ground X 1⌺+g state
was very small. Since then new experimental schemes are
being devised, in order to increase the efficiency of produc-
tion of cold calcium molecules. In particular, one possible
scheme concerns the photoassociative spectroscopy to the
A 1⌺+u state.31
The assignment of the photoassociation spectrum requires a detailed knowledge of the rovibrational levels close
to the dissociation limit. However, the experimental data on
the A state recorded thus far17 are far from complete. Therefore, the experimental spectrum corresponding to the A 1⌺+u
← X 1⌺+g transitions in Ca2 was recently revisited by Tiemann
and collaborators.31 In parallel, we started a theoretical study
of the spectroscopy of the calcium dimer in the A 1⌺+u , c 3⌸u,
and a 3⌺+u manifolds by ab initio methods. In the present
paper we report a fully relativistic and nonadiabatic treatment of the dynamics of Ca2 in the A state. The plan of this
paper is as follows. In Sec. II we briefly outline the nonadiabatic theory of nuclear motions in diatomic molecules. We
derive the coupled equations for the rovibrational wave functions and the expressions for the transition intensities in the
nonadiabatic framework. In all derivations we assume that
the relativistic effects can be described with the Breit-Pauli
Hamiltonian, so the spin-orbit coupling and the scalar relativistic terms are fully included in our theory. We also briefly
sketch the applications of this theory to the case of the calcium dimer in the A 1⌺+u , c 3⌸u, and a 3⌺+u manifolds. In Sec.
III we describe the computational methods used to compute
the potential energy curves, transition moments, and spinorbit and nonadiabatic coupling matrix elements. In Sec. IV
we present the numerical results, and show that we can explain 共semi兲quantitatively the observed spectrum. Finally, in
Sec. V we conclude our paper.
II. THEORY
A. Coupled equations for the nonadiabatic wave
function of a diatomic molecule
We consider an N-electron diatomic molecule. Let us
denote a vector connecting the atoms. The molecular Hamiltonian in the body-fixed coordinate system attached to the
center of mass of the nuclei with the z axis parallel to the
vector R can be written as32–34
H=
1 2 J2 + 共L + S兲2 − 2J · 共L + S兲
p +
2␮ R
2␮R2
1
+ He + HCG + HSO +
2M
冉兺 冊
N
pi
2
,
共1兲
i=1
where M = M A + M B and M A and M B denote the nuclear
masses, ␮ is the reduced mass of the diatomic, pR is the
radial momentum operator, J, S, and L are vectors of the
total, electron spin, and electronic orbital angular momenta,
respectively, pi denotes the momentum of the electron i, He
is the clamped-nuclei Born-Oppenheimer Hamiltonian, HCG
is the so-called Cowan-Griffin Hamiltonian35 collecting the
scalar one-electron Darwin and mass-velocity terms of the
Breit-Pauli Hamiltonian,36 and HSO denotes the Breit-Pauli
spin-orbit coupling term including all one- and two-electron
terms.36
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114315-3
J. Chem. Phys. 125, 114315 共2006兲
Theoretical spectroscopy of calcium dimer
Due to the isotropy of space the total wave function of a
p
, is an eigenfunction of the square of
diatomic molecule, ⌿JM
the total angular momentum J2 and its projection on the
space-fixed Z axis JZ:
J
2
p
⌿JM
= J共J +
p
1兲⌿JM
,
p
JZ⌿JM
=
p
M⌿JM
.
共2兲
It also has a well defined parity p = ± 1 with respect to the
space-fixed inversion E쐓,
E
쐓
p
⌿JM
=
p
共兵r其 , R兲 deThe total wave function of the diatomic ⌿JM
pends on the coordinates of the N electrons, denoted for simplicity by 兵r其 = 共r1 , . . . rN兲, and on the vector R = 共R , ␽ , ␸兲. It
p
can be represented as
follows directly from Eq. 共2兲 that ⌿JM
1
H⍀⍀ =
1 2
p
2␮ R
+
J共J + 1兲 + S2 − ⍀2 − Sz2 + L2x + L2y + L+S− + L−S+
2␮R2
1
+ He + HCG + HSO +
2M
−1/2
p
共␺J⍀
共兵r其,R兲
쐓
,
共7兲
i=1
⫻
关J共J + 1兲 − ⍀共⍀ ± 1兲兴1/2
共L⫿ + S⫿兲,
2␮R2
共8兲
쐓
共J兲
JzD共J兲
M⍀ = ⍀D M⍀ ,
共4兲
where D共J兲
⍀M 共␸ , ␽ , 0兲 is the Wigner rotation matrix, and ␽ and
␸ are the Euler angles defining the orientations of the bodyfixed axes with respect to the space-fixed axes. The fact that
p
is a linear combination of Wigner’s D
the wave function ⌿JM
matrices is a direct consequence of Eq. 共2兲. The inversion
operation changes R → −R and ri → −ri. In terms of the polar
angles of R the transformation R → −R is equivalent to
共R , ␽ , ␸兲 → 共R , ␲ − ␽ , ␲ + ␸兲. Using the well-known properp
of Eq. 共4兲 is
ties of the D matrices one easily finds that ⌿JM
p
parity adapted, if ␺J⍀ fulfills the following conditions:
p
p
共兵r其,R兲 = 共− 1兲⍀共− 1兲␴␺J,−⍀
共兵r其,R兲.
E쐓␺J⍀
pi
2
H⍀⍀±1 = − 关1 + ␦⍀0 + ␦⍀±1,0兴−1/2
쐓
쐓
J
␴
⫻D共J兲
M⍀共␸, ␽,0兲 + p共− 1兲 共− 1兲
p
⫻D共J兲
M,−⍀共␸, ␽,0兲␺J,−⍀共兵r其,R兲兲,
冉兺 冊
N
where L± = Lx ± iLy and a similar definition applies to S±. In
Eqs. 共7兲 and 共8兲 the components of the angular momentum
operators are defined in the body-fixed coordinate system.
When deriving Eqs. 共6兲–共8兲 we have used the fact that
J
兺 共1 + ␦⍀0兲
冑2 ⍀=−J
共6兲
where EJp is the total energy of the diatomic. The blocks of
the Hamiltonian H⍀⍀⬘ are given by the following equations:
共3兲
p
p⌿JM
.
p
共兵r其,R兲 =
⌿JM
p
p
p
p
H⍀⍀␺J⍀
+ H⍀⍀−1␺J⍀−1
+ H⍀⍀+1␺J⍀+1
= EJp␺J⍀
,
共5兲
쐓
J±D共J兲
M⍀
= 关J共J + 1兲 − ⍀共⍀ ⫿ 1兲兴
쐓
D共J兲
M⍀⫿1 .
共9兲
The coupled equations for the internal wave function, Eq.
共6兲, differ somewhat from the equations derived in Refs. 32
and 33 because in the present paper the total wave function
p
is parity adapted, see Eq. 共4兲. It is worth noting that the
⌿JM
total angular momentum J and the total electronic angular
momentum L + S have the same projections on the bodyfixed z axis:
p
p
= ⍀␺J⍀
.
共Lz + Sz兲␺J⍀
共10兲
To proceed further let us introduce the Born-Huang
p
expansion37 of the wave function ␺J⍀
:
p
␺J⍀
共兵r其,R兲
=兺兺
S
The additional index ␴ appearing in Eqs. 共4兲 and 共5兲 is related to the transformation of the body-fixed electronic coordinates with respect to the space-fixed inversion. It can easily
be shown that the action of E쐓 on the body-fixed vector r is
equivalent to the action of the reflexion in the body-fixed yz
plane, ␴v. For ⍀ = 0 the values of ␴ are just 0 and 1 and
define the so-called ⫾ parities of the ⌺ states. For the degenerate case, i.e., for ⍀ ⫽ 0, they depend on the choice of the
p
共兵r其 , R兲 but can be fixed in
phases in the wave function ␺J⍀
such a way that Eq. 共5兲 remains valid for any ⍀. This point
is mostly related to the labeling of the electronic states, and
is discussed in more detail in the next paragraph. The quantity p̃ = p共−1兲J共−1兲␴ appearing in Eq. 共4兲 is often called spectroscopic parity. States with p̃ = ± 1 will be referred to as the
e and f states.
p
共兵r其 , R兲 are
The internal “body-fixed” wave functions ␺J⍀
solutions of the following coupled equations:
1/2
n
S
S⌺⌳
pS⌺⌳
␾n⍀
共兵r其;R兲␹nJ⍀
共R兲,
兺
兺
⌺=−S ⌳
共11兲
pS⌺⌳
共R兲 denotes the rovibrational wave function and
where ␹nJ⍀
S⌺⌳
␾n⍀ 共兵r其 ; R兲 the electronic wave functions satisfying the
clamped-nuclei Schrödinger equation:
S⌺⌳
S⌺⌳
= VS兩⌳兩
He␾n⍀
n 共R兲␾n⍀ .
共12兲
Here VS兩⌳兩
n 共R兲 is the potential energy for the nuclear motions.
