Core-mass nonadiabatic corrections to molecules

Core-mass nonadiabatic corrections to molecules: H 2 , H 2 + , and isotopologuesLeonardo G. Diniz, Alexander Alijah, and José Rachid Mohallem Citation: The Journal of Chemical Physics 137, 164316 (2012); doi: 10.1063/1.4762442 View online: http://dx.doi.org/10.1063/1.4762442 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/137/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Modelling non-adiabatic effects in H 3 + : Solution of the rovibrational Schrödinger equation with motion-dependent masses and mass surfaces J. Chem. Phys. 141, 154111 (2014); 10.1063/1.4897566 A time-independent Hermitian Floquet approach for high harmonic generation in H 2 + and HD + : Effect ofnonadiabatic interaction in HD + J. Chem. Phys. 132, 234314 (2010); 10.1063/1.3448636 Nonadiabatic corrections to rovibrational levels of H 2 J. Chem. Phys. 130, 164113 (2009); 10.1063/1.3114680 The Al + – H 2 cation complex: Rotationally resolved infrared spectrum, potential energy surface, androvibrational calculations J. Chem. Phys. 127, 164310 (2007); 10.1063/1.2778422 Hydrogen molecular ion and molecule in two dimensions J. Chem. Phys. 118, 2197 (2003); 10.1063/1.1531103

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THE JOURNAL OF CHEMICAL PHYSICS 137, 164316 (2012)

Core-mass nonadiabatic corrections to molecules: H2, H+2 , and

isotopologuesLeonardo G. Diniz,1,2,a) Alexander Alijah,3,b) and José Rachid Mohallem1,c)

1Laboratório de Átomos e Moléculas Especiais, Departamento de Física, ICEx, Universidade Federal deMinas Gerais, P. O. Box 702, 30123-970 Belo Horizonte, MG, Brazil2Centro Federal de Educação Tecnológica de Minas Gerais, Unidade VII, Av. Amazonas 1193,CEP 35.183-006 Timóteo, MG, Brazil3Groupe de Spectrométrie Moléculaire et Atmosphérique, GSMA, UMR CNRS 7331, Université de ReimsChampagne-Ardenne, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse B.P. 1039,51687 Reims Cedex 2, France

(Received 16 July 2012; accepted 8 October 2012; published online 31 October 2012)

For high-precision calculations of rovibrational states of light molecules, it is essential to includenon-adiabatic corrections. In the absence of crossings of potential energy surfaces, they can be incor-porated in a single surface picture through coordinate-dependent vibrational and rotational reducedmasses. We present a compact method for their evaluation and relate in particular the vibrationalmass to a well defined nuclear core mass derived from a Mulliken analysis of the electronic den-sity. For the rotational mass we propose a simple, but very effective parametrization. The use ofthese masses in the nuclear Schrödinger equation yields numerical data for the corrections of a muchhigher quality than can be obtained with optimized constant masses, typically better than 0.1 cm−1.We demonstrate the method for H2, H+

2 , and singly deuterated isotopologues. Isotopic asymmetrydoes not present any particular difficulty. Generalization to polyatomic molecules is straightforward.© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4762442]

I. INTRODUCTION

The Born-Oppenheimer (clamped nuclei) approxima-tion1–3 is the starting point for most of the calculations inmolecular physics and chemistry. It permits the separation ofelectronic and nuclear motions, and leads to the picture of anelectronic potential energy surface on which the nuclei move.4

The nuclei are considered fixed when solving the electronicSchrödinger equation. Since here the parametric dependenceof the electronic wavefunction on the nuclear coordinates isneglected, two errors arise: The first is the missing expec-tation value of the nuclear kinetic energy operator over theelectronic wavefunction, which, when included, gives rise tothe adiabatic correction to the potential energy surface. Thesecond error is related to the neglect of the off-diagonal el-ements of the nuclear kinetic energy operator and hence theso-called non-adiabatic effects. In a complete picture, the nu-clei would move simultaneously, with a certain probability,on all electronic surfaces. Post Born-Oppenheimer effects aremost important for light hydrogenic molecules.

