Calibration-quality adiabatic potential energy surfaces for H3(+) and its isotopologues.
ABSTRACT Calibration-quality ab initio adiabatic potential energy surfaces (PES) have been determined for all isotopologues of the molecular ion H(3)(+). The underlying Born-Oppenheimer electronic structure computations used optimized explicitly correlated shifted Gaussian functions. The surfaces include diagonal Born-Oppenheimer corrections computed from the accurate electronic wave functions. A fit to the 41,655 ab initio points is presented which gives a standard deviation better than 0.1 cm(-1) when restricted to the points up to 6000 cm(-1) above the first dissociation asymptote. Nuclear motion calculations utilizing this PES, called GLH3P, and an exact kinetic energy operator given in orthogonal internal coordinates are presented. The ro-vibrational transition frequencies for H(3)(+), H(2)D(+), and HD(2)(+) are compared with high resolution measurements. The most sophisticated and complete procedure employed to compute ro-vibrational energy levels, which makes explicit allowance for the inclusion of non-adiabatic effects, reproduces all the known ro-vibrational levels of the H(3)(+) isotopologues considered to better than 0.2 cm(-1). This represents a significant (order-of-magnitude) improvement compared to previous studies of transitions in the visible. Careful treatment of linear geometries is important for high frequency transitions and leads to new assignments for some of the previously observed lines. Prospects for further investigations of non-adiabatic effects in the H(3)(+) isotopologues are discussed. In short, the paper presents (a) an extremely accurate global potential energy surface of H(3)(+) resulting from high accuracy ab initio computations and global fit, (b) very accurate nuclear motion calculations of all available experimental line data up to 16,000 cm(-1), and (c) results suggest that we can predict accurately the lines of H(3)(+) towards dissociation and thus facilitate their experimental observation.
- [Show abstract] [Hide abstract]
ABSTRACT: An accurate description of the complicated shape of the potential energy surface (PES) and that of the highly excited vibration states is of crucial importance for various unsolved issues in the spectroscopy and dynamics of ozone and remains a challenge for the theory. In this work a new analytical representation is proposed for the PES of the ground electronic state of the ozone molecule in the range covering the main potential well and the transition state towards the dissociation. This model accounts for particular features specific to the ozone PES for large variations of nuclear displacements along the minimum energy path. The impact of the shape of the PES near the transition state (existence of the "reef structure") on vibration energy levels was studied for the first time. The major purpose of this work was to provide accurate theoretical predictions for ozone vibrational band centres at the energy range near the dissociation threshold, which would be helpful for understanding the very complicated high-resolution spectra and its analyses currently in progress. Extended ab initio electronic structure calculations were carried out enabling the determination of the parameters of a minimum energy path PES model resulting in a new set of theoretical vibrational levels of ozone. A comparison with recent high-resolution spectroscopic data on the vibrational levels gives the root-mean-square deviations below 1 cm(-1) for ozone band centres up to 90% of the dissociation energy. New ab initio vibrational predictions represent a significant improvement with respect to all previously available calculations.The Journal of Chemical Physics 10/2013; 139(13):134307. · 3.12 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: A Lanczos algorithm with a non-direct product basis was used to compute energy levels of H+3, H2D+, D2H+, D+3, and T+3 with J values as large as 46, 53, 66, 66, and 81. The energy levels are based on a modified potential surface of M. Pavanello et al. that is better adapted to the ab initio energies near the dissociation limit.Molecular Physics 09/2013; 111(16-17):2606-2616. · 1.67 Impact Factor - SourceAvailable from: José Rachid MohallemLeonardo G. Diniz, José Rachid Mohallem, Alexander Alijah, Michele Pavanello, Ludwik Adamowicz, Oleg L. Polyansky, Jonathan Tennyson[Show abstract] [Hide abstract]
ABSTRACT: Using the core-mass approach, we have generated a vibrational-mass surface for the triatomic H3+. The coordinate-dependent masses account for the off-resonance nonadiabatic coupling and permit a very accurate determination of the rovibrational states using a single potential energy surface. The new, high-precision measurements of 12 rovibrational transitions in the ν2 bending fundamental of H3+ by Wu et al. [ Phys. Rev. A 88 032507 (2013)] are used to scale this surface empirically and to derive state-dependent vibrational and rotational masses that reproduce the experimental transition energies to 10−3 cm−1. Rotational term values for J≤10 are presented for the two lowest vibrational states and equivalent transitions in D3+ considered.Physical Review A 09/2013; 88(3). · 3.04 Impact Factor
Page 1
Calibration-quality adiabatic potential energy surfaces for H3+ and itsCalibration-quality adiabatic potential energy surfaces for H3+ and its
isotopologuesisotopologues
Michele Pavanello, Ludwik Adamowicz, Alexander Alijah, Nikolai F. Zobov, Irina I. Mizus et al.
Citation: J. Chem. Phys. 136136, 184303 (2012); doi: 10.1063/1.4711756
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THE JOURNAL OF CHEMICAL PHYSICS 136, 184303 (2012)
Calibration-quality adiabatic potential energy surfaces for H+
and its isotopologues
3
Michele Pavanello,1,a)Ludwik Adamowicz,2,b)Alexander Alijah,3,c)Nikolai F. Zobov,4
Irina I. Mizus,4Oleg L. Polyansky,4,d)Jonathan Tennyson,5,e)Tamás Szidarovszky,6
and Attila G. Császár6,f)
1Gorlaeus Laboratories, Leiden Institute of Chemistry, Leiden University, P.O. Box 9502, 2300 RA Leiden,
The Netherlands
2Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA
3Groupe de Spectrométrie Moléculaire et Atmosphérique, GSMA, UMR CNRS 7331, Université de Reims
Champagne-Ardenne, France
4Institute of Applied Physics, Russian Academy of Science, Ulyanov Street 46, Nizhnii Novgorod
603950, Russia
5Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT,
United Kingdom
6Laboratory of Molecular Structure and Dynamics, Institute of Chemistry, Eötvös University, H-1518
Budapest 112, P.O. Box 32, Hungary
(Received 9 March 2012; accepted 17 April 2012; published online 10 May 2012)
Calibration-quality ab initio adiabatic potential energy surfaces (PES) have been determined for
all isotopologues of the molecular ion H+
computations used optimized explicitly correlated shifted Gaussian functions. The surfaces include
diagonal Born–Oppenheimer corrections computed from the accurate electronic wave functions. A
fit to the 41655 ab initio points is presented which gives a standard deviation better than 0.1 cm−1
when restricted to the points up to 6000 cm−1above the first dissociation asymptote. Nuclear mo-
tion calculations utilizing this PES, called GLH3P, and an exact kinetic energy operator given in
orthogonal internal coordinates are presented. The ro-vibrational transition frequencies for H+
H2D+, and HD+
complete procedure employed to compute ro-vibrational energy levels, which makes explicit al-
lowance for the inclusion of non-adiabatic effects, reproduces all the known ro-vibrational levels
of the H+
magnitude) improvement compared to previous studies of transitions in the visible. Careful treat-
ment of linear geometries is important for high frequency transitions and leads to new assignments
for some of the previously observed lines. Prospects for further investigations of non-adiabatic ef-
fects in the H+
rate global potential energy surface of H+
global fit, (b) very accurate nuclear motion calculations of all available experimental line data up
to 16000 cm−1, and (c) results suggest that we can predict accurately the lines of H+
dissociation and thus facilitate their experimental observation. © 2012 American Institute of Physics.