The summation index S runs over all possible values of the
total electronic spin 共i.e., over all spin multiplicities of the
electronic states兲, n runs over all possible nonrelativistic dissociation limits of the diatomic in the Born-Oppenheimer
approximation for a given spin multiplicity, ⌺ is the projection of the total electronic spin S on the body-fixed z axis,
and ⌳ runs over all possible projections of the electronic
angular momentum L on the body-fixed z axis. Note that
the total angular momentum J and the total electronic
angular momentum L + S have the same projections on the
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114315-4
J. Chem. Phys. 125, 114315 共2006兲
Bussery-Honvault et al.
body-fixed z axis, so the summations over ⌺ and ⌳ are such
that ⍀ = ⌺ + ⌳. Note that in the Born-Oppenheimer approximation neglecting the spin-orbit coupling the potential for
the nuclear motions does not depend on the sign of the quantum number ⌳ and on the quantum numbers ⌺ and ⍀. Note
pS⌺⌳
共R兲 does not
also that the rovibrational wave function ␹nJ⍀
pS⌺⌳
depend on the sign of ⌺, ⌳, and ⍀, so we have ␹nJ⍀
pS−⌺−⌳
= ␹nJ−⍀ 共R兲.
For ⍀ = 0 the electronic wave functions are eigenfunctions of the reflection in the yz plane, ␴v, with the eigenvalues 共−1兲␴, ␴ = 0 or 1. For the degenerate case ⍀ ⫽ 0 the phase
of the electronic wave functions is arbitrary, but it can be
fixed in such a way that
S,−⌺,−⌳
S⌺⌳
E쐓␾n⍀
共兵r其;R兲 = 共− 1兲⍀共− 1兲␴␾n,−⍀
共兵r其;R兲.
共13兲
This is related to the fact that E쐓 acts on the electronic bodyfixed coordinates as the reflection in the yz plane, ␴v. Since
␴v commutes with the electronic Hamiltonian He, the elec-
冉
tronic states can always be chosen to be eigenfunctions of He
and ␴v 共but for ⍀ ⫽ 0 they are not eigenfunctions of Jz兲, or to
be eigenfunctions of He and Jz, and fulfill condition 共13兲.
S⌺⌳
is an eigenfunction of the opThe wave function ␾n⍀
2
erators S , Sz, and Lz:
S⌺⌳
S⌺⌳
S2␾n⍀
= S共S + 1兲␾n⍀
,
S⌺⌳
S⌺⌳
Sz␾n⍀
= ⌺␾n⍀
,
共14兲
S⌺⌳
S⌺⌳
Lz␾n⍀
= ⌳␾n⍀
.
S⌺⌳
The operators S± act on ␾n⍀
in the usual way:
S⌺±1⌳
S⌺⌳
= 关S共S + 1兲 − ⌺共⌺ ± 1兲兴1/2␾n⍀±1
.
S±␾n⍀
共15兲
Inserting expansion 共11兲 into Eq. 共6兲, multiplying from the
⌺⬘⌳⬘
共兵r其 ; R兲兲쐓, and performing the integration
left with 共␾nS⬘⬘⍀
⬘
over the electronic coordinates 兵r其 we arrive at the following
parity-blocked coupled equations for the rovibrational wave
S⌺⌳
for a given parity block:
functions ␹nJ⍀
冊
J共J + 1兲 + S共S + 1兲 − ⍀2 − ⌺2
1 2
pS⌺⌳
pR + VS兩⌳兩
+
− EJp ␹nJ⍀
n
2␮
2␮R2
⌺⌳⍀
− N共−兲
⍀ N⌺−1⌳⍀−1
关J共J + 1兲 − ⍀共⍀ − 1兲兴1/2关S共S + 1兲 − ⌺共⌺ − 1兲兴1/2 pS⌺−1⌳
␹nJ⍀−1
2␮R2
⌺⌳⍀
− N共+兲
⍀ N⌺+1⌳⍀+1
关J共J + 1兲 − ⍀共⍀ + 1兲兴1/2关S共S + 1兲 − ⌺共⌺ + 1兲兴1/2 pS⌺+1⌳
␹nJ⍀+1 +
2␮R2
1
1 S⌺⌳ ⳵ S⌺⌳ ⳵
S⌺⌳
S⌺⌳
兩 兩 ␾ n⬘⍀
兩HCG兩␾n⬘⍀ 典 +
− 具␾n⍀
+ 具␾n⍀
具␾S⌺⌳兩
␮
⳵R
⳵R
2M n⍀
典
冉兺 冊 兩
2
N
pi
i=1
␾nS⌺⌳
⬘⍀
兺
n⬘
典+
冉
1 S⌺⌳ 2 S⌺⌳
具␾ 兩p 兩␾ 典
2␮ n⍀ R n⬘⍀
S⌺⌳
S⌺⌳ 2
具␾n⍀
兩Lx + L2y 兩␾n⬘⍀ 典
2␮R2
冊
␹npS⌺⌳
⬘J⍀
S⌺⌳−1
⌺⌳⍀
+ N⌺⌳−1⍀−1
兺
S⌺⌳
关S共S + 1兲 − ⌺共⌺ + 1兲兴1/2具␾n⍀
兩L+兩␾n⬘⍀−1 典
2␮R2
n⬘
pS⌺+1⌳−1
␹n⬘J⍀
S⌺⌳+1
⌺⌳⍀
+ N⌺⌳+1⍀+1
兺
S⌺⌳
关S共S + 1兲 − ⌺共⌺ − 1兲兴1/2具␾n⍀
兩L−兩␾n⬘⍀+1 典
2␮R2
n⬘
pS⌺−1⌳+1
␹n⬘J⍀
S⌺⌳−1
−
⌺⌳⍀
N共−兲
⍀ N⌺⌳−1⍀−1
兺
S⌺⌳
兩L+兩␾n⬘⍀−1 典
关J共J + 1兲 − ⍀共⍀ − 1兲兴1/2具␾n⍀
2␮R2
n⬘
pS⌺⌳−1
␹n⬘J⍀−1
S⌺⌳+1
−
⌺⌳⍀
N共+兲
⍀ N⌺⌳+1⍀+1
兺
S⌺⌳
关J共J + 1兲 − ⍀共⍀ + 1兲兴1/2具␾n⍀
兩L−兩␾n⬘⍀+1 典
2␮R
n⬘
2
⌺⌳⍀
S⌺⌳
+ N⌺+1⌳−1⍀
兩HSO兩␾n⬘⍀
兺 具␾n⍀
典␹n⬘J⍀
⌺⌳⍀
S⌺⌳
+ N⌺−1⌳+1⍀
兩HSO兩␾n⬘⍀
兺 具␾n⍀
典␹n⬘J⍀
S+1⌺+1⌳−1
n⬘
S−1⌺−1⌳+1
n⬘
⌺⌳⍀
S⌺⌳
+ N⌺−1⌳+1⍀
兩HSO兩␾n⬘⍀
兺 具␾n⍀
S⌺−1⌳+1
n⬘
+
S⌺⌳ pS⌺⌳
S⌺⌳
兩HSO兩␾n⬘⍀ 典␹n⬘J⍀ = 0,
兺 具␾n⍀
pS+1⌺+1⌳−1
pS⌺−1⌳+1
⌺⌳⍀
S⌺⌳
+ N⌺−1⌳+1⍀
兩HSO兩␾n⬘⍀
兺 具␾n⍀
典␹n⬘J⍀
⌺⌳⍀
S⌺⌳
+ N⌺+1⌳−1⍀
兩HSO兩␾n⬘⍀
兺 具␾n⍀
典␹n⬘J⍀
S+1⌺−1⌳+1
n⬘
pS−1⌺−1⌳+1
典␹n⬘J⍀
pS⌺⌳+1
␹n⬘J⍀+1
S−1⌺+1⌳−1
n⬘
⌺⌳⍀
S⌺⌳
+ N⌺+1⌳−1⍀
兩HSO兩␾n⬘⍀
兺 具␾n⍀
S⌺+1⌳−1
n⬘
pS+1⌺−1⌳+1
pS−1⌺+1⌳−1
pS⌺+1⌳−1
典␹n⬘J⍀
共16兲
n⬘
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114315-5
J. Chem. Phys. 125, 114315 共2006兲
Theoretical spectroscopy of calcium dimer
coupling schemes, e.g., Hund’s case 共b兲 or 共c兲. It should be
noted that whatever the coupling scheme taken the system of
coupled equations is exact. However, depending on the
strength of the nondiagonal terms numerical solution of the
coupled equations may be more or less complicated.
where for simplicity we used the following short-hand notation:
−1/2
,
N共±兲
⍀ = 关1 + ␦⍀0 + ␦⍀±1,0兴
⌺⌳⍀
N ⌺⬘⌳⬘⍀⬘
= 关1 + ␦⌺0␦⌳0␦⍀0 + ␦⌺⬘0␦⌳⬘0␦⍀⬘0兴
共17兲
−1/2
.