In a now classical article, Kołos and Wolniewicz5 devel-oped a non-adiabatic theory for diatomic molecules. In theirapplication to H2, in the same paper, they suggested a trialfunction that depends on nuclear and electronic coordinates,including the inter-electronic distance. Their theory was sub-sequently applied to H2, D2, and T2,6 and at this stage rela-tivistic corrections were also included. Extensions to higher

a)[email protected])[email protected])[email protected].

rotational, vibrational, and electronic states of H2 and iso-topologues followed,7–9 as well as applications to HeH+10

and to H+2 and HD+.11 Bishop and Cheung12 and Bishop and

Wetmore13 studied H+2 and deuterated and tritiated isotop-

logues with an accurate, non-adiabatic variational approachwith relativistic and radiative corrections included.

Alternatives to these non-adiabatic methods have beendeveloped by Herman and Asgharian14 and Bunker andMoss.15, 16 If the electronic state of interest is well separatedfrom the others, the effect of non-adiabatic coupling can bemerged into an effective Hamiltonian for this state with amodified effective potential energy surface and coordinate-dependent reduced vibrational and rotational masses. Bunkerand Moss15, 16 derived, for diatomics and triatomics, analyt-ical expressions for the correction terms arising from such atreatment. As these terms are quite complicated, they replacedthe coordinate-dependent vibrational and rotational massesby constant masses which they fitted to reproduce experi-mental data. Accurate results were reported for H2, H+

2 , andisotopologues.17–19 Their concept has proved quite powerfulalso for H+

3 , where non-adiabatic effects have successfullybeen simulated through the use of Moss masses for H+

2 .20–22

In a direct application of Bunker and Moss’ correction for-mulae, Schwenke evaluated the various terms ab initio forH+

2 , HD+,23 H2, and water24 and demonstrated the practica-bility of this formalism. Jaquet and Khoma25 applied Her-man and Asgharian’s theory to H+

2 and H2. Coordinate de-pendent masses for the hydrogen molecule have also beenevaluated through vibrational or rotational g factors by Baket al.26 and by Selg.27 Pachucki and Komasa28 calculated non-adiabatic energies of H2 to ultra-high accuracy using their

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164316-2 Diniz, Alijah, and Mohallem J. Chem. Phys. 137, 164316 (2012)

adiabatic perturbative theory which, as they point out, isequivalent to the formalism of Bunker and Moss. They re-port effective, R-dependent vibrational and rotational reducedmasses. In a recent paper, Komasa and co-workers29 comple-mented this work by providing relativistic and, in particular,quantum electrodynamics corrections which, at this level ofaccuracy, can no longer be excluded. Korobov reported suchterms for the H+

2 ion.30 Highly accurate data of all the boundstates of HD were provided by Pachucki and Komasa.31

Conceptually different approaches have been developedthat do not start from the Born-Oppenheimer separation.Korobov32 and Hilico et al.33 solved the H+

2 problem varia-tionally to high accuracy as a three-body Coulomb problem.Bubin et al.34–36 transformed the vibrational Hamiltonian ofH2 and isotopologues into a three-body problem, by eliminat-ing the centre-of-mass motion, which was then solved exactly.

These and other articles demonstrate the impressive ac-curacy that ab initio calculations have reached nowadays forfew-electron diatomic systems, an accuracy which is closeto the experimental one. In the present article we focus onthe computation of the non-adiabatic shifts of rovibrationaleigenvalues which are, together with the shifts due to theadiabatic correction, the most important corrections to theBorn-Oppenheimer based eigenvalues. Ab initio evaluationof these shifts is possible, but is cumbersome and requireshighly specialised numerical codes that cannot be applied eas-ily to the general molecule. Hence, some authors have de-veloped simpler models, including empirical ones. Alijah andcolleagues37, 38 have proposed a method for the evaluation ofthe non-adiabatic shifts as rovibrational expectation values ofan operator containing the derivative of the potential. In a re-cent publication, Kutzelnigg39 analysed the problem from thelinear combination of atomic orbitals (LCAO) point of viewand derived analytical expressions for coordinate-dependentreduced vibrational and rotational masses for H2 and H+