[http://dx.doi.org/10.1063/1.4711756]
3. The underlying Born–Oppenheimer electronic structure
3,
2are compared with high resolution measurements. The most sophisticated and
3isotopologues considered to better than 0.2 cm−1. This represents a significant (order-of-
3isotopologues are discussed. In short, the paper presents (a) an extremely accu-
3resulting from high accuracy ab initio computations and
3towards
I. INTRODUCTION
However simple the molecular ion H+
contains five quantum particles and thus the non-adiabatic
treatment of its spectra within relativistic or even non-
relativistic quantum mechanics is still not within reach. With
the latest developments in the appropriate protocols and
codes, variational non-adiabatic treatments of three-body sys-
tems within non-relativistic quantum mechanics are becom-
ing commonplace. There are certain evolving techniques
3may appear, it
a)Electronic mail: m.pavanello@chem.leidenuniv.nl.
b)Electronic mail: ludwik@u.arizona.edu.
c)Electronic mail: alexander.alijah@univ-reims.fr.
d)Electronic mail: oleg@theory.phys.ucl.ac.uk.
e)Electronic mail: j.tennyson@ucl.ac.uk.
f)Electronic mail: csaszar@chem.elte.hu.
whereby four-particle systems, such as the isotopologues of
the H2molecule, can be treated in a non-adiabatic and non-
relativistic fashion. Larger systems can also be treated but
only to a rather limited extent.1
When adiabatic quantum mechanical computations,
based on the separation of nuclear and electronic degrees of
freedom, are employed to determine the high-resolution spec-
tra of small molecules, the following factors need to be inves-
tigated when the precision and the accuracy of the computed
results is determined: (a) the electronic and nuclear Hamilto-
nians used for the computations; (b) the accuracy of the fun-
damental constants; (c) the accuracy and precision of the elec-
tronicenergies computed over agrid;(d)thesizeandextentof
the grid; (e) the number of electronic surfaces treated; (f) the
accuracy of the fitting of the PES employing a suitable func-
tional form and interpolating between the computed points;
0021-9606/2012/136(18)/184303/14/$30.00 © 2012 American Institute of Physics
136, 184303-1
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Page 3
184303-2Pavanello et al. J. Chem. Phys. 136, 184303 (2012)
(g) the choice of the masses in the nuclear motion compu-
tations; (h) the accuracy and precision of the variational nu-
clear motion treatment; and (i) the treatment of non-adiabatic
effects.
The H+
mark system of quantum physical and quantum chemical
interest; for example, a number of PESs have been com-
puted for it using explicitly correlated electronic structure
techniques.2–5The accuracy of these and other6PESs devel-
oped for the H+
titude of high-resolution spectroscopy data available for the
H3 − nD+
region (n = 0–3) ab initio transitions significantly more accu-
rate than about 1 cm−1can only be computed by treatment
of non-Born–Oppenheimer effects. For transitions below
5000cm−1,twooftheauthorsdeveloped anaccurateabinitio
model,23which reproduced the spectra of H+
ated isotopologues to better than 0.1 cm−1on average. How-
ever, a series of subsequent experimental studies on H+
H2D+, and D2H+(Refs. 19–22) which extend the measure-
ments to much higher frequencies have so far proved harder
to model to high accuracy theoretically. These and other ob-
servations scattered in the literature suggest that there are sig-
nificant remaining difficulties with non-Born–Oppenheimer
effects.21Furthermore, particular care needs to be exercised
with treatment of the system above the barrier to linearity.11,24
H+
medium25and the atmospheres of gas giants26,27and has
even been detected in supernova remnants.28Its spectroscopic
detection in space relied on ab initio calculations29which
have since helped to provide tabulations of key transitions,30
extensive line lists,31,32partition functions,33and cooling
functions.34There are aspects of these data which need im-
proving: for example, the available cooling functions are not
reliable at low temperatures which may be important for stud-
ies of primordial chemistry.35
Our aim is to generate adiabatic PESs with underlying
ab initio, Born-Oppenheimer energies of an accuracy on the
order of 10−8Eh, which corresponds to sub 0.01 cm−1and,
for convenience, energies are largely given in cm−1below.
Such an accuracy level is not achievable with the use of
any of the black-box electronic structure packages employ-
ing basis sets composed of one-electron functions. The need
to move beyond the standard one-electron basis set approach
has presented a long-standing challenge for electronic struc-
ture computations. In the past two decades several computa-
tional quantum chemistry groups responded to this challenge
and became involved in a sort of “a race for the highest accu-
racy” for the H+
racy in the H+
accuracy one wants to achieve in the computational model-
ing of the ro-vibrational spectrum of the H+
accurate and complete first-principles spectrum would allow
for better assignment of the experimental spectrum, which is
increasingly better measured and characterized. Ultimately,
assignment of the Carrington bands36is the goal of compu-
tational studies of the rotational-vibrational spectra of H+
Among the milestones that have been particularly rele-
vant for studies of the high-resolution spectrum of H+
3molecular ion has long been used as a bench-
3isotopologues can be tested against a mul-
nsystem.7–22It turns out that even in the mid-infrared
3and its deuter-
3,9–12
3is a key astronomical species in the interstellar
3molecular ion. The need for the high accu-
3PES calculations is partially related to a higher
3system. A more
3.
3iso-
topologues one should mention the PES generated by Meyer
et al.6(hereafter referred to as MBB) using the full configu-
ration interaction method and, by modern standards, a fairly
small basis set. The MBB PES includes 69 grid points with
the energy reaching up to about 25000 cm−1above the bot-
tom of the PES. The accuracy of the MBB PES was later
improved by more precise computations performed with the
CI method involving configuration functions multiplied by r12
pre-factors.3An even higher accuracy, claimed to be as high
as 0.02 cm−1at each point of the 69-point grid, was achieved
by Cencek et al.37in their computations performed with
explicitly correlated Gaussian functions, augmented with adi-
abatic and relativistic corrections. Recently, even further im-
proved energies and more accurate calculations of the H+
vibrational spectrum were reported by Bachorz et al.38They
produced total energies of H+
0.02 cm−1at over 5000 PES points including many located
above the barrier to linearity.
Even though significant progress has been made in the
calculations of the H+
siderable room for improvement, especially in the peripheral
regions of the PES corresponding to dissociative geometrical
configurations. Better electronic structurecalculations need to
involvelonger,moreaccurateexpansionsofthewavefunction
intermsofthebasisfunctionsandmoreeffectiveoptimization
of their parameters. Such an improved optimization applied to
the wave function expanded in terms of explicitly correlated
shifted Gaussian functions (ECSGs) was recently developed
and presented by the Adamowicz group.5,39In Ref. 39 the
method was applied for the variational determination of the
H+
lateral triangle with a bond length of 1.65 bohr). It was shown
that using up to 1000 Gaussians in the wave function expan-
sion, the best variational energy upper bound ever for this sys-
tem could be obtained. The best result of –1.343 835 625 02
Ehexhibits an unprecedented precision of below 10−10Eh. In
Ref. 5, Pavanello et al. developed a procedure that allowed
for the energy calculation to be carried out at multiple H+
ometrical points, using for each point the wave function ob-
tained for a nearby point with the Gaussian centers shifted in
thedirectionoftheshiftednuclei.Withthatprocedure theyre-
calculatedtheenergiesatall69MBBgridpointsandachieved
the accuracy of 5 × 10−8Eh(about 0.01 cm−1), which is al-
most an order of magnitude improvement over the previous
best literature computations.37
Recently, we have computed a new, ultra-high accuracy
potential energy surface and used it to assign newly observed
H+
In that study, the fully ab initio calculations presented re-
produced the observed transition frequencies to within about
0.1 cm−1. Here we give full details of this work, show that the
model is equally applicable to the deuterated isotopologues of
H+
the entire visible region of the spectrum. Section II details the
electronic structure calculations, Sec. III discusses our fitting
to the ab initio grid points, and Sec. IV presents the results
of our nuclear motion calculations. Possible future develop-
ments and our conclusions are discussed in Sec. V. Extensive
data have been placed in the electronic archive40to aid those
3ro-
3with a claimed accuracy of
3ground-state PES, there is still con-
3ground-state energy at its equilibrium structure (an equi-
3ge-
3transitions in the mid-visible region of the spectrum.13
3, and present predicted vibrational band origins covering
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Page 4
184303-3Pavanello et al. J. Chem. Phys. 136, 184303 (2012)
interested in high accuracy spectroscopic and related studies
on H+
3and its isotopologues.