B. Transition intensities and selection rules
The line strength of a transition from a state 共J⬙ p⬙兲 to a
state 共J⬘ p⬘兲 is given by
When deriving the system of coupled equations we have
used the definition of pR and the fact that
S⌺⌳ pS⌺⌳
pS⌺⌳ 2 S⌺⌳
S⌺⌳ 2 pS⌺⌳
␹nJ⍀ 兲 = ␹nJ⍀
pR␾n⍀ + ␾n⍀
pR␹nJ⍀
pR2 共␾n⍀
−2
pS⌺⌳
S⌺⌳
⳵␹nJ⍀
⳵␾n⍀
⳵R
⳵R
S共J⬘ p⬘ ← J⬙ p⬙兲 = 共2J⬙ + 1兲共2J⬘ + 1兲
⫻
冏兺
兺
1
⍀⬙=−J⬙ ⍀⬘=⍀⬙,⍀⬙±1
⫻ 兺 共2S + 1兲 兺
S
n⬙,n⬘
兺 共− 1兲⍀⬘
␴=−1
冉
J⬙
1
M ⬙=−J⬙ M ⬘=−J⬘
共19兲
where ␮m denotes the mth spherical component of the electric dipole moment operator in the space-fixed coordinate
system. Using the well-known40 transformation of ␮ from
the body-fixed to the space-fixed systems of axes,
1
␮m =
兺 Dm共1兲␴ 共␸, ␽,0兲␮␴ ,
␴=−1
쐓
共20兲
representation 共4兲 of the wave function, some well-known
integrals involving Wigner’s D functions, and the properties
of the 3j symbols, we get
1
S共J⬘ p⬘ ← J⬙ p⬙兲 = 共2J⬙ + 1兲共2J⬘ + 1兲共p⬘ p⬙ − 1兲2
4
冏兺
J⬙
⫻
⫻
冉
J⬘
⍀⬙=−J⬙ ⍀⬘=−J⬘
J⬙
1
共− 1兲⍀⬘
兺 ␴兺
=−1
1
J⬘
− ⍀⬙ ␴ ⍀⬘
冊
p
p
具␺J⬙⬙⍀⬙兩␮␴兩␺J⬘⬘⍀⬘典
冏
2
,
共21兲
where the expression between round brackets is a 3j symbol.
Note that Eq. 共21兲 does not vanish only for p⬘ = −p⬙. Inserting into Eq. 共21兲 the Born-Huang expansion 共11兲 and using
the fact that the electric dipole moment operator is diagonal
in S and ⌺, we get
1
J⬘
− ⍀⬙ ␴ ⍀⬘
p S⌺⍀ −⌺
具␹n⬙⬙J⬙⍀⬙⬙ 兩␮␴共n⬘S⍀⬘
J⬘
兺 兺 m=−1
兺 兩具⌿Jp⬙⬙M⬙兩␮m兩⌿Jp⬘⬘M⬘典兩2 ,
共18兲
.
The system of coupled differential equations for the
pS⌺⌳
rovibrational wave functions ␹nJ⍀
, Eq. 共16兲, represents the
main result of this section. It shows how the rovibrational
wave functions are coupled through the relativistic spin-orbit
coupling term and through different radial and angular coupling terms. It is worth noting that the terms involving the
N
pi兲2, pR2 , and 共L2x + L2y 兲 operators have nonzero diHCG, 共兺i=1
agonal elements. These diagonal terms represent the scalar
relativistic correction to the clamped-nuclei potential 共involving the HCG operator兲 and the so-called adiabatic correction 共involving all remaining operators quoted above兲.34,38
Other terms are purely nondiagonal and represent nonadiabatic couplings. The couplings through the electronic terms
N
pi兲2, through the radial terms pR2 and ⳵ / ⳵R, and
HCG and 共兺i=1
through the angular terms 共L2x + L2y 兲 couple states with equal
S, ⌺, ⌳, ⍀, and different n. They are often referred to as
radial couplings and are responsible for the so-called homogeneous perturbations in the spectra of diatomic molecules.
Other terms couple states with different ⌳ and/or ⌺ and/or ⍀
and/or S, so they are called angular couplings, and they lead
to heterogeneous perturbations.39
We wish to end this section by saying that Eq. 共16兲 was
derived by tacitly assuming that we work in the coupling
corresponding to Hund’s case 共a兲,40,41 i.e., that the quantum
numbers S, ⌺, and ⌳ are almost conserved, so the spin-orbit,
angular, and radial couplings represent small corrections. For
strong couplings it may be advantageous to choose other
J⬙
J⬙
S共J⬘ p⬘ ← J⬙ p⬙兲 =
←
冊
−p S⌺⍀ −⌺
n⬙S⍀⬙兲兩␹n⬘J⬙⬘⍀⬘ ⬘ 典
冏
2
,
共22兲
where the electronic transition dipole moment ␮␴共n⬘S⍀⬘ ← n⬙S⍀⬙兲 is defined as
⬙−⌺兩␮ 兩␾S⌺⍀⬘−⌺典.
␮␴共n⬘S⍀⬘ ← n⬙S⍀⬙兲 = 具␾nS⌺⍀
␴ n⬘⍀⬘
⬙⍀⬙
共23兲
It follows directly from the properties of the 3j symbols that the selection rule ⌬J = 0 , ± 1 holds. Moreover, for weak coupling
an additional approximate selection rule should hold: ⌬⍀ = 0 , ± 1.
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114315-6
J. Chem. Phys. 125, 114315 共2006兲
Bussery-Honvault et al.
C. Application to the spectroscopy of the calcium
dimer
In this section we will briefly describe how the theory
derived above can be applied to the spectroscopy of the calcium dimer in the A 1⌺+u 共1 1S + 1 1D兲, a 3⌺+u 共1 1S + 1 3 P兲, and
c 3⌸u共1 1S + 1 3 P兲 manifolds. We will limit the coupled equations to these states. The a 3⌺+u and c 3⌸u states have the
same dissociation limit 1 1S + 1 3 P in the nonrelativistic approximation, so they will be coupled by the spin-orbit and
angular couplings. The potential energy curve for the state
A 1⌺+u crosses the curve of the c 3⌸u state, so we expect
strong spin-orbit coupling between these states. Other states
dissociating into 1 1S + 1 1D lie far above the triplet manifold, so their coupling with the a 3⌺+u and c 3⌸u states will be
less pronounced. Finally, the spin-orbit coupling of the A
state with the triplet states corresponding to 1 1S + 1 3D will
共2S+1兲兩⌳兩u
Hdiag
⬅
1 2
J共J + 1兲 + S共S + 1兲 − ⍀2 − ⌺2
pR + VS兩⌳兩
共R兲
+
n
2␮
2␮R2
+
S⌺⌳
S⌺⌳
具␾n⍀
兩HCG兩␾n⍀
典
1
+
具␾S⌺⌳兩
2M n⍀
冉兺 冊
N
pi
i=1
2
S⌺⌳
兩␾n⍀
典+
The angular 共Coriolis-type兲 couplings between J, S, and L
will be denoted by CJL, CJS, and CLS:
共0兲
共R兲
CJL
=2
共2兲
共R兲 =
CJL
共0兲
共R兲 =
CJS
−1/2 关J共J
+ 1兲兴1/2L共R兲
,
2␮R2
关J共J + 1兲 − 2兴 L共R兲
,
2␮R2
共25兲
S⌺⌳ 2
S⌺⌳
兩Lx + L2y 兩␾n⍀
典
1 S⌺⌳ 2 S⌺⌳ 具␾n⍀
具␾n⍀ 兩pR兩␾n⍀ 典 +
.
2
2␮
2␮R
关J共J + 1兲兴
2␮R2
共26兲
He =
共27兲
2 关J共J + 1兲 − 2兴
2␮R2
CLS共R兲 =
21/2L共R兲
,
2␮R2
1/2
1/2
,
共28兲
S⌺⌳+1
S⌺⌳
具␾n⍀+1
兩L+兩␾n⍀
典
=
S⌺⌳+1
S⌺⌳
具␾n⍀
兩L−兩␾n⍀+1
典.
共30兲
Finally, the spin-orbit coupling matrix elements A, ␨, and ␰
are given by
A共R兲 = 具c 3⌸u共⌺ = ± 1,⌳ = ± 1兲兩HSO兩c 3⌸u共⌺ = ± 1,⌳
= ± 1兲典,
␰共R兲 = 具c 3⌸u共⌺ = ⫿ 1,⌳ = ± 1兲兩HSO兩A 1⌺+u 典,
共31兲
共32兲
冢
1 +
⌺u
Hdiag
0
0
␰
0
0
⌸u
Hdiag
+A
共2兲
− CJS
0
共2兲
CJL
0
共2兲
− CJS
⌸u
Hdiag
共0兲
− CJS
− ␨ + CLS
3
3
3
␰
0
共0兲
− CJS
⌸u
Hdiag
−A
0
共2兲
CJL
− ␨ + CLS
共0兲
− CJL
共0兲
− CJL
3 +
⌺u
Hdiag
冣
,
共34兲
while for f spectroscopic parity block and even values of J
the corresponding matrix reads
共29兲
where L共R兲 denotes the following matrix elements of the L±
operators:
共33兲
With this notation the 5 ⫻ 5 Hamiltonian matrix for the e
spectroscopic parity block and odd values of J reads
1/2
,
共24兲
␨共R兲 = 具c 3⌸u共⌺,⌳ = − 1兲兩HSO兩a 3⌺+u 共⌺ − 1兲典.
1/2
共2兲
共R兲 =
CJS
L共R兲 =
also be neglected. This means that the Hamiltonian matrix
corresponding to Eq. 共6兲 will be of dimension 10. Obviously,
since the coupled equations are decoupled in parity p, we
will have to consider two blocks of 5 ⫻ 5 corresponding to
e / f spectroscopic parities. The first row of the He/f matrix
will correspond to the A 1⌺+u state. The next three rows correspond to the c 3⌸u state with 兩⍀兩 = 2, 1, and 0, respectively.