2 . Ap-plying this method to H+

2 and D+2 , Jaquet and Kutzelnigg40

reported accurate non-adiabatic shifts.Mohallem and collaborators, starting from the finite nu-

clear mass approach,41–43 suggested a different formalism,based on a separation of motions of atomic cores and valenceelectrons, and obtained an empirical expression of the nuclearcore mass44 which uses the diagonal elements of the Mullikendensity matrix. In the present work we have further investi-gated the concept of core masses, extending it to the rotatingmolecule, i.e., the general case J �= 0. Application to H2, H+

2 ,and isotopologues yields accurate results of the same qualityas those obtained by Jaquet and Kutzelnigg.

II. THEORY

As shown by Bunker and Moss15 by means of a contacttransformation, an effective non-adiabatic Hamiltonian can beconstructed in the form

H = − ∇2R

2μvib(R)+ J (J + 1)

2μrot(R)R2+ W (R), (1)

where μvib(R) and μvib(R) are coordinate-dependent effectivevibrational and rotational masses. W (R) is a potential consist-ing of three parts: the Born-Oppenheimer potential, W0(R),

the adiabatic correction, WA(R), i.e., the expectation value ofthe nuclear kinetic energy operator over the electronic wavefunction, and a non-adiabatic correction, WNA(R),

W (R) = W0(R) + WA(R) + WNA(R). (2)

Relativistic and radiative corrections may be added to the W0

term as they are mass-independent. Pachucki and Komasa28

developed this theory further, reported accurate expressionsfor the effective masses and potential functions, and computedthe relativistic and radiative corrections for H2 and deuteratedisotopologues. They then calculated the rovibrational states toeven more significant figures than Wolniewicz.

An interesting connection with this approach appeared inRef. 44 in which the molecular Hamiltonian in the molecularreference frame is partitioned as (taking a diatomic moleculeAB for instance)

H = −∇2R,θ,φ

2μcore+ HA + HB + Hval. (3)

This equation assumes a separation of motion of the atomiccores A and B (a core there means a nucleus plus the coreelectrons) and the motion of the valence electrons. The au-thors then worked out a set of coupled equations for the coremotion in which the R-dependent core reduced mass μcore ap-pears automatically instead of the nuclear reduced mass asone assumes the electronic state decoupling.

The reduced mass μcore that appears in the nuclear equa-tion (3) is appropriate for vibrations, i.e., μcore

vib ≡ μcore. Onemust be aware that the rotational and vibrational motions havequite different frequencies so that when the kinetic energy op-erator is split into its vibrational and rotational parts, a rota-tional reduced core mass μcore

rot should be introduced such thatμcore

rot < μcorevib , just as in the treatment by Bunker and Moss,

see Eq. (1).The question now arises as to how to separate the core

contributions of the electrons from the valence contributionsand hence how to define the core masses. There is no uniqueway of doing this. Mohallem et al.44 proposed an empiricalsolution within the LCAO approach, inspired from Mulliken’spopulation analysis and the recent work by Kutzelnigg.39 Ac-cording to Mulliken,45 the diagonal populations, nII, give thefractions attached to their corresponding nuclei, while the off-diagonal populations are interpreted as due to the valenceelectrons which build up the chemical bond. Hence, Mo-hallem and co-workers define the R-dependent core mass as-sociated to the nucleus A as (from here on we will drop thesuperscript core to simplify the notation)

mA(R) = mA + nAA(R). (4)

Note that as the molecule dissociates, the nuclear coremasses approach the masses of the atomic fragments. The R-dependent core masses are then used to compute the equiva-lent of the effective, R-dependent reduced vibrational mass ofthe Bunker and Moss approach, see Eq. (1). For a diatomicAB,

1

μvib(R)= 1

mA(R)+ 1

mB(R). (5)

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164316-3 Diniz, Alijah, and Mohallem J. Chem. Phys. 137, 164316 (2012)

Considering now the rotational mass, it may, to a good ap-proximation, be left as a constant, 1