II. ELECTRONIC STRUCTURE COMPUTATIONS
The adiabatic composite PESs of the H+
considered in this work were constructed by the addition
of three independently generated surfaces: (a) the non-
relativistic Born–Oppenheimer (BO) energy surface, (b) the
diagonal Born–Oppenheimer correction (DBOC) surface, and
(c) the relativistic correction (RC) surface. Generation of each
of these surfaces is based on two steps, computation of the ab
initio BO energies, DBOCs, and RCs on a grid of nuclear ge-
ometries, and subsequent fitting with properly chosen fit func-
tions.
3isotopologues
A. BO energies
The following two features distinguish our present H+
PES computations from the ones published in the literature:
(1) we calculate the BO energies of H+
ing 41 655 nuclear geometries, the densest and most extended
grid ever used in computations for the H+
unprecedented high accuracy achieved in the BO energy cal-
culation at every grid point.
Let us now explain in some detail the approach we use
in the present study to determine BO energies for H+
spatial component of the variational wave function, ?M(r),
of an n–electron system is expanded in terms of a set of M
ECSGs, {gk}k = 1, ...M, as
?
The ECSGs are the following functions:
3
3on a grid contain-
3ion; and (2) the
3. The
?M(r) =
M
k=1
Ckgk(r).
(1)
gk(r) = exp[−(r − sk)?Ak(r − sk)],
where r and skare 3n dimensional vectors of the electronic
Cartesian coordinates and of the coordinates of the Gaussian
shift, respectively, and the prime denotes the vector transpo-
sition. Akis a symmetric matrix of the Gaussian exponential
coefficients defined as
(2)
Ak= Ak⊗ I3,
(3)
with I3being the 3 × 3 identity matrix and ⊗ denoting the
Kronecker product. We represent the Ak matrix in the fol-
lowing Cholesky-factorized form: Ak= L?
lower triangular matrix. This factorization automatically as-
sures that Akis a positive definite matrix and gk(r) is a square-
integrable function regardless of the particular choice of the
Lkmatrix elements. The Lkmatrix together with the shift vec-
tor, sk, uniquely define the kth ECSG basis function, gk.
In the calculation, the proper permutational symmetry
needs to be implemented in the wave function so that it satis-
fies the Pauli exclusion principle. For the ground singlet state
of H+
respect to permutations of the coordinates of the two electrons
which is achieved by adding to each gkbasis function (1) a
kLk, where Lkis a
3, the spatial wave function needs to be symmetric with
function with permuted coordinates of the shifts vector and
permuted elements of the Akmatrix.
The total variational energy is calculated by solving the
secular equation
HC = SCE,
(4)
where the elements of the Hamiltonian and overlap matri-
ces, H and S, are Hkl= ?gk|ˆ H|gl? and Skl= ?gk|gl?, respec-
tively. The diagonal matrix E comprises the BO energies of
the ground and excited states. Those energies are functions of
the nonlinear parameters of the ECSG basis functions, i.e., the
Lkmatrix elements and the elements of the skvectors.
Equation (4) is solved every time the nonlinear parame-
ters are changed by the routine that runs the variational energy
minimization. The optimization is carried out by the mini-
mization of the energy functional with respect to the nonlin-
ear parameters. For this purpose we use the truncated Newton
minimum-search routine of Nash.41The input to the routine
consists of three items: the values of the nonlinear parameters,
the corresponding energy, and the energy gradient. In our cal-
culations the gradient, which comprises the first derivatives of
the energy with respect to the nonlinear parameters, is deter-
mined using the formulas obtained by analytical differentia-
tion of the energy with respect to those parameters. The for-
mulas involve the first derivative of the error function needed
in the potential energy derivatives. The error-function deriva-
tive is obtained by numerical differentiation.42
The use of analytical energy gradients in the variational
energy minimization sets our work apart from other works
where the ECSG basis functions were utilized,37,38and where
the gradient was approximated numerically. The efficiency of
the optimization is significantly improved by the use of the
analytical gradient. For instance, the best H+
energy obtained with our gradient-based method39is two or-
ders of magnitude more accurate than the one obtained with
the procedure that involved the numerical derivatives.43
3variational BO
B. DBOCs
Beside the ECSG calculation of the total energy, other
molecular properties have also been subjected to ECSG cal-
culations. Properties, such as the transition dipole moments44
and post-BO energy corrections,45have been computed. The
calculations of molecular properties and post-BO energy cor-
rections are often more sensitive to the shortcomings of Gaus-
sians than the total energy. These shortcomings include that
Gaussians do not properly describe Kato’s cusp condition46
and that they fade too fast at large distances, faster than re-
quired by the asymptotic conditions for the exact solutions of
the Schrödinger electronic equation.
In the BO approximation, the nuclear and electronic mo-
tions are not dynamically coupled. However, as the nuclei
havefinitemasses,itisimportantinveryaccuratecalculations
to account for small energy effects that are due to this cou-
pling. Most of the adiabatic correction to the total energy of a
molecular system is recovered by the DBOC. In this work, in
calculating the adiabatic correction we follow the procedure
described in the work of Cencek and Kutzelnigg.47The pro-
cedure is based on the approach proposed by Handy et al.,48
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Page 5
184303-4Pavanello et al.J. Chem. Phys. 136, 184303 (2012)
which starts with the following expression for the adiabatic
correction, Ead:
Ead= −1
2
K
?
α=1
1
Mα
3
?
iα=1
??|
∂2
∂Q2
iα
|??,
(5)
where Mαis the mass of nucleus α, Qiαis the ith coordinate
of the nucleus α, and K is the number of the nuclei in the
system. In Eq. (5), instead of directly differentiating the elec-
tronic wave function with respect to the nuclear coordinate
Qiα, the derivative is approximated numerically as
???Qiα+1
?Qiα
The calculation of the wave functions at a molecular geom-
etry where the Qiαcoordinate of the α nucleus was shifted
by ?Qiα(i.e., ?(Qiα±1
the linear expansion coefficients through solving Eq. (4). In
this work, the DBOCs have been calculated using ?Qiα= 5
× 10−4bohr. In addition, the calculation of the shifted wave-
function, ?(Qiα±1
of the ECSGs in the basis set to the shifted position of the α
nucleus. Given optimal nonlinear parameters (Gaussian expo-
nents and centers) for the ECSG basis set at a certain nuclear
configuration, {Qik}, determined by the optimization routine,
the ECSG basis set for the nuclear configuration shifted by
1
2?Qiαneeds to be found. As the positions (and the exponen-
tial parameters Lk) of the Gaussians are expected to change
when the nuclear configuration changes, a relation needs to
be found between those two changes to effectively calculate
the derivative (see Eq. (6)). Unfortunately, this relation is not
known in a functional form. This complicates the calculation
of the DBOCs.
To evaluate the derivatives in Eq. (6), six independent BO
energy calculations per atom need to be performed at each
PES grid point. If such calculations were carried out in the
same way as the BO energy calculations, over 255000 BO
calculations, each involving full optimizations of Lk’s and
sk’s, instead of the 41 655 calculations, would be needed.
Thiswouldbecomputationallyunfeasible.Therefore,instead,
along the lines of the work of Cencek et al.,47a shifting pro-
cedure was devised in this work to determine the shift of the
Gaussian centers for a particular shift of the coordinates of the
nuclei. Below we describe the procedure emphasizing the fea-
tures which make the procedure more general in comparison
to the procedure proposed by Cencek et al.47
Let us first introduce 3 three-dimensional vectors, Q1,
Q2, and Q3, containing the coordinates of the three nuclei of
H+
φI, φII, and φIII, that have the following shifts of the Gaussian
centers:
?Qi
where i is equal to either 1, 2, or 3. The functions are called
ionic because in Eq. (7) both Gaussian centers coincide with
the position of a nucleus. With that we can approximate any
basis function, φk, with a product of the three ionic functions
∂?