Finally, the last row of this matrix corresponds to the a 3⌺+u
state with 兩⍀兩 = 1. The first three rows of the H f/e matrix will
correspond to the c 3⌸u state with 兩⍀兩 = 2, 1, and 0, respectively, while the next two rows correspond to the a 3⌺+u state
with 兩⍀兩 = 0 and 1, respectively. Let us introduce several
short-hand notations. The diagonal part of the Hamiltonian
matrix including the potential, the centrifugal term, the relativistic correction, and the diagonal adiabatic correction will
共2S+1兲兩⌳兩u
,
be denoted by Hdiag
Hf =
冢
3
⌸u
Hdiag
−A
共0兲
− CJS
共0兲
− CJS
⌸u
Hdiag
0
␨ − CLS
共0兲
− CJL
0
␨ − CLS
共0兲
− CJL
3
共2兲
− CJS
共0兲
− CJL
␨ − CLS
−
共2兲
CJS
3
0
共2兲
CJL
−
共0兲
CJL
␨ − CLS
⌸u
Hdiag
+A
3 +
⌺u
0
Hdiag
共0兲
− CJL
共0兲
− CJS
−
共0兲
CJS
3 +
⌺u
Hdiag
冣
.
共35兲
Note that in the experiment only the P and R branches corresponding to transitions from the ground state rovibrational
level J⬙ to the e rovibrational levels with J⬘ = J⬙ ± 1 of the
excited electronic state were observed, so only the first He
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114315-7
Theoretical spectroscopy of calcium dimer
matrix needs to be considered. The transitions corresponding
to the Q branch, i.e., connecting the level J⬙ of the ground
electronic state with the levels J⬘ = J⬙ of the excited electronic state, were not observed, so the matrix H f does not
need to be considered.
The coupled equations 共16兲 with the Hamiltonian matrix
He will be solved in four approximations. In the first step we
will separate the nuclear and electronic motions in the three
states and neglect the diagonal 共adiabatic兲 correction for the
nuclear motion, i.e., we will treat the dynamics within the
Born-Oppenheimer approximation. In the second step we
will consider the nuclear motion problem within the adiabatic approximation. Again, the electronic and nuclear motions will be separated, but the diagonal term will be added
to the potential. This diagonal correction will describe how
the fast electrons follow adiabatically the slow nuclei, in the
spirit of the mean-field theory. Note that within the BornOppenheimer and adiabatic approximations the rovibrational
levels are obtained for a given potential energy curve, so
they are in a one-to-one correspondence with the electronic
state considered. In the third step we will include the spinorbit coupling, i.e., we will consider the He matrix with
terms diagonal in ⍀. This approximation neglects the angular
couplings of J and L, and J and S. In this approximation the
electronic motion is described by the potential energy curves
obtained by diagonalizing the matrix He/f with nondiagonal
terms in ⍀ neglected for each value of the distance R and
solving the nuclear motion problem separately for the five
spin-orbit coupled curves. Obviously, the rovibrational energy levels obtained in this way do not correspond to the
clamped-nuclei electronic states, since the latter are mixed
through the spin-orbit coupling terms. Finally, the highest
level of sophistication is to solve the coupled equations 共16兲
with the full Hamiltonian matrix He/f . Note that in all cases
the relativistic correction corresponding to the scalar CowanGriffin Hamiltonian HGC is included in the potentials.
For a given approximation scheme Eq. 共16兲 can be
solved numerically by direct integration of the coupled differential equations, by diagonalization of the Hamiltonian
matrix in a finite basis of analytical radial functions, or by
using the discrete variable representation 共DVR兲. In the
present work we used the standard DVR approach. Computational details concerning ab initio electronic structure calculations are presented in the following section.
III. COMPUTATIONAL DETAILS
A. Potential energy curves, adiabatic and relativistic
corrections, and the matrix elements of the
spin-orbit and rotational couplings
The experimental data on the a, c, and A states suggest
that these states are rather strongly bound. Therefore, the
corresponding potential energy curves have been obtained by
supermolecular methods. In this work we follow the approach introduced in our previous paper on the calcium
dimer.4 Contributions to the potential corresponding to the
core-core and core-valence electronic correlations with a frozen core of five orbitals 共ten electrons兲 on each calcium atom
were computed using the linear response theory within the
J. Chem. Phys. 125, 114315 共2006兲
coupled-cluster
singles
and
doubles
共LRCCSD兲
framework,42 while the valence-valence contribution was obtained from the full configuration interaction 共FCI兲 calculations, correlating the two valence electrons on each calcium.
Thus, the three potential energy curves V共2S+1兲兩⌳兩u were constructed according to the formula
共2S+1兲兩⌳兩u
共2S+1兲兩⌳兩u
共2S+1兲兩⌳兩u
+ Vc-v
+ Vv-v
.
V共2S+1兲兩⌳兩u = Vc-c
共36兲
共2S + 1兲
关Vc-c 兩⌳兩u兴
and core-valence
The sum of the core-core
共2S + 1兲
关Vc-v 兩⌳兩u兴 contributions to the potential were obtained from
the linear response CCSD calculations:
共2S+1兲兩⌳兩u
共2S+1兲兩⌳兩u
LRCCSD
+ Vc-v
= 关EAB
共20e兲 − EALRCCSD共10e兲
Vc-c
LRCCSD
共4e兲
− EBLRCCSD共10e兲兴 − 关EAB
− EALRCCSD共2e兲 − EBLRCCSD共2e兲兴,
共37兲
LRCCSD
共Ne兲 denote the energies of the dimer comwhere EAB
puted with the LRCCSD method and N electrons correlated,
and similar definition applies to atomic energies appearing in
Eq. 共37兲, except that for atoms only half of the electrons are
in the valence shell. The valence-valence correlation contribution was computed from the expression
共2S+1兲兩⌳兩u
FCI
= 关EAB
共4e兲 − EAFCI共2e兲 − EBFCI共2e兲兴
Vv-v
共38兲
where the energies appearing on the right-hand side denote
the FCI energies of the dimer AB and of the atoms A and B,
with four and two electrons correlated, respectively. The
LRCCSD calculations were performed with the DALTON
program,43 while the FCI calculations were done with the
MOLCAS code.44
It follows directly from Eq. 共23兲 that the transitions from
the ground X 1⌺+g state to the A, a, and c manifolds are governed by the electric transition dipole moment from the X to
the A state, and the corresponding transition dipole moment
is given by
␮共A ← X兲 = 具X 1⌺+g 兩␮z兩A 1⌺+u 典,
共39兲
where ␮z is the z component of the electric dipole moment
operator for the Ca2 dimer. In the present calculations the
electric transition dipole moment ␮共A ← X兲 was computed as
the first residue of the linear response function using the
LRCCSD approach42 of the DALTON program,43 as used in
the computations of the excited state potential.
The potential energy curves for the a, c, and A states and
the transition dipole moment were computed for 15 distances
R ranging from 4 to 15 bohrs with the 关9s7p5d2f2g兴 basis
set. Full specification of this basis can be found in Ref. 4.
The full basis of the dimer was used in all atomic calculations in order to correct the interaction energies from the
basis set superposition error 共BSSE兲 according to the Boys
and Bernardi formulation.45
In the calculations of the relativistic corrections, spinorbit coupling elements, and angular couplings we employed
the multireference configuration interaction method restricted to single and double excitations 共MRCISD兲. The orbitals were optimized using the complete active space selfconsistent field 共CASSCF兲 method. In the CASSCF
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114315-8
J. Chem. Phys. 125, 114315 共2006兲
Bussery-Honvault et al.
calculations, as well as in the subsequent MRCISD calculations, 18 lowest orbitals were frozen, and the active space
consisted of 18 active orbitals. Only four valence electrons
were correlated. The frozen orbitals correspond to the
1s2s2p3s3p set on each atom, and the active space is composed of 4s4p3d orbitals on each atom. The relativistic corrections and the angular coupling matrix elements were directly obtained from the MRCI one-particle density and
transition density matrices, respectively. In the calculations
of the spin-orbit coupling matrix elements we have used the
full HSO Hamiltonian, i.e., we included both the one- and
two-electron spin-orbit integrals. The calculations of the
spin-orbit coupling matrix followed essentially Ref. 46. The
coupling matrix elements were computed for the same set of
distances as the potential energy curves. The MRCISD calculations of the relativistic corrections and of the spin-orbit
and angular coupling matrix elements were performed with
the MOLPRO program.47 Due to some limitations of the MOL46
PRO integral program we have used entirely uncontracted
basis 共15s13p5d2f兲 and we omitted the orbitals of g symmetry from the basis.
The diagonal corrections appearing in Eq. 共24兲 cannot be
computed with the standard ab initio programs available on
the market. Some parts of the corrections could be computed,
but in view of the results reported in Ref. 38 this would
break the correct long-range behavior. Therefore, in the
present work we decided to limit the diagonal terms to 共LA
+ LB兲2 − ⌳2, and approximate it by its separated-atoms limit:
S⌺⌳
S⌺⌳
具␾n⍀
兩共LA + LB兲2 − ⌳2兩␾n⍀
典 ⬇ L共L + 1兲 − ⌳2 ,
共40兲
where L is the quantum number corresponding to the
separated-atoms limit. Since in our case the united-atoms and
separated-atoms limits correspond to the same value of L,
L = 1, the approximation proposed above should work reasonably well. Note that other terms present in the diagonal
correction behave like R−6M −1 or R−8M −1,38 so their effect on
the potential should be minor.