μrot(R) ≈ 1mA

+ 1mB

, sincethe rotational non-adiabatic effects are much smaller than thevibrational ones. Such an approach would work well for lowvalues of J. However, when the molecule rotates more vigor-ously, centrifugal stretching of the bond occurs until it even-tually breaks, and, thus, at large R, the nuclear rotational massapproaches the atomic mass. This suggests a parametrizationof the rotational mass that makes it pass from the nuclear massnear the equilibrium distance to the atomic mass at dissocia-tion. The following formula:

mA;rot(R) = mA + a{1 − [1 + eα(R−RT )]−1}, (6)

satisfies these requirements. Here, a is the number of electronsof the atomic fragment (for example, a = 1 for H2, a = 1

2 forH+

2 ), while α and RT, the turning points of this function, areempirical parameters. The effective, R-dependent rotationalreduced mass in Eq. (1) is then obtained in the usual way, buttaking the rotational nuclear masses defined above.

The separation of the motions of atomic cores andvalence electrons has led us to a physically motivatedparametrization of the reduced vibrational and rotationalmasses. This, however, is not restricted to any particular par-titioning of the molecular Hamiltonian and is expected to per-form well also in conventional Born-Oppenheimer type calcu-lations. It is much less demanding than the rigorous formalismof Bunker and Moss.15 Schwenke, who applied their methodto H+

2 , HD+,23 and to H2 and water24 had to overcome somesevere difficulties. In the present work, we have studied withour theory H2 and H+

2 and some of their deuterated isotopo-logues.

III. RESULTS AND DISCUSSION

The conventional electronic Schrödinger equation wassolved with the GAMESS package46, 47 at the FCI/cc-pV5Zlevel of theory, and ab initio energies and Mulliken popula-tions generated on an equidistant grid for a range of internu-clear distances, R. For H2, this range was 0.2a0 ≤ R ≤ 12.0a0

and spacing �R = 0.05a0. For H+2 , the range was extended to

0.2a0 ≤ R ≤ 32.0a0 with the same increment.The rovibrational wavefunctions were computed with

the Fourier grid Hamiltonian method, implemented in theFGHEVEN (Fourier Grid Hamiltonian with an even num-ber of grid points) computational routine,49 and the non-adiabatic corrections evaluated as difference between calcu-lations with effective masses and nuclear masses. Our ownBorn-Oppenheimer potential energy curves were employedin this work by default. In a second series of calculations,we used highly accurate curves taken from the literature,from Wolniewicz9, 48 for H2 and from Bishop and Wetmore13

for H+2 . The calculated non-adiabatic shifts do not depend

strongly on the quality of the potential curves, the differencebeing of the order of only 0.01 cm−1.

The effective mass problem was solved iteratively foreach vibrational state v,

m(i+1)A,v = mA +

∫nAA(R)

[χ (i)

v (R)]2

dR, (7)

where χ (i)v (R) is the vibrational wave function computed with

the mass m(i)A,v . Starting from the nuclear mass, this proce-

dure generally converges in the first step. The results wereconfirmed through direct, non-iterative calculations with ourown extended Numerov code which can handle coordinate-dependent masses. The iterative method lends itself to com-putations with numerical codes that cannot cope with variablemasses and might be a good choice for applications to poly-atomic molecules.

It is well known that Mulliken populations depend morestrongly on the basis set used in the electronic structure calcu-lation than the electronic energies themselves. Small errors ofthe populations can be corrected posterior through light scal-ing of the calculated rovibrational energy shifts.

A. Vibrational states

As we have discussed above, far away from regions withstrong non-adiabatic coupling, the non-adiabatic correctioncan be interpreted as due to a partial participation of the elec-trons in the nuclear motion, which would increase their vibra-tional mass. In a recent publication, Kutzelnigg39 applied theLCAO theory to this problem and derived functional forms forthe coordinate-dependent vibrational and rotational masses.In his work, the vibrational reduced mass of a homonucleardiatomic molecule AA is obtained as, using our notation,

1

μvib(R)= 1

μ

(1 + A(R)

mA

)+ O

(1/m2

A

), (8)

where A(R) is an R-dependent correction function whose ex-plicit form, which is different for H2 and H+

2 , is given inRef. 39. We note here that A(R) is negative. In order to re-late his formula to ours, we convert the above expression intothe corresponding one for the effective nuclear mass, whichis, of course, just twice the reduced mass, to obtain

mA(R) = m2A

mA + A(R)= mA − mAA(R)

mA + A(R). (9)

Comparing this expression with Eq. (4), we find

nAA(R) = − mAA(R)

mA + A(R)= −A(R) + O(A(R)2/mA).