∂Qiα
2?Qiα
?− ??Qiα−1
2?Qiα
?
.
(6)
2?Qiα)) involves recomputation of
2?Qiα), involves adjusting the positions
3. Next, we introduce three two-electron “ionic” functions,
si=
Qi
?
,
(7)
introduced above
φk= φIφIIφIII= exp
?
3
?
i=1
(−r?¯Air + 2r¯Aisi− s?
i¯Aisi)
?
,
(8)
where¯Aiis Ai⊗ I3. By equating like terms in Eq. (8) one
gets
3
?
3
?
i=1
¯Ai=¯Ak,
(9)
i=1
¯Aisi=¯Aksk,
(10)
s?
k¯Aksk=
3
?
i=1
s?
i¯Aisi,
(11)
where sk is the 3n-dimensional (i.e., six-dimensional for
H+
Eqs. (9)and(10) become
3) Gaussian shift vector. By assuming that ¯Ai= ai¯Ak,
3
?
i=1
ai= 1,
(12)
3
?
i=1
aisi= sk.
(13)
With that Eq. (11) is automatically satisfied. Notice that
Eq. (13) is actually composed of two independent equations,
one for the x coordinates and one for the y coordinates. For
nonlinear geometries of H+
topredicttheECSG shiftvectors forthenew geometrical con-
figuration of the nuclei. The procedure involves the following
steps.
3, Eqs. (12)and(13) are sufficient
1. For each ECSG construct the three auxiliary functions
φI − IIIby using the H+
vectors as shown in Eq. (7).
Solve the three independent equations, Eqs. (12) and
(13), to obtain the values of the a1, a2, and a3param-
eters.
Determine the new Gaussian shift vector by inverting
Eq. (13) for the new, changed H+
±1
However, Eqs. (12)and(13) are not independent when
the H+
by making use of those equations in Eqs. (11)and(13) which
do not zero out in this situation. In addition, Eq. (11) needs to
be simplified (approximated) by “decoupling” the parts corre-
sponding to the different electrons in order to make Eqs. (9)–
(11) specific to each Gaussian center. In the “decoupling” we
assume that the off-diagonal terms in¯Akare small compared
to the diagonal terms. This turns Eq. (11) into an equation
that constrains the squares of the x coordinates of the Gaus-
sian centers to the square of the corresponding x coordinate of
the α nucleus
3nuclear coordinates as the shift
2.
3.
3geometry, i.e., Qi
2?Qi.
3geometry becomes linear. The linear case is dealt with
a1+ a2+ a3= 1,
(14)
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Page 6
184303-5Pavanello et al.J. Chem. Phys. 136, 184303 (2012)
a1x1+ a2x2+ a3x3= xα,
(15)
a1x2
1+ a2x2
2+ a3x2
3= x2
α,
(16)
where we have assumed that the linear H+
With that, even for a linear H+
equations, Eqs. (9)–(11), is non-singular and can be solved.
3lies on the x axis.
3configuration, the system of
III. FITTING OF THE PES
A. The grid
An appropriate choice of the grid is important for the
quality of the surface fit. Traditionally, the energy is calcu-
lated at points located around the stationary points of the po-
tential, which is a good strategy for the construction of a lo-
cal, i.e., non-global, PES. The renowned PES parametrization
by Meyer et al.,6the MBB surface, is based on 69 artfully
selected configurations around the minimum of the potential
which extend up to 25000 cm−1. Their parametrization has
been used by others, such as Röhse et al.3or Jaquet et al.49
Bachorz et al.38started from the MBB grid to which they in-
cluded three additional sets of configurations. For construct-
ing a global potential energy surface it is essential to have a
grid spanning the complete configuration space. Such a grid
may be constructed in a systematic manner in hyperspherical
coordinates, in particular, in the so-called democratic hyper-
spherical coordinates.50,51Of these coordinates, the hyperra-
dius, ρ, describes the overall size of the system, while the two
hyperangles, θ and φ, describe its geometrical shape.
In the present work we have generated a very dense
grid using the following ranges of the parameters and the
corresponding step sizes:
ρ : 1 ≤ ρ ≤ 20,
θ : 0◦≤ θ ≤ 90◦,
φ : 30◦≤ φ ≤ 90◦,
?ρ = 0.1,
?θ = 5◦,
?φ = 5◦.
(17)
These generate a grid consisting of 44 885 points. However,
the configurations with one or more internuclear distances
being smaller than 0.7a0were eliminated leaving a total of
42 498 points. An additional 843 points were also discarded
leaving a final grid of 41 655 points.53Ab initio data for these
points are given in the electronic archive.40As can be inferred
from the range of the φ angle, these points span only one sixth
of the surface. The remaining parts of the surface were ob-
tained by symmetry considerations. Due to the very small ρ
step size, our grid is much denser than the grid used before by
Viegas et al.,52which comprised 9985 points. Figure 1 illus-
trates the density of the grid.
B. BO surface
The three lowest singlet states of H+
connected due to avoided crossings between the ground state
and the first excited state and a conical intersection line be-
tween the first and second excited states. Viegas et al.,52
who first constructed global potential energy surfaces of those
three states, showed that the diatomics in molecules, DIM,
approach54is a good starting point for an accurate description
of the surfaces. In the present work we follow their approach
to generate a global, high-quality potential energy surface of
the electronic ground state. In the DIM approach, the PES is
obtained by diagonalization of the following 3 × 3 matrix,
3are intrinsically
H(R) =
⎡
⎣
⎢
E(R1) + ?(R2,R3)
?(R3)
?(R2)
?(R3)
?(R2)
?(R1)
E(R2) + ?(R3,R1)
?(R1)
E(R3) + ?(R1,R2)
⎤
⎦,
⎥
(18)
where
E(Ri) = V[H2,X1?+
g](Ri),
(19)
?(Rj,Rk) =1
2[V[H+
2,X2?+
g](Rj) + V[H+
2,A2?+
u](Rj)
+V[H+
2,X2?+
g](Rk) + V[H+
2,A2?+
u](Rk)] − 2EH,
(20)
and
?(Rk) =1
In the above equations, V denotes potential energy curves
of H2 or H+
EH = −0.5Eh, and i, j, and k are the nuclear indices. R1
is the distance between nuclei 2 and 3, etc., and R is a
three element vector with coordinates R1, R2, and R3. In
2[V[H+
2,X2?+
g](Rk) − V[H+
2,A2?+
u](Rk)].
(21)
2and EH is the energy of the 1s state of H,
our approach we use the H2 and H+
et al.,52which are based on the accurate ab initio ener-
gies of Wolniewicz55,56(H2(X1?+
(H+
The advantage of the DIM PES representation is that it
correctly describes the degeneracies within the three states
and within the dissociation channels. However, it is not accu-
rate at small distances because only a limited number of the
diatomic states are used and the overlap between the atomic
and diatomic fragments forming H+
curacy can be improved by adding a three-body term V(3)(R)
either to the adiabatic or diabatic potential matrices.59,60In
the latter approach, which we employ here, care is needed
in order not to spoil the symmetry properties of the DIM
matrix. The terms inserted to the diagonal elements of this
matrix must be identical for the three matrix elements and
fully symmetric with respect to the permutation of the nuclei.
2potentials of Viegas
g)), Bishop and Wetmore57
2(A2?+
2(X2?+
g)), and Peek58(H+
u)).
3are neglected.54The ac-
Downloaded 05 Jun 2012 to 157.181.190.20. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Page 7
184303-6Pavanello et al.J. Chem. Phys. 136, 184303 (2012)
FIG. 1. Two-dimensional cut of the resulting PES as an illustration of a scope and density of a grid. The plot is in Jacobi coordinates for a fixed angle of 90◦.
Distances are in bohr with the horizontal axis giving the diatomic H-H distance and the vertical axis the distance of the H+to diatomic center-of-mass.
Hence the diagonal matrix elements are changed to
Hii→ Hii+ V(3)(R1,R2,R3).