The computed potential energy for the A 1⌺+u state was
fitted to the expression
V
共R兲 = 共A1 + B1R兲e
−␣1R
+ 共A2 + B2R兲e
−␣2R
X
X
X共R兲 = X⬁ + 共AX1 + BX1 R兲e−␣1 R + 共AX2 + BX2 R兲e−␣2 R
nf
+
兺
n=n
0
Xn
f n共␤X,R兲,
Rn
共42兲
where the symbol X stands for ␮, A, ␨, ␰, and L. For ␮共R兲,
␰共R兲, A共R兲, and ␨共R兲, the leading power n0 is equal to 6, and
for L共R兲, n0 = 3. For ␮共R兲, ␰共R兲, A共R兲, and ␨共R兲, n f = 8, while
for L共R兲, n f = 5. The atomic values X⬁ were fixed at their
experimental or theoretical values:
A
兩1 3 P典 = 53 cm−1 ,
A⬁ = ␨⬁ = 具1 3 P兩HSO
␰⬁ = 0,
L⬁ = 冑2,
共43兲
C2n
C5
f 5共␤,R兲 − 兺 2n f 2n共␤,R兲,
R5
n=3 R
IV. NUMERICAL RESULTS AND DISCUSSION
A. Potential energy curves and coupling matrix
elements
5
+
The transition moment ␮共R兲, the spin-orbit coupling matrix elements, A共R兲, ␨共R兲, and ␰共R兲, and the angular coupling
L共R兲 were fitted to the following generic expression:
while the remaining parameters were adjusted to the ab initio
points.
B. Analytical fits
1⌺+u
FIG. 1. Clamped-nuclei potential energy curves for the A 1⌺+u , c 3⌸u, and
a 3⌺+u states of Ca2.
共41兲
2
2
2
5
where 兵Ai其i=1
, 兵Bi其i=1
, 兵␣i其i=1
, ␤, and 兵C2n其n=3
were adjusted
to the computed points, and f n is the damping function in the
Tang-Toennies form.49 The leading long-range coefficient C5
describing the first-order resonant interaction of Ca共1 1S兲 and
Ca共1 1D兲 was not fitted but computed in the same basis set
and at the same level of the theory as those for the total
excited state potential. Its numerical value is taken from Ref.
48. The potentials for the c 3⌸u and a 3⌺+u states were fitted
to the same expression, but since for singlet-triplet interactions the first-order resonance energy vanishes, the C5 constant was set equal to zero.31
The computed Born-Oppenheimer potential energy
curves for the A 1⌺+u , c 3⌸u, and a 3⌺+u states are reported in
Fig. 1, while their main characteristics can be found in Table
I. An inspection of Fig. 1 and Table I shows that the A state
is by far the most strongly bound state. Its binding energy
amounts to 8991 cm−1. The a state is also strongly bound
with a binding energy of 6879 cm−1, while the c state is a
weakly bound, van der Waals-type state, characterized by a
binding energy of 924 cm−1. These large binding energies for
the A 1⌺+u and a 3⌺+u states can qualitatively be explained by
the strong bonding character of the 2␴g and 3␴g orbitals,
while the weak bonding of the c 3⌸u state is primarily due to
the antibonding character of the 1␲u orbital. Note that the
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114315-9
J. Chem. Phys. 125, 114315 共2006兲
Theoretical spectroscopy of calcium dimer
TABLE I. Spectroscopic characteristics of the excited state potentials for the
calcium dimer in the clamped-nuclei and spin-orbit coupled picture. The
binding De and dissociation D0 energies 共both in cm−1兲 of the clampednuclei and spin-orbit coupled potentials are given with respect to the nonrelativistic 3 P + 1S and spin-orbit coupled 3 P0 + 1S, 3 P1 + 1S, and 3 P2 + 1S
atomic dissociations, respectively. The equilibrium distances Re are given in
bohr.
This paper
Theor.a
Expt.b
a 3⌺+u
Re
De
D0
7.2
6879
6808
7.0
7604
2495
A 1⌺+u
Re
De
D0
6.8
8991
8917
6.7
5290
6.9
8694
Re
De
8.0
924
7.5
1575
艌1167
a1u
Re
De
D0
7.2
6880
6808
A0+u
Re
De
D0
6.8
2405
2331
c0+u
Re
De
D0
8.0
966
944
c1u
Re
De
D0
8.1
924
896
Re
De
8.1
875
a0−u
Re
De
7.2
c0−u
Re
De
7.2
c 3⌸ u
c2u
a
Reference 11.
References 17–19.
b
curves of the A and c states cross around R = 8.2 bohrs, so we
can expect important spin-orbit couplings between these two
states.
Also reported in Table I are the experimental binding
energies for the A, c, and a states. Our binding energy for the
A state is in a very good agreement with the experimental
determinations of Hofmann and Harris17 and Gondal et al.18
The theoretical value is only 3% larger than the experimental
estimates. Such a good agreement does not hold for the a and
c states. Our binding energy for the a state is almost three
times larger than the experimental value of Gondal et al.19
FIG. 2. Spin-orbit coupling matrix elements of the A 1⌺+u , c 3⌸u, and a 3⌺+u
states of Ca2.
However, we think that in the experiment reported in Ref. 19
the transitions observed correspond to the spin-orbit coupled
state A0+u and not to the a state. We will discuss this point in
more detail below. Finally, the binding energy of the c state
estimated in Ref. 17 from the deperturbation of spectra of the
A state is ⬇25% deeper than the present ab initio value. At
first glance, it seems that the present theoretical value is not
correct. Indeed, the core-valence correlation was obtained
from the LRCCSD calculations that depart from the ground
singlet state and reach the triplet state with a spin-flip operator. Although no literature results would suggest that such a
procedure is not accurate, one may think that the singlet state
is not a good starting point for calculations on a triplet state.
In order to check this point we performed MRCISD calculations of the valence-core correlation contribution with the
Davidson correction. With this approach we get a binding of
936 cm−1 at Re = 8.0, very close to the value reported in Table
I. Given the fact that two very different methods lead to
results that agree within 1.3%, we believe that the present
theoretical determination is correct. Moreover, as will be
shown in Sec. IV C, with the present c state potential we are
able to explain semiquantitatively all the spectroscopic features observed in Ref. 17, so either the experimental data are
not very sensitive to the details of the c state curve or the
present ab initio potential is essentially correct.
In a recent paper Czuchaj et al.11 reported calculations of
all curves for the excited states of Ca2 using the MRCISD
approach with four correlated electrons, and all remaining
described by a pseudopotential. As shown in Table I their
results strongly disagree with the present determinations.
However, the results reported in Ref. 11 showed the reversed
1
P- 1D state ordering at the atomic level, so the molecular
results may not be very accurate.
The spin-orbit coupling matrix elements as functions of
the interatomic distance are reported in Fig. 2. An inspection
of Fig. 2 shows that the spin-orbit matrix elements show
strong variations with R at small interatomic distances and
start to be almost constant around the minima. Note that the
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114315-10
Bussery-Honvault et al.
J. Chem. Phys. 125, 114315 共2006兲
FIG. 4. Transition dipole moment for the A 1⌺+u ← X 1⌺+g transitions 共upper
panel兲 and the angular coupling matrix elements 具c 3⌸u 兩 L+ 兩 a 3⌺+u 典 共lower
panel兲 for the calcium dimer.
FIG. 3. Spin-orbit coupled potential energy curves for the A 1⌺+u , c 3⌸u, and
a 3⌺+u states of Ca2 共upper panel兲 and their zoom 共lower panel兲 in the vicinity
of the avoided crossing between the A0+u and c0+u spin-orbit states.
values of A and ␨ are very close to the atomic data, so most
probably the latter could be used in the calculations. Interestingly, the value of the coupling between the A and c
states, ␰, is also almost constant over a wide range of distances and starting from R = 18 bohrs it shows a slow decay
to zero. It should be stressed that the derivation of an approximate expression for ␰ reported in Ref. 17 is not correct.
The authors of Ref. 17 arrive at a constant value of ␰ that
they relate to the atomic spin-orbit coupling. However, ␰
must vanish at large distances due to different parities of the
atomic 1D and 3 P states.
If we take into account the spin-orbit interactions, the
clamped-nuclei picture is no longer valid, and it is better to
use the spin-orbit 共SO兲 coupled states related to four possible
dissociation limits: 1S + 3 P0, 1S + 3 P1, 1S + 3 P2, and 1S + 1D.
The corresponding SO coupled states will be denoted by a0−u ;
a1u and A0+u ; c0−u , c1u, and c2u; and c0+u , respectively. The
potential energy curves are shown in Fig. 3, while their spectroscopic characteristics are reported in Table I. Note that the
curves for the a0−u and a1u states originate from the spinorbit interaction of the a state with the c state, so their well
depths and equilibrium distances are not far from those of
the pure a state, see Table I. Similarly, the c0−u , c1u, and c2u
states originate from the coupling of the c state with the a
state, and the corresponding curves closely follow the curve
for the pure c state. The states A0+u and c0+u originate from the
spin-orbit coupling of the A and c states. In the SO coupled
picture the corresponding curves show an avoided crossing
around R = 8.2 bohrs. It is interesting to note that the well
depth of the A0+u state, 2405 cm−1, is not far from the value
of 2495 cm−1 reported in Ref. 19 and assigned to the a state.