(10)These functions are displayed in Figure 1 for H+

2 . They con-verge to the asymptotic value of 1

2 , signifying that half ofan electron mass is attached to the nucleus. Near the equi-librium distance, Req = 2.0 a0, the two values are practicallyidentical. However, on its approach to the asymptotic value,Kutzelnigg’s correction function passes through a maximumand hence attains values larger than the asymptotic value.In contrast, our function increases monotonously so that theeffective nuclear mass never exceeds the atomic mass. InTable I we compare the non-adiabatic corrections to the vi-brational levels of H+

2 obtained here with exact ab initio data.Wolniewicz and Poll11 were chosen as a reference, as they re-port the non-adiabatic shifts of the rovibrational energies withreference to such energies obtained within the traditional adi-abatic approximation. Hence these data can be used directly.Moss,17 in contrast, presents shifts within his partitioned

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164316-4 Diniz, Alijah, and Mohallem J. Chem. Phys. 137, 164316 (2012)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12

frac

tio

n o

f th

e el

etro

nic

mas

s

R/a0

nAA(R)

-A(R)

FIG. 1. H+2 : Fraction of the electronic mass attached to the core according to

the present model [nAA(R)] and that of Ref. 40 [−A(R)], as a function of theinternuclear distance.

adiabatic approach. Since in this approach the molecule dis-sociates to the exact limit while in the traditional adiabaticapproach it dissociates to the adiabatic limit, there is a smalldifference of the order of (me/mp)2.

Table I also provides a comparison with other models, inparticular those by Jaquet and Kutzelnigg,40 by Moss17 usinghis constant, optimized vibrational mass, and by Alijah, An-drae, and Hinze.38 Only the highest vibrational state, v = 19,has been excluded as it is located only 0.741 cm−1 below dis-sociation and would require an extended coordinate range of

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0 2 4 6 8 10 12 14 16 18

no

nad

iab

atic

-co

rrec

tio

ns

/ cm

-1

v

exactΔSCM

ΔMM

FIG. 2. H+2 (J = 0): Non-adiabatic corrections to the vibrational states. Com-

parison of our scaled core mass model (SCM) results with those obtained withthe constant Moss mass (MM), here μvib = 0.5037518 u, and the exact databy Wolniewicz and Poll.11

up to 80 a050 for a reliable calculation, while its non-adiabatic

shift is extremely small. As the mean deviations of thedifferent models show, see Table I, the present method givesmore accurate results than can be obtained with an optimizedconstant mass or with the empirical correction by Alijahet al. When we scale our data with a constant factor, themean deviation is lowered considerably, making the qualityof our data comparable to those of Jaquet and Kutzelnigg. AsFigure 2 demonstrates, our scaled data are almost on top

TABLE I. H+2 (J = 0): Non-adiabatic shifts of the vibrational energy levels. The exact data are from Wolniewicz

and Poll.11 The other columns show deviations from the exact data, �X = X − exact, where X = AM denotesthe atomic mass (which means here the proton mass plus one half of the electron mass), MM Moss mass, i.e.,an optimized vibrational mass (here μvib = 0.5037518 u17), AH Alijah et al.,38 using their empirical formula,JK the Finkelstein-Horowitz (FH) approach of Jaquet and Kutzelnigg.40 The JK data were calculated from theirTable I as the difference between their FH and AD data, such that the difference becomes �JK = [their(“FH− exact”) − their(“AD − exact”)] − our “exact”. Finally, �CM compares with the present results using theeffective core mass, while �SCM are our scaled results (here with a factor of f = 1.082, see Table V). For each ofthe approximate methods, the root mean square (rms) is reported in the last line. All data are given in cm−1.