The corresponding corrections to the off-diagonal elements
are
Hij= ?(Rk) −1
The three-body term in Eq. (23) is squared to make the off-
diagonal element, Hij, negative (see Eq. (21)).
Let us now describe the functional form of the three-body
terms. They are represented as polynomials of the following
threefunctionsofthestandard(seeRef.23,forexample)sym-
metry coordinates, Si,
(22)
2[˜V(3)(R1,R2,R3)]2.
(23)
?1= Sa,?2= S2
x+ S2
y,?3= Sy
?S2
y− 3S2
x
?
(24)
known as the integrity basis functions.61Any product of these
functions is totally symmetric with respect to permutations of
the three nuclei. A detailed discussion on this point can be
found in the book by Murrell et al.62
The three-body terms are then written as
V(3)(R) = PI(?1,?2,?3)T (?1).
They are products of polynomial PI,
(25)
PI(?1,?2,?3) =
?
i+2j+3k≤I
cijk?i
1?j
2?k
3,
(26)
and the following range-determining factor:
T(x) = [1 + eγ(x−x0)]−1,
(27)
which damps the three-body terms at large distances where
the DIM approximation takes over. The summation in
Eq. (26) includes all possible terms up to order I with respect
to the symmetry coordinates Si. The symmetry coordinates,
Si, are expressed in terms of expansion coordinates ˜Ri(see
Viegas et al.52for details), for which we use, following Meyer
et al.,6the Morse displacement coordinates
˜Ri= [1 − e−βα(Ri/R0,α−1)]/βα.
(28)
Downloaded 05 Jun 2012 to 157.181.190.20. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Page 8
184303-7Pavanello et al.J. Chem. Phys. 136, 184303 (2012)
TABLE I. Parameters of expansion functions: for the two three-body terms,
the order of the polynomial and the resulting number of linear coefficients is
given together with the numerical values of the nonlinear parameters.
TermOrder #Coef
γ
R0
β
x0
Diagonal
Off-diagonal
15
13
174
123
0.3
0.3
2.50
2.50
1.0
1.0
10.0
12.0
The analytical expression for the potential energy surface
contains linear expansion parameters cijk of the three-body
terms and nonlinear parameters of the range-determining fac-
tors and the Morse expansion functions. The initial values of
the nonlinear parameters were taken from our previous fit52
and the linear parameters were determined by least-squares
fitting. Next, the nonlinear parameters were adjusted manu-
ally by a trial-and-error procedure. A summary of the pro-
cedure is given in Table I. The ab initio data points used to
generate the fit describe the energy region up to 6000 cm−1
above the dissociation limit. A few configurations have been
omitted as they were found to spoil the overall quality of the
fit. These were asymptotic configurations for which at least
two internuclear distances are bigger than 9.0a0. Such con-
figurations are well described by the DIM approach without
the three body corrections (i.e., by the pure DIM representa-
tion). In the FORTRAN code, where the PES fit is calculated,
the representation of the potential that includes the three-body
terms is automatically replaced by the pure DIM representa-
tion for asymptotic configurations.
The quality of the fit generated in this work can be eval-
uated based on Fig. 2, which shows the deviation between
calculated points and their representation by the fit. Two fur-
ther figures are given in the electronic archive40which give
two-dimensional cuts through the full PES. For most con-
figurations, the fit switches correctly to the asymptotic pure
DIM representation. Unphysical behavior is observed for a
few configurations located well above the fitted energy range.
This range would require a description in terms of three elec-
tronic states, which we will attempt in future work.
Theoverallroot mean
− E(i))2/N)
= 7840 points with 297 parameters in the fitting function
is rms = 0.097 cm−1. As mentioned, the 7840 points span
only one sixth of the whole PES, the complete surface is
obtained based on 47 040 points. Of the calculated points not
used in generating the fit most correspond to configurations
with high energies. These points will be needed in our future
work concerning the H+
dissociation threshold.
An important aspect of this fit is the very large number of
ab initio points used, which allows the PES to be fully deter-
mined at all geometries within the region of interest, and its
correct representation at linear geometries. Both these issues
have been discussed in Ref. 63.
square, rms = (?N
1(V(i)
1
2, for the PES representation obtained from N
3metastable states located above the
C. Correction surfaces
The DBOC points are fitted not globally, but only up
to 30000 cm−1as here we only consider nuclear motion
up to this energy. The fit to the symmetric corrections
was performed using polynomials in terms of symmetry
coordinates23Sa, Sx, and Sy. Ninety-eight parameters were fit-
ted to about 4000 points giving an analytic surface which re-
produced the points with a standard deviation of 0.007 cm−1.
For the mixed isotopologues, the DBOC obeys special
0.1
0.2
0.3
0.4
0.5
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
0 5000 10000 15000 20000
Energy / cm-1
25000 30000 35000 40000
Ediss
Deviation / cm-1
<= 3 rms
outlier
excluded
FIG. 2. An illustration of the quality of the analytical fit of the ground-state H+
energy above the PES minimum. The vertical line marks the dissociation energy.
3PES: residuals for the ab initio grid points used in the fit as function of their
Downloaded 05 Jun 2012 to 157.181.190.20. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Page 9
184303-8Pavanello et al. J. Chem. Phys. 136, 184303 (2012)
-0.2
0
0.2
0 2000 4000 6000 8000
Eexp (cm-1)
10000 12000 14000 16000
Eexp - Ecalc (cm-1)
H3
+
-0.2
0
0.2
0 1000 2000 3000 4000 5000 6000 7000 8000
Eexp - Ecalc (cm-1)
Eexp (cm-1)
H2D+
-0.2
0
0.2
0 1000 2000 3000 4000 5000 6000 7000 8000
Eexp - Ecalc (cm-1)
Eexp (cm-1)
D2H+
FIG. 3. Residuals (observed minus calculated) for all observed states with J = 0, 1, 2, 3 for H+
potential, the non-adiabatic model,23and DVR3D.
3, H2D+, and D2H+. Calculations are performed with the GLH3P
symmetry rules64and an extra surface was fitted using a stan-
dard functional form.23
The data for the relativistic correction were taken from
the work of Bachorz et al.38and fitted up to 30000 cm−1
using an analytical surface with six constants23and a stan-
dard deviation of 0.007 cm−1. This surface was then added to
the symmetric DBOC. These fits are given in the electronic
archive.40We call our final PES, including the correction sur-
faces, GLH3P.
IV. NUCLEAR MOTION COMPUTATIONS
We used the GLH3P PES in calculations of ro-vibrational
energy levels. Calculations considering both rotational and vi-
brational motion were performed up to 17000 cm−1, the re-
gion covered by available experimental studies, and for J = 0
up to 25000 cm−1, to the end of the visible region. The cal-
culations used the BO surface as well as relativistic and adi-
abatic corrections, as well as non-adiabatic corrections which
are discussed below. The studies considered H+
D2H+. We do not consider D+
only available for the bending fundamental.
To make direct comparisons with the extensive results
available fromhigh-resolutionmolecularspectroscopyforH+
(Refs. 8–11, and 13) and the mixed isotopologues H2D+and
3, H2D+, and
3here as experimental data are
3
D2H+,14–21a series of variational rotation-vibration computa-
tions were performed using the adiabatic PESs of this study
and exact kinetic energy operators. These computations uti-
lized the DVR3D program suite65and previously tested basis
sets, the D2FOPI code66–68with appropriate basis sets to deal
with singularities present in the ro-vibrational Hamiltonian,68
and the Hyperspherical harmonics code.69,70
Both DVR3D and D2FOPI have been adapted to allow
for both vibrational and rotational non-adiabatic effects.71Ini-
tially, all nuclear motion calculations used nuclear masses, the
preferred choice when mass-dependent adiabatic surfaces are
available. The variational procedures employed, without the
non-adiabatic corrections, give energies which agree within
0.01 cm−1, in line with previous72comparisons. In contrast to
the electronic structure calculations, all nuclear-motion calcu-
lations presented here were performed on desktop computers.