Thus, the spectrum observed in Ref. 19 corresponds probably to the A0+u state.
The variation of the transition dipole moment ␮共A
← X兲 and of the nonadiabatic matrix element coupling the a
and c states is reported in Fig. 4. An inspection of this figure
shows that similarly to the spin-orbit coupling matrix elements, both the transition dipole moment and the nonadiabatic coupling show strong variations at small interatomic
distances and become almost constant around the minima.
The nonadiabatic matrix element reaches very fast its atomic
value of 冑2, while the transition dipole moment decays to
zero. It is worth noting that around the minimum of the A
state the transition dipole moment is quite important and
amounts to 10 D, so the observed transitions from the X to
the A state should have appreciable intensities.
B. Rovibrational energy levels
In Table II we report the positions of the rovibrational
levels for J = 1. The levels were calculated at three different
levels of sophistication: in the adiabatic approximation, i.e.,
separately for each clamped-nuclei potential energy curve
including the diagonal 共adiabatic兲 correction, separately for
each SO coupled state, and finally in the full nonadiabatic
framework. Also reported in this table is the assignment of
the rovibrational levels to different spin-orbit coupled states.
It should be noted that when the nonadiabatic effects are
taken into account the rovibrational levels can no longer be
uniquely assigned to the electronic or spin-orbit coupled
Downloaded 29 Sep 2007 to 128.135.12.127. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
114315-11
J. Chem. Phys. 125, 114315 共2006兲
Theoretical spectroscopy of calcium dimer
TABLE II. Adiabatic, spin-orbit coupled, and nonadiabatic rovibrational energy levels in the manifold of the a,
c, and A electronic states of Ca2 共in cm−1兲 for J = 1.
n
Adiabatic
Nonadiabatica
xA0+
xc1u
xc0+
xa1u
Assignment
1
]
38
39
40
41
42
43
44
45
46
47
]
68
69
70
71
72
73
75
76
77
79
80
81
82
84
85
86
87
89
90
91
92
94
95
96
98
99
100
102
103
104
105
107
108
109
111
112
113
115
116
117
118
120
121
122
124
125
−6807.55
]
−2392.52
−2330.81
−2300.38
−2209.79
−2186.88
−2120.78
−2045.61
−2033.34
−1947.50
−1906.91
]
−949.41
−921.21
−896.38
−895.02
−879.99
−861.10
−841.99
−842.35
−802.84
−789.32
−791.40
−760.72
−746.46
−738.37
−742.18
−691.98
−694.69
−689.15
−648.93
−639.43
−641.66
−643.19
−604.90
−595.90
−588.82
−562.62
−551.87
−540.18
−527.35
−522.07
−509.59
−493.53
−483.27
−469.04
−448.89
−446.21
−430.24
−413.11
−406.30
−410.90
−393.18
−377.33
−365.77
−357.87
−345.51
−327.33
−6807.79
]
−2393.14
−2330.94
−2301.03
−2210.47
−2187.05
−2121.48
−2045.81
−2034.08
−1948.27
−1907.15
]
−944.39
−922.96
−895.98
−892.30
−879.58
−862.98
−841.57
−837.52
−804.88
−788.87
−788.17
−761.59
−748.67
−737.89
−737.13
−694.39
−691.27
−688.63
−649.75
−642.06
−641.10
−639.00
−601.33
−595.29
−591.71
−560.09
−551.22
−543.36
−528.58
−517.06
−508.87
−497.05
−480.37
−468.24
−452.81
−443.96
−429.35
−414.28
−410.69
−406.19
−392.19
−374.75
−370.72
−356.75
−343.31
−332.94
0.00
]
0.00
99.99
0.00
0.00
99.99
0.00
99.99
0.00
0.00
99.98
]
6.16
0.00
0.00
6.43
88.83
0.00
0.00
3.56
0.00
0.00
1.47
91.30
0.00
0.00
6.74
0.00
0.59
0.00
37.49
0.00
0.00
61.05
1.04
0.00
0.00
1.80
0.00
0.00
80.94
16.85
0.00
0.00
0.49
0.00
0.00
1.91
0.00
73.72
0.00
23.85
0.00
0.42
0.00
0.00
0.84
0.00
0.00
]
0.03
0.00
0.03
0.03
0.00
0.03
0.00
0.04
0.04
0.00
]
0.00
0.19
99.99
0.00
0.00
0.22
99.99
0.00
0.26
99.97
0.02
0.00
0.30
99.97
0.01
0.36
0.00
99.98
0.00
0.43
99.98
0.00
0.00
99.98
0.51
0.00
99.98
0.62
0.00
0.00
99.97
0.76
0.00
99.96
0.94
0.00
99.96
0.00
1.17
0.00
99.94
0.00
1.47
99.93
0.00
1.87
0.00
]
0.00
0.01
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.02
]
93.84
0.00
0.00
93.57
11.17
0.00
0.00
96.43
0.00
0.02
98.51
8.70
0.00
0.01
93.25
0.00
99.41
0.00
62.51
0.00
0.00
38.95
98.96
0.00
0.00
98.20
0.00
0.00
19.06
83.15
0.00
0.00
99.51
0.00
0.00
98.09
0.00
26.28
0.00
76.15
0.00
99.58
0.00
0.00
99.16
0.00
100.00
]
99.97
0.00
99.97
99.97
0.00
99.97
0.00
99.96
99.96
0.00
]
0.00
99.81
0.01
0.00
0.00
99.78
0.01
0.00
99.74
0.01
0.00
0.00
99.70
0.01
0.00
99.64
0.00
0.01
0.00
99.57
0.02
0.00
0.00
0.02
99.49
0.00
0.02
99.38
0.00
0.00
0.03
99.24
0.00
0.04
99.06
0.00
0.04
0.00
98.83
0.00
0.06
0.00
98.53
0.07
0.00
98.13
a1u共v = 0兲
u
u
a1u共v = 37兲
A0+u 共v = 0兲
a1u共v = 38兲
a1u共v = 39兲
A0+u 共v = 1兲
a1u共v = 40兲
A0+u 共v = 2兲
a1u共v = 41兲
a1u共v = 42兲
A0+u 共v = 3兲
c0+u 共v = 0兲
a1u共v = 56兲
c1u共v = 0兲
0.25A0+u 共v = 11兲 + 0.97c0+u 共v = 1兲
0.94A0+u 共v = 11兲 + 0.33c0+u 共v = 1兲
a1u共v = 57兲
c1u共v = 1兲
c0+u 共v = 2兲
a1u共v = 58兲
c1u共v = 2兲
c0+u 共v = 3兲
0.95A0+u 共v = 12兲 + 0.29c0+u 共v = 4兲
a1u共v = 59兲
c1u共v = 3兲
0.26A0+u 共v = 12兲 + 0.96c0+u 共v = 4兲
a1u共v = 60兲
c0+u 共v = 5兲
c1u共v = 4兲
0.61A0+u 共v = 13兲 + 0.79c0+u 共v = 6兲
a1u共v = 61兲
c1u共v = 5兲
0.78A0+u 共v = 13兲 + 0.62c0+u 共v = 6兲
c0+u 共v = 7兲
c1u共v = 6兲
a1u共v = 62兲
c0+u 共v = 8兲
c1u共v = 7兲
a1u共v = 63兲
0.90A0+u 共v = 14兲 + 0.44c0+u 共v = 9兲
0.41A0+u 共v = 14兲 + 0.91c0+u 共v = 9兲
c1u共v = 8兲
a1u共v = 64兲
c0+u 共v = 10兲
c1u共v = 9兲
a1u共v = 65兲
c0+u 共v = 11兲
c1u共v = 10兲
0.86A0+u 共v = 15兲 + 0.51c0+u 共v = 12兲
a1u共v = 66兲
0.49A0+u 共v = 15兲 + 0.87c0+u 共v = 12兲
c1u共v = 11兲
c0+u 共v = 13兲
a1u共v = 67兲
c1u共v = 12兲
c0+u 共v = 14兲
a1u共v = 68兲
Downloaded 29 Sep 2007 to 128.135.12.127. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
114315-12
J. Chem. Phys. 125, 114315 共2006兲
Bussery-Honvault et al.