v exact �AM �MM �AH �JK �CM �SCM

0 − 0.132 − 0.022 0.004 0.001 0.035 0.034 0.0261 − 0.333 − 0.103 − 0.028 − 0.036 0.050 0.050 0.0272 − 0.516 − 0.167 − 0.049 − 0.062 0.057 0.063 0.0263 − 0.682 − 0.216 − 0.062 − 0.079 0.058 0.073 0.0234 − 0.831 − 0.251 − 0.065 − 0.085 0.056 0.081 0.0205 − 0.964 − 0.273 − 0.060 − 0.083 0.052 0.087 0.0166 − 1.079 − 0.283 − 0.049 − 0.073 0.048 0.092 0.0117 − 1.176 − 0.282 − 0.031 − 0.056 0.044 0.095 0.0068 − 1.254 − 0.272 − 0.009 − 0.033 0.042 0.096 0.0019 − 1.310 − 0.253 0.017 − 0.006 0.042 0.095 − 0.004

10 − 1.342 − 0.227 0.043 0.024 0.044 0.092 − 0.01011 − 1.346 − 0.197 0.069 0.053 0.048 0.087 − 0.01712 − 1.318 − 0.162 0.092 0.081 0.053 0.079 − 0.02213 − 1.254 − 0.122 0.115 0.104 0.059 0.072 − 0.02514 − 1.146 − 0.075 0.135 0.119 0.062 0.068 − 0.02015 − 0.988 − 0.022 0.152 0.123 0.062 0.070 − 0.00616 − 0.773 0.031 0.159 0.113 0.058 0.078 0.02117 − 0.495 0.081 0.152 0.080 0.047 0.095 0.06218 − 0.188 0.111 0.125 − 0.055 0.040 0.111 0.105rms 0.189 0.090 0.075 0.051 0.082 0.033

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164316-5 Diniz, Alijah, and Mohallem J. Chem. Phys. 137, 164316 (2012)

of the line that interpolates the exact results. Jaquet andKutzelnigg present results at two different levels of theirmodel: First, the LCAO approximation with a constantorbital exponent, and, second, the LCAO approximation withoptimized orbital exponent at each bond-length R, i.e., theFinkelstein-Horowitz (FH) approach.51 The latter improvesthe results of the simple LCAO approach by one order ofmagnitude, but is more difficult to implement. We note herethat our method yields the same precision as FH, but doesnot require any extra effort compared to standard ab initiocalculations.

In a recent publication,44 two of the present authors re-ported non-adiabatic shifts of H2. If we also scale their data,with a simple factor f = 1.239, they can be brought into ex-cellent agreement with the exact ones of Wolniewicz,9 seeFigure 3. This figure also shows what accuracy can be ob-tained with the best constant mass, optimized by Bunker andMoss,19 (μ = 0.5038920 u). In Table II we still compare ourresults to those obtained by Alijah and Hinze37, 38 and bySchwenke.24 Schwenke, calculating ab initio the correctionterms of the Bunker and Moss formalism, did approach theexact data to within 20% which he could improve by scal-ing. More recently, Pachucki and Komasa28 investigated thisproblem by means of their non-adiabatic perturbative theory,which is equivalent to the theory of Bunker and Moss, andobtained results in excellent agreement with Wolniewicz. Nodata are yet available from Kutzelnigg’s approach, though heshows in Ref. 39 a plot of the R-dependent correction of thevibrational mass. Mass asymmetry in molecules like HD+ orHD gives rise to an additional non-adiabatic term, see, forexample, Ref. 52, coupling states with gerade to those withungerade symmetry, which complicates the application ofmethods based on effective Hamiltonians. To the best of ourknowledge, Kutzelnigg’s formalism has not been applied tothese systems. Schwenke,23 utilizing the formalism of Bunkerand Moss,15 noticed that the errors of the HD+ data were big-

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

0 2 4 6 8 10 12 14

no

nad

iab

atic

-co

rrec

tio

ns

/ cm

-1

v

exactΔSCM

ΔMM

FIG. 3. H2 (J = 0): Non-adiabatic corrections to the vibrational states. Com-parison of our scaled results (SCM) with those obtained with the constantMM, here μvib = 0