The numerical calculations with the hyperspherical har-
monics code of Wolniewicz proceed in two steps: First, for
each value of the angular momentum, J, and each symmetry,
?, the corresponding hyperspherical harmonics are generated
and the potential matrix is set up in this basis, for each hy-
perradial grid point. In the present work, we calculate the ma-
trix at 300 points within 0.5a0≤ ρ ≤ 6.0a0. In Wolniewicz’s
algorithm,69the size of the hyperspherical harmonic expan-
sion is controlled by a single input parameter, Kmax, which
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Page 10
184303-9Pavanello et al. J. Chem. Phys. 136, 184303 (2012)
TABLE II. Difference between observed11and calculated line frequencies for higher J states of H+
are reassigned; for these the final column gives the obs.−calc. difference for the previous assignment, which is
also given.
3. Three lines
Band
(v1,v2,l)
Obs.
cm−1
Calc.
cm−1
Obs.−Calc.
cm−1
o.− c., old assig.
cm−1
Line
tP(3, 0)
R(6, 6)
nR(2, 2)
nR(2, 1)
nR(3, 3)
P(6, 6)
tQ(2, 1)
tQ(3, 1)
R(5, 5)
nP(2, 1)
tR(4, 4)
nP(3, 2)
P(4, 3)
nP(5, 5)l
tQ(3, 0)
tQ(1, 0)
R(4, 3)
nP(1, 1)
nP(3, 3)
nP(2, 2)
tR(3, 3)
tR(2, 2)
P(4, 3)
P(4, 4)
Q(5, 0)
tR(1, 1)
+6Q(3, 0)
Q(5, 3)
nQ(1, 1)
nQ(2, 1)
nP(4, 4)u
tR(4, 3)
tR(3, 2)
P(3, 2)
Q(4, 3)u
tR(2, 1)
P(3, 3)
tR(1, 0)
P(2, 2)
nQ(3, 2)u
nQ(2, 2)
P(2, 1)
Q(3, 2)u
+6Q(2, 1)
nQ(4, 2)u
P(3, 0)
P(1, 1)
+6Q(3, 1)
tR(3, 1)
P(5, 3)l
P(3, 1)u
+6Q(4, 1)
Q(1, 0)
nR(1, 1)
nQ(4, 3)
Q(4, 2)u
(2,2,2)
Unknown
(0,4,4)
(0,4,4)
(0,4,4)
Unknown
(2,2,2)
(2,2,2)
Unknown
(2,2,2)
(2,2,2)
(2,2,2)
(2,2,2)
(2,2,2)
(2,2,2)
(2,2,2)
Unknown
(2,2,2)
(2,2,2)
(2,2,2)
(2,2,2)
(2,2,2)
(0,5,1)
(0,5,1)
Unknown
(2,2,2)
(0,5,3)
Unknown
(2,2,2)
(2,2,2)
(2,2,2)
(2,2,2)
(2,2,2)
(0,5,1)
(0,5,1)
(2,2,2)
(0,5,1)
(2,2,2)
(0,5,1)
(2,2,2)
(2,2,2)
(0,5,1)
(0,5,1)
(0,5,3)
(2,2,2)
(0,5,1)
(0,5,1)
(0,5,3)
(2,2,2)
(0,5,1)
(0,5,1)
(0,5,3)
(0,5,1)
(2,2,2)
(2,2,2)
(0,5,1)
10322.235
10329.307
10366.546
10367.184
10454.539
10462.405
10467.800
10468.544
10496.287
10496.571
10497.078
10507.396
10528.992
10558.882
10560.443
10568.209
10573.997
10581.256
10583.688
10586.424
10609.077
10621.634
10624.888
10632.042
10639.058
10641.024
10657.149
10666.604
10669.815
10671.864
10686.611
10690.240
10705.364
10705.894
10710.311
10725.953
10730.107
10752.150
10752.369
10760.627
10766.108
10766.320
10779.136
10789.844
10793.060
10798.691
10798.785
10803.820
10805.800
10811.027
10813.699
10816.758
10831.677
10845.089
10847.551
10855.172
10322.2110.02
10366.433
10367.038
10454.394
10462.493
10467.701
10468.442
10496.122
10496.562
10496.886
10507.406
10529.038
10559.012
10560.363
10568.131
10573.837
10581.218
10583.719
10586.424
10608.913
10621.479
10624.814
10632.078
10638.824
10640.897
10656.968
10666.415
10669.754
10671.801
10686.622
10690.062
10704.971
10705.756
10710.128
10725.807
10730.035
10752.042
10752.278
10760.490
10766.070
10766.182
10779.092
10789.709
10793.021
10798.490
10798.652
10803.595
10805.622
10810.882
10813.528
10816.537
10831.526
10844.994
10847.629
10854.962
0.11
0.15
0.15
−0.09
0.10
0.10
0.17
0.01
0.19
−0.01
−0.05
−0.13
0.08
0.08
0.16
0.04
−0.03
0.00
0.16
0.16
0.07
−0.04
0.23
0.13
0.18
0.19
0.06
0.06
−0.01
0.18
0.39a
0.14
0.18
0.15
0.07
0.11
0.09
0.14
0.04
0.14
0.04
0.13
0.04
0.20
0.13
0.23
0.18
0.15
0.17
0.22
0.15
0.09
−0.08
0.21
Downloaded 05 Jun 2012 to 157.181.190.20. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Page 11
184303-10Pavanello et al. J. Chem. Phys. 136, 184303 (2012)
TABLE II. (Continued.)
Band
(v1,v2,l)
Obs.
cm−1
Calc.
cm−1
Obs.−Calc.
cm−1
o.− c., old assig.
cm−1
Line
P(6, 6)
nR(2, 1)
tR(3, 0)
Q(3, 0)
Q(1, 1)
P(5, 5)
+6R(1, 1)
Q(2, 2)
+6R(2, 2)
Q(3, 3)
P(4, 3)
nR(2, 2)u
nR(3, 1)u
P(4, 4)
R(6, 6)
R(1, 1)l
R(5, 5)
R(4, 4)
R(3, 3)u
Q(2, 1)u
P(3, 3)
nR(4, 3)
R(5, 0)
R(3, 2)u
R(1, 0)
R(1, 1)u
R(2, 1)l
Q(3, 0)
R(2, 2)l
Q(1, 0)
P(6, 6)
Q(3, 3)
−6P(5, 5)u
−6P(4, 4)
P(4, 3)
R(4, 3)
tQ(1, 0)
P(3, 3)
tQ(3, 3)
R(3, 0)
tQ(1, 1)
nR(3, 3)
tR(1, 0)
R(1, 0)
Q(1, 0)
R(3, 3)
Q(1, 0)
R(2, 0)
R(2, 3)
(3,1,1)
(2,2,2)
(2,2,2)
(0,5,1)
(0,5,1)
(3,1,1)
(0,5,3)
(0,5,1)
(0,5,1)
(0,5,1)
(3,1,1)
(2,2,2)
(2,2,2)
(3,1,1)
Unknown
(0,5,1)
Unknown
Unknown
(0,5,1)
(0,5,1)
(3,1,1)
(2,2,0)
Unknown
(0,5,1)
(0,5,1)
(0,5,1)
(0,5,1)
(3,1,1)
(0,5,1)
(3,1,1)
Unknown
(3,1,1)
(0,5,1)
(0,5,3)
(0,5,5)
Unknown
(0,6,2)
(0,6,2)
(1,4,4)
Unknown
(1,4,4)
(1,4,2)
(1,4,4)
(0,6,2)
(0,6,4)
Unknown
(0,7,1)
hot
hot
10874.681
10934.327
10935.358
10935.631
10939.559
10953.026
10963.072
10964.605
10964.792
10968.257
11015.488
11019.351
11024.705
11033.268
11036.111
11044.146
11046.569
11048.996
11053.686
11071.117
11111.798
11114.428
11114.628
11195.630
11228.601
11244.353
11246.707
11278.517
11304.480
11318.080
11331.112
11358.855
11422.627
11482.938
11494.835
12331.180
12419.140
12502.614
12525.302
12536.621
12623.171
12658.335
12897.888
13056.013
13597.367
13606.093
13676.446
10827.764
11265.189
10875.095
10934.204
10935.113
10935.477
10939.374
10953.644
10962.870
10964.418
10964.574
10968.110
11015.619
11019.157
11025.223
11033.421
−0.41
0.12
0.25
0.15
0.18
−0.62a
0.20
0.19
0.22
0.15
−0.13
0.19
−0.52a
−0.15
−0.12 Q(3, 0)
0.52 tR(3, 0)
11043.931
11046.415
11048.794
11053.424
11070.892
11111.726
11114.293
11114.454
11195.343
11228.321
11244.085
11246.405
11278.537
11304.199
11318.099
11331.214
11358.915
11422.689
11482.967
11494.892
12330.898
12419.124
12502.659
12525.250
12536.423
12623.057
12658.114
12897.786
13055.763
13597.389
13605.820
13676.197
10827.500
0.22
0.15
0.20
0.26
0.23
0.07
0.14
0.17
0.28
0.28
0.27
0.30
−0.02
0.28
−0.02
−0.10
−0.06
−0.06
−0.03
−0.06
0.28
0.02
−0.05
0.05
0.20
0.11
0.22
0.10
0.25
−0.02
0.27
0.25
0.26
−1.33 P(3,3)
aProbable misassigned line.