TABLE II. 共Continued.兲
n
Adiabatic
Nonadiabatica
xA0+
xc1u
xc0+
xa1u
Assignment
126
127
129
130
131
132
134
135
136
138
139
140
142
143
−324.30
−315.44
−300.42
−291.01
−292.48
−287.11
−256.84
−262.41
−260.53
−235.68
−224.82
−234.08
−212.56
−207.50
−323.03
−313.75
−299.34
−297.41
−291.03
−284.58
−264.19
−260.73
−258.42
−233.75
−233.33
−232.12
−210.80
−205.19
0.00
5.22
92.02
0.00
0.00
1.68
0.00
0.00
0.28
0.58
0.00
0.00
1.45
0.00
99.91
0.00
0.00
2.40
99.88
0.00
3.12
99.83
0.00
0.00
4.11
99.77
0.00
99.69
0.00
94.78
7.98
0.00
0.00
98.32
0.00
0.00
99.71
99.42
0.00
0.00
98.55
0.00
0.09
0.00
0.00
97.60
0.12
0.00
96.88
0.16
0.00
0.00
95.89
0.23
0.00
0.31
c1u共v = 13兲
0.23A0+u 共v = 16兲 + 0.97c0+u 共v = 15兲
0.96A0+u 共v = 16兲 + 0.28c0+u 共v = 15兲
a1u共v = 69兲
c1u共v = 14兲
c0+u 共v = 16兲
a1u共v = 70兲
c1u共v = 15兲
c0+u 共v = 17兲
c0+u 共v = 18兲
a1u共v = 71兲
c1u共v = 16兲
c0+u 共v = 19兲
c1u共v = 17兲
u
u
a
The SO-coupled rovibrational energies differ from the nonadiabatic values at most on the last significant figure
reported in this table.
states. However, if the coupling is weak, the levels can still
be assigned with the labels corresponding to the A, a, and c
states. In the present paper we will strongly rely on such an
assignment. The levels numbered with a single label v can
mostly be assigned to a single state. In some cases the mixing is so strong that such an assignment is not possible.
However, it is still possible to give an interpretation of these
levels. This interpretation follows from the fact that the inp
共兵r其 , R兲 is normalized and that the
ternal wave function ␺J⍀
S⌺⌳
are orthonormal. It follows
electronic wave functions ␾n⍀
p
from the integration of 兩␺J⍀共兵r其 , R兲兩2 over 兵r其 and R that
1=兺兺
S
n
S
兺兺
⌺=−S ⌳
冕
⬁
pS⌺⌳
兩␹nJ⍀
共R兲兩2R2dR.
共44兲
0
In some cases series 共44兲 will be dominated by a single term.
In such a case we can say that the corresponding rovibrational wave function can be assigned to a single electronic
state. When several terms on the right-hand side of Eq. 共44兲
give significant contributions, we can assign the rovibrational levels to the electronic states corresponding to these
terms.
A more detailed information on the strongly mixed levels can be obtained by looking at the overlap between the
rovibrational wave functions obtained in the BornOppenheimer approximation, i.e., computed for a given electronic state with the nonadiabatic rovibrational wave functions. Assuming that the overlap between the BornOppenheimer rovibrational wave functions of various
electronic states is small 共we have checked that this is true in
our case兲, we can represent the nonadiabatic rovibrational
wave function as a combination of the Born-Oppenheimer
rovibrational wave functions. The magnitude of these coefficients will tell us what the contribution of various electronic
states is to the nonadiabatic rovibrational wave function.
An inspection of Table II shows that the levels assigned
to the 1u components of the a 3⌺+u and c 3⌸u states are almost
unperturbed. Indeed, the internal wave functions for these
states are dominated by a single term corresponding to the a
or c state, respectively. This is clearly shown by the values of
xa1u and xc1u which represent the value of the integral in Eq.
共44兲 in percentage. These values are always very close to
100%. Even for highly vibrationally excited states xa1u and
xc1u are always larger than 95%. This means that the spinorbit coupling and the nonadiabatic coupling are small. Indeed, comparison of the energy levels corresponding to the
adiabatic and SO coupled potential energy curves shows that
the effect of the spin-orbit coupling is very small. It contributes a fraction of a wave number for low lying rovibrational
states. The spin-orbit effects become more important for high
values of v, but even in this case the difference between the
adiabatic and SO coupled energy levels amounts to a few
wave numbers. By contrast, the nonadiabatic effects are
completely negligible even for highly excited levels, and
they do not affect the energy levels on the digits reported in
Table II.
The same picture is true for the energy levels of the A0+u
state that are located below the avoided crossing with the c0+u
state. Comparison of the adiabatic and SO coupled energies
shows that this is indeed the case; the energies differ by less
than 1 cm−1. Just above the avoided crossing all levels of the
A 1⌺+u state become heavily perturbed by the levels of the
c 3⌸u state and cannot be assigned to a single electronic or
spin-orbit coupled state. This picture is valid for all levels of
the A state lying above the avoided crossing. It is interesting
to note that the pattern of levels of the c state is somewhat
different. Some energy levels are strongly perturbed by the A
state and some are not. This behavior is due to the differences in the vibrational constants of the A and c states. The
density of states for the c state is much higher than for the A
state, so some rovibrational levels of the c state are relatively
separated from those of the A state. This is schematically
illustrated in Fig. 5, where the vibrational energy levels of
the A and c states are drawn on the corresponding SO
coupled potential energy curves. An inspection of Fig. 5
shows that all levels of the A state are very close to some
levels of the c state. By contrast, some levels of the c state
are quite well separated from the levels of the A state. It is
interesting to note that the levels of the c state that are un-
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114315-13
J. Chem. Phys. 125, 114315 共2006兲
Theoretical spectroscopy of calcium dimer
TABLE III. Nonadiabatic vibrational energy levels EJv and vibrational
quanta ⌬G共v + 1 / 2兲 共both in cm−1兲 computed from the scaled A state potential for J = 1. The experimental vibrational quanta are taken from Ref. 17.
FIG. 5. Nonadiabatic rovibrational energy levels assigned to the A0+u 共solid
line兲 and c0+u 共dashed line兲 states of Ca2 for J = 1.
perturbed are relatively well described in the adiabatic approximation, the errors being of the order of a few wave
numbers. We wish to end this section by saying that the
pattern of the levels and the importance of the spin-orbit and
nonadiabatic effects are very similar for higher values of J.
C. Comparison with the experiment
As discussed in Sec. IV A the well depth of the ab initio
potential energy curve for the A state agrees within 3% with
the experimental value.17 Unfortunately, the 3% error in the
well depth introduces two additional vibrational levels that
are not observed in the experiments. Therefore, before making more detailed comparisons between theory and experiment we decided to scale the A state potential by 0.98 and
performed full nonadiabatic calculations of the energy levels,
line positions, and intensities in the A 1⌺+u 共v⬘ = 14兲
← X 1⌺+g 共v⬙ = 6兲 band of the calcium dimer. The energy levels and the corresponding vibrational quanta are compared in
Table III with the experimental data of Hofmann and
Harris.17 An inspection of this table shows that the theoretical and experimental vibrational quanta agree reasonably
well. Of course, with a simple scaling of the potential we did
not expect to reach the full agreement with the experiment.
However, the pattern is very well reproduced, and the perturbations showing up in the values of ⌬G共v + 1 / 2兲 appear
for the same values of v as in the experiment. Obviously, the
irregular behavior of the vibrational quanta is due to perturbations by the c state, and our analysis of the wave functions
shows that the vibrational levels corresponding to v = 9 – 16
are indeed strongly mixed with the levels of the c state.
Similar agreement is obtained for the A 1⌺+u 共v⬘ = 14兲
← X 1⌺+g 共v⬙ = 6兲 spectral band. The computed transition frequencies and line strengths are reported in Table IV and illustrated in Fig. 6. An inspection of Table IV shows that the
transition frequencies computed from the ab initio potential
are 120– 140 cm−1 lower than the experimental ones. This is
due to the fact that the well depth of the A state potential is
roughly 300 cm−1 too deep. With a simple scaling of the
potential we get the transition frequencies in a fair agreement
with the experiment. Indeed, the errors in the transition frequencies are reduced to 15– 20 cm−1. More importantly, the
v
EJv
xA0+
xc0+
⌬G共v + 1 / 2兲
Expt.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
−2161.63
−2019.15
−1879.34
−1742.11
−1607.40
−1475.14
−1345.27
−1217.79
−1092.92
−974.94
−861.24
−743.76
−631.36
−512.65
−400.25
−284.76
−168.69
99.95
99.94
99.93
99.92
99.89
99.85
99.77
99.58
98.85
81.69
64.81
81.17
66.80
79.40
54.19
69.90
49.56
0.05
0.06
0.07
0.08
0.11
0.15
0.23
0.42
1.15
18.31
35.19
18.83
33.20
20.60
45.81
30.10
50.44
142.48
139.81
137.23
134.71
132.26
129.87
127.48
124.87
117.98
113.70
117.48
112.40
118.71
112.40
115.49
116.07
122.37
118.70
114.61
113.74
113.56
116.53
113.24
113.92
114.62
u
u
experimental data show that transitions to the rovibrational
levels of the c state that are heavily mixed with the A state
levels should be observed, and this feature is reproduced in
our calculations. Moreover, the values of the level mixing,
xA0+, and of the line strengths computed from the ab initio A
u
state potential roughly explain the experimental observations. Indeed, in most cases the perturbed levels of the c state
that are not observed in the experiment17 have either a too
small component from the A0+u state 共small value of xA0+兲 or
u
a small value of the line strength S共J⬘ ← J⬙兲.
It should be stressed that we obtained such a semiquantitative agreement with the experiment despite the fact that
the spectroscopic constants corresponding to the ab initio
potential energy curve for the c state are quite different from
the constants obtained by deperturbation of the spectra.17
Hofmann and Harris17 observed 34 vibrational levels for the
c state, with v = vi , vi + 1 , . . . , vi + 33, but they were not able to
determine the value of vi. Consequently, the spectroscopic
constants for the c state reported in Ref. 17 were obtained for
vi = 0. Our results suggest that vi should be equal to 2. The
well depth of the c state potential reported in Ref. 17 is 25%
larger, but it is not clear from the Hofmann and Harris analysis how the value of the well depth is sensitive to the choice
of the value of vi. At any rate, the present calculations show
that it is possible to reproduce all the important features of
the experimental data with a c state potential quite different
from the one inferred from the experimental data. This would
suggest that either the spectroscopic constants reported in
Ref. 17 strongly depend on the value of vi, so they cannot be
considered as a final characterization of the c state potential,
or that the experimental data are not very sensitive to the
details of the c state potential. Another conclusion that could
be drawn from the present study is that the ab initio potential
for the c state is essentially correct. Obviously, more experimental work on the deperturbation of the spectrum is needed
to answer these questions.
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114315-14
J. Chem. Phys. 125, 114315 共2006兲
Bussery-Honvault et al.
TABLE IV. Transition frequencies 共in cm−1兲, line strengths 共in a.u.兲, and assignments of the upper levels for the P and R branches of the A 1⌺+u 共v⬘ = 14兲
← X 1⌺+g 共v⬙ = 6兲 band of the calcium dimer. The experimental transition frequencies are taken from Ref. 17.
P branch
J⬙
0
0
2
2
4
4
6
6
8
8
10
10
12
12
14
14
16
16
18
18
20
20
22
22
24
24
26
26
28
28
30
30
32
32
Theor.
15 483.51
15 495.03
15 483.47
15 494.88
15 483.52
15 494.74
15 483.66
15 494.62
15 483.88
15 494.52
15 484.19
15 494.45
15 484.55
15 494.42
15 484.97
15 494.45
15 485.41
15 494.56
15 485.85
15 494.78
15 486.26
15 495.14
15 486.59
15 495.68
15 486.83
15 496.42
15 486.96
15 497.36
15 486.99
15 498.52
15 486.92
15 500.03
共15 632.81兲
共15 641.73兲
共15 632.71兲
共15 641.54兲
共15 632.76兲
共15 641.43兲
共15 632.84兲
共15 641.36兲
共15 633.01兲
共15 641.24兲
共15 633.30兲
共15 641.15兲
共15 633.56兲
共15 641.13兲
共15 633.87兲
共15 641.14兲
共15 634.01兲
共15 641.26兲
共15 634.11兲
共15 641.40兲
共15 634.22兲
共15 641.64兲
共15 634.76兲
共15 641.87兲
共15 634.97兲
共15 642.68兲
共15 635.08兲
共15 643.17兲
共15 635.12兲
共15 644.55兲
共15 635.04兲
共15 646.11兲
Expt.
15 613.48
15 613.46
15 613.48
15 613.61
15 613.80
15 614.02
15 614.23
15 621.22
15 614.38
15 621.34
15 614.38
15 621.68
15 614.23
15 622.24
15 623.08
15 624.13
15 625.37
15 626.80
15 628.35
15 630.00
R branch
S共J⬘ ← J⬙兲
74.78
21.32
74.42
21.75
73.66
22.47
72.59
23.65
71.07
25.86
69.11
27.72
66.34
30.78
63.02
33.88
58.70
38.29
53.97
43.23
48.10
48.71
41.99
55.23
35.47
61.44
29.27
67.31
23.89
69.12
18.94
72.99
xA0+
u
0.304
0.043
0.303
0.044
0.300
0.046
0.296
0.050
0.290
0.055
0.282
0.063
0.272
0.073
0.258
0.086
0.239
0.104
0.217
0.125
0.190
0.151
0.160
0.179
0.131
0.207
0.104
0.231
0.082
0.250
0.064
0.155
Theor.
15 483.75
15 495.27
15 484.03
15 495.44
15 484.41
15 495.63
15 484.87
15 495.83
15 485.68
15 496.02
15 486.05
15 496.31
15 486.74
15 496.60
15 487.48
15 496.95
15 488.24
15 497.39
15 489.00
15 497.93
15 489.73
15 498.61
15 490.38
15 499.47
15 490.94
15 500.53
15 491.39
15 501.80
15 491.74
15 503.27
15 491.99
15 505.10
15 492.16
15 506.84
共15 628.73兲
共15 640.44兲
共15 628.99兲
共15 640.56兲
共15 629.32兲
共15 640.71兲
共15 629.81兲
共15 640.89兲
共15 630.71兲
共15 641.53兲
共15 631.11兲
共15 641.79兲
共15 631.65兲
共15 642.10兲
共15 632.32兲
共15 641.51兲
共15 632.73兲
共15 641.87兲
共15 633.04兲
共15 643.31兲
共15 633.77兲
共15 644.02兲
共15 634.41兲
共15 645.33兲
共15 635.02兲
共15 646.37兲
共15 635.59兲
共15 647.76兲
共15 635.92兲
共15 648.29兲
共15 636.14兲
共15 650.17兲
共15 636.27兲
共15 651.92兲
Expt.
S共J⬘ ← J⬙兲
xA0+
15 613.76
74.78
21.32
74.42
21.75
73.66
22.47
72.59
23.65
71.07
25.86
69.11
27.72
66.34
30.78
63.02
33.88
58.70
38.29
53.97
43.23
48.10
48.71
41.99
55.23
35.47
61.44
29.27
67.31
23.89
69.12
18.94
72.99
15.11
81.02
0.304
0.043
0.303
0.044
0.300
0.046
0.296
0.050
0.290
0.055
0.283
0.063
0.273
0.073
0.259
0.087
0.241
0.105
0.218
0.127
0.191
0.153
0.161
0.181
0.132
0.209
0.105
0.234
0.083
0.253
0.065
0.158
0.051
0.280
15 614.02
15 614.38
15 614.85
15 615.36
15 615 .90
15 616.46
15 623.44
15 616.92
15 623.89
15 617.25
15 624.45
15 617.45
15 626.60
15 626.60
15 627.98
15 629.56
15 631.31
15 633.18
15 635.16
15 637.13
u
V. CONCLUSIONS
In this paper we have reported a theoretical study of the
spectroscopy of the calcium dimer in the A 1⌺+u , c 3⌸u, and
a 3⌺+u manifolds in an ab initio fully nonadiabatic framework. Our results can be summarized as follows:
共1兲
共2兲
共3兲
In the fully nonadiabatic framework, the rovibrational
wave functions describing the nuclear motions in diatomic molecules can be obtained from a system of
coupled differential equations. The rovibrational wave
functions corresponding to various electronic states are
coupled through the relativistic spin-orbit coupling interaction and through different radial and angular coupling terms.
In the nonadiabatic theory the transition intensities can
be written in terms of the ground state rovibrational
wave function and bound rovibrational wave functions
of the excited electronic states that are electric dipole
connected with the ground state.
The characteristics of the potential energy curve for the
FIG. 6. Electronic spectrum corresponding to P 共upper panel兲 and R 共lower
panel兲 branches of the A 1⌺+u 共v⬘ = 14兲 ← X 1⌺+g 共v⬙ = 6兲 band of the calcium
dimer. The assignment of the transitions with the values of J⬙ is given on the
top of each line.
Downloaded 29 Sep 2007 to 128.135.12.127. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
114315-15
共4兲
共5兲
共6兲
共7兲
A 1⌺+u state of Ca2 are in a very good agreement with
the experimental values deduced from the spectroscopic data.17,18
The spin-orbit matrix elements coupling the a 3⌺+u and
c 3⌸u states are almost constant over a wide range of
interatomic distances and very close to the atomic values. The value of the spin-orbit constant coupling the A
and c states is also almost constant, but it decays to
zero at large interatomic distances. The transition dipole moment connecting the ground X state with the A
state shows the very same behavior with the interatomic distance R. Around the minimum the transition
dipole moment is very large, suggesting very intense
transitions between the ground state and the A state.
The rovibrational energy levels corresponding to spinorbit coupled states a1u and c1u, dissociating into the
1
S + 3 P1 and 1S + 3 P2 atoms, respectively, are almost unperturbed, and the corresponding energies are very
close to the energies obtained in the adiabatic approximation. Similar behavior is observed for the levels of
the A0+u state dissociating into 1S + 3 P1 atoms lying below the avoided crossing with the c0+u state.
The rovibrational levels of the A state lying above the
avoided crossing with the c state are all heavily perturbed by the rovibrational states of the c state. These
perturbations are exclusively due to the spin-orbit coupling. In all cases the nonadiabatic effects were shown
to be negligible.
The pattern of the levels and the predicted line positions in the A 1⌺+u 共v⬘ = 14兲 ← X 1⌺+g 共v⬙ = 6兲 band of the
calcium dimer are in a semiquantitative agreement with
the experiment despite the noticeable differences in the
characteristics of the c 3⌸u state.
ACKNOWLEDGMENTS
The authors would like to thank Professor Bogumil Jeziorski and Professor Eberhard Tiemann for many useful discussions and for reading and commenting on the manuscript.
This work was supported by the Centre National de la Recherche Scientifique 共CNRS兲, by the French and Polish governments through the Programme d’Actions Integrées 共PAI兲
Polonium 共Contract No. 05941VJ兲, by the European Research Training Network “Molecular Universe” 共Contract
No. MRTN-CT-2004-512302兲, and by the Polish Scientific
Research Council 共KBN兲 within Grant No. 4-T09A-071-22.
The authors also thank the Institut du Développement et des
Ressources en Informatique Scientifique 共IDRIS兲 for providing the authors with the computer power.
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