is the maximum value of the grand angular momentum. We
have used here Kmax= 140, which, for J = 0, yields 444
functions of symmetry A?
and 852 functions of symmetry E?. The basis has then been
contracted to a convenient size N as described by Wolniewicz
and Hinze.70The ro-vibrational eigenvalues are obtained in
the second step, where a system of N coupled equations in
1, 408 functions of symmetry A?
2,
the hyperradius, ρ, is integrated numerically using the ma-
trix Numerov algorithm with a step size of ?ρ = 0.01a0and
an interval 0.5a0≤ ρ ≤ 6.0a0. The potential matrices are
interpolated by cubic splines. Numerical tests were made to
guarantee the convergence of the eigenvalues with respect to
boundaries of the integration interval and the step size. More
critical is the number of coupled equations, N. While N = 100
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Page 12
184303-11Pavanello et al. J. Chem. Phys. 136, 184303 (2012)
is appropriate for the lowest eigenvalues (up to 10000 cm−1),
it has to be increased for the higher ones. We made numeri-
cal tests with N up to N = 250. For the calculations in sym-
metry E?, N = 200 was found to give fully converged results
for eigenvalues up to 30000 cm−1. Note that there are 129
eigenvalues up to this energy. The calculations in symme-
tries A?
the density of states is lower (86 and 45 states, respectively).
N = 150 was found to be sufficient. Our reported eigenvalues
are converged with respect to all these parameters to 10−9Eh.
The CPU time on a single processor (Intel Xenon X5650)
is about 12 min for the preparation of the potential matri-
ces, step 1, and roughly 30 s per eigenvalue with N = 100,
100 s with N = 150, 250 s with N = 200 and 400 s with
N = 250 in step 2. The precise CPU time depends on the den-
sity of states and time needed to find upper and lower bounds
for each eigenvalue.
Calculations of ro-vibrational energies up to 30000 cm−1
(25000 cm−1when allowance is made for the zero-point en-
ergy) were performed with and without non-adiabatic cor-
rections. First, the BO surface plus adiabatic corrections was
used. The discrepancy between theory and experiment proved
to be within 2 cm−1for observed transition frequencies up to
13 000 cm−1.9–12As all the parts of the calculations are per-
formed with an accuracy of approaching 0.01 cm−1, the only
possible source of this discrepancy could be non-adiabatic
effects. We therefore used a simple method to allow for non-
adiabatic correction as developed in Ref. 23: the kinetic en-
ergy (KE) operator was modified by using different vibra-
tional and rotational masses. The vibrational mass was taken
as intermediate between nuclear and atomic, and equal to
1.007 537 u, a value taken from studies of H+
For the deuterated species we also employed the formula of
Moss, see Ref. 23 for details. The rotational mass was taken
to be equal to the nuclear mass. As only J up to 3 was con-
sidered, we did not have to modify the rotational mass as for
low J rotational non-adiabatic effects are negligible. The dif-
ference in the two masses leads to an additional kinetic en-
ergy operator term23which is zero when the two masses are
equal.
Table III gives results of these calculations for transitions
observed in the visible. After re-assigning four previously
misassigned transitions, the maximum deviation is about
0.2 cm−1and the standard deviation is less than 0.1 cm−1.
Detailed comparison with the newly observed transitions in
the mid-visible region were given in Ref. 13 and are of simi-
lar quality.
The real test of any beyond-Born–Oppenheimer model is
that it should be capable of giving results of similar quality
for all isopotologues of a system. In this context the asym-
metric isotopologues, H2D+and D2H+, provide a particularly
stringent test since an accurate calculation of the splitting of
the degenerate H+
topic substitution requires non-Born–Oppenheimer terms of
a lower symmetry than the BO PES.74,75The results of the
calculations for energy levels up to J = 3 for H+
D2H+are given in the electronic archive.40For H2D+and
D2H+we undertook a systematic check on the labelling of the
energy levels usingthe rigidrotor decomposition procedure.76
1and A?
2can be done with less coupled equations, as
2by Moss.73
3ν2bending mode upon asymmetric iso-
3, H2D+, and
TABLE III. Selected calculated transition frequencies of H+
umn I-rotational assignment, II-observed line center,10,12III-calculated fre-
quency using the model of Ref. 23, IV-observed – calculated in this model,
V-observed–calculatedusinganuclearmassmodel.Notethefourreassigned
lines.
3in cm−1. Col-
IIIs IIIIVV
Q(1,0)
P(1,1)
R(1,0)a
R(2,2)b
P(1,1)
R(1,1)
R(1,1)
Q(1,1)
R(1,1)
R(1,1)
R(1,1)a
P(1,1)b
Q(1,0)
P(1,1)
P(1,1)
R(1,0)
R(1,0)
Q(1,1)
R(1,0)a
P(1,0)b
Q(1,0)a
P(1,0)b
10831.681
10798.777
10752.161
10752.161
12413.247
12588.951
12620.223
12373.526
12381.137
12678.688
13332.903
13332.903
13638.251
15058.681
15130.480
15450.112
15643.052
15716.813
16506.139
16506.139
16660.216
16660.216
10831.520
10798.650
10752.040
10755.590
12413.280
12588.910
12620.080
12373.270
12381.070
12678.520
13332.850
13332.393
13638.430
15058.490
15130.430
15450.190
15643.000
15716.590
16505.920
16495.969
16660.210
16670.796
0.16
0.13
0.12
−0.80
−0.84
−1.04
−3.55
−1.17
−1.28
−1.10
−1.05
−1.06
−1.06
−1.32
0.51
−1.31
−1.25
−1.34
−1.54
−1.44
−1.34
−1.38
−10.17
−1.55
−10.58
0.03
0.04
0.14
0.26
0.07
0.17
0.05
0.18
0.19
0.05
0.08
0.05
0.22
0.21
0.00
aNew assignment on the basis of the calculations presented here.
bPrevious assignment.
These largely confirmed the previous labels, even in cases
where these were not particularly secure.21
A summary of differences between the published ob-
served and calculated levels with J = 0, 1, 2, and 3 is pre-
sented in Fig. 3. As seen from the figure, the results for all
isotopologues are of similar high quality. All, however, show
a small, systematic shift which places the calculated levels
slightly too low. This discrepancy could be easily decreased
further by slight reduction in the effective vibrational mass re-
ducing the maximum deviation to about 0.1 cm−1and a stan-
dard deviation of about 0.05 cm−1. However, this would de-
stroy the purely ab initio character of this model and was not
pursued here. A further reduction in this discrepancy would
require more sophisticated modeling of non-adiabatic effects;
we return to this point below.
Tables II and III compare our ab initio calculations with
observations for various high-lying states of H+
our new calculations significantly better than previous studies
but also we are able to make (re-)assignments for a number of
lines which have previously proved problematic. For exam-
ple, the transition at 13 676 cm−1was reproduced so badly, a
discrepancy of over 3 cm−1, by the previous calculations that
Bachorz et al.38concluded that the reason of such discrepancy
was not yet known. Our calculations reproduce this line with
an accuracy similar to the others. As indicated in the table,
there are three lines which are probably misassigned. To con-
firm this will require further calculations with higher J levels;
this will form part of a future study.
3. Not only are
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Page 13
184303-12Pavanello et al.J. Chem. Phys. 136, 184303 (2012)
TABLE IV. J = 0 vibrational band origins calculated using the hyperspher-
ical method in full symmetry.
v1
v2
??
n
E/cm−1
0
0
1
0
0
1
2
0
0
0
1
1
2
0
0
3
1
1
0
1
2
2
0
0
3
0
0
1
1
4
2
0
0
2
1
2
3
3
1
1
0
0
0
1
4
2
1
2
0
1
1
3
0
5
0
0
3
0
2
0
1
0
2
2
1
0
3
3
3
2
2
1
4
4
0
3
3
4
3
2
2
5
5
1
5
5
4
4
0
3
6
6
3
4
3
2
2
5
5
6
7
7
5
1
4
5
4
7
6
6
3
8
0
6
8
3
6
4
0
1
0
0
2
1
0
1
3
3
0
2
1
0
2
0
1
3
4
3
0
2
1
3
1
3
5
0
2
0
1
0
2
3
4
3
0
2
1
3
4
1
3
3
1
0
5
2
3
0
2
1
0
0
6
2
3
6
4
A?
E?
A?
A?
E?
E?
A?
E?
A?
A?
A?
E?
E?
A?
E?
A?
E?
A?
E?
A?
A?
E?
E?
A?
E?
A?
E?
A?
E?
A?
E?
A?
E?
A?
E?
A?
A?
E?
E?
A?
E?
E?
A?
A?
E?
A?
E?
E?
A?
A?
E?
E?
A?
A?
A?
E?
A?
A?
E?
1
0
0
1
2
1
2
3
3
4
0
5
4
5
6
6
7
7
8
8
1
9
9
10
10
11
2
12
11
13
12
14
13
15
14
16
3
15
17
18
16
19
20
17
4
21
18
22
23
5
19
24
25
20
21
22
26
23
6
27
0.00
2521.30
3178.29
4778.15
4997.89
5554.20
6262.13
7005.97
7285.56
7492.78
7769.23
7870.23
8488.01
9001.57
9113.04
9251.91
9653.70
9968.94
9997.18
10210.33
10593.19
10645.31
10862.75
10923.36
11323.12
11529.24
11658.31
11814.52
12079.40
12146.46
12303.33
12382.15
12477.39
12590.53
12697.40
12832.17
13288.91
13318.19
13395.20
13405.25
13592.25
13702.58
13725.43
13756.61
14055.01
14198.51
14218.02
14478.22
14566.28
14666.18
14890.47
14901.82
14909.87
14940.15
15080.18
15122.64
15168.30
15190.71
15214.80
1
1
1
1
2
1
1
1
1
2
1
1
2
1
1
1
1
2
1
1
1
2
1
2
1
1
1
1
1
2
TABLE IV. (Continued.)
v1
v2
??
n
E/cm−1
1
3
2
4
4
2
2
6
3
5
2
2
5
5
4
3
1
0
2
3
3
E?
A?
E?
A?
E?
A?
A?
28
7
29
24
30
25
8
15335.96
15373.68
15791.11
15877.85
15888.28
15925.04
15969.60
2
1
1
2
Finally,TableIVgivesourpredictedvibrationalbandori-
gins for H+
band origins we predict to lie below 25000 cm−1is given
in the electronic archive.40The results presented were ob-
tained with using the Hyperspherical harmonics code since
this works in full symmetry but are closely mirrored by calcu-
lations performed using the programs D2FOPI and DVR3D.
Vibrational assignments have been included where possible;
however, H+
leading to an early onset of classical chaos and a loss of ap-
proximate vibrational quantum numbers.77These predicted
band origins provide a starting point for further, higher fre-
quency experimental studies.
3for states up to 16000 cm−1; a full list of the 263
3is well known to show very strong couplings
V. SUMMARY AND CONCLUSIONS
We present a highly accurate global PES of H+
ion, which resulted from extra high accuracy non-relativistic
electronic structure BO calculation in a grid of 41 655 points
which give high density coverage of all H+
and well above dissociation and global fit of these ab initio
points using three PESs of interacting electronic states. Adi-
abatic corrections to the BO approximation have also been
computed.
We used three independent nuclear motion codes to cal-
culate rotation-vibration lines to compare our calculated line
centers to experimentally known ones. We have done it both
without and with modeling of non-adiabatic effects. Lines
known experimentally are both below and above the bar-
rier to linearity. The calculated discrepancy between theory
and experiment proved to be better than 0.1 cm−1on aver-
age, once both adiabatic and non-adiabatic corrections to the
Born–Oppenheimer approximation are explicitly included in
the calculation. We are confident that the presented results
will allow us to predict accurately the rotation-vibration levels
of H+
observation of the lines close to dissociation.
Overall, for the ground-state PES of H+
logues, the largest remaining source of error in the prediction
of ro-vibrational transition frequencies lies in the treatment of
non-adiabatic effects and, possibly, inclusion of effects due to
quantum electrodynamics.86
The present work uses the Polyansky–Tennyson23ap-
proach of a fixed, effective vibrational mass to represent
non-adiabatic effects. Furthermore, this treatment includes
no allowance for rotational non-adiabatic effects which
will result in it becoming increasingly less accurate with
3molecular
3geometries up to
3in vicinity of dissociation and assist the experimental
3and its isotopo-
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Page 14
184303-13Pavanello et al.J. Chem. Phys. 136, 184303 (2012)
rotational excitation, J. It is well known from studies on di-
atomic systems that non-adiabatic effects are geometry de-
pendent and, therefore, high-accuracy treatments require the
use of coordinate dependent masses.78,79Bunker and Moss80
formulated a method for treating non-adiabatic effects in tri-
atomic molecules but so far studies that have considered
non-adiabatic effects in triatomic systems have been very
limited.81–84For example, an empirically motivated approach
due to Schiffels et al.82introduces energy-dependent correc-
tions to the band origins. Recently, ab initio methods for cal-
culating effective, coordinate-dependent rotational and vibra-
tional masses have been proposed.79,85A logical follow-up to
the present study is to implement such an approach for the H+
system.
In short, this paper presents (a) an extremely accurate
global potential energy surface of H+
curacy ab initio calculations and a global fit, (b) very accurate
calculations for all available experimental energy levels up to
16000 cm−1above the ground state, (c) results that suggest
we can predict accurately the lines of H+
tion and thus facilitate their experimental observation.
3
3resulting from high ac-
3towards dissocia-
ACKNOWLEDGMENTS
The computations of the ab initio energies utilized a
1392-core SGI Altix ICE 8200 (2.83 GHZ quad-core Xeon,
“Harpertown”), 2 GB memory/core at the High Performance
Computing center at The University of Arizona. This work
was also supported by the computational center of the Univer-
sité de Reims Champagne-Ardenne. M.P. has been supported
for part of the work by a Marie Curie Fellowship (PIIF-GA-
2009-254444). We also thank the Royal Society, the ERC (un-
der Advanced Investigator Project 267219), the Scientific Re-
search Fund of Hungary (Grant No. OTKA NK83583), the
European Union and the European Social Fund under Grant
No. TÁMOP-4.2.1/B-09/1/KMR-2010-0003, and the Russian
Fund for Fundamental Studies for their support for aspects of
this project.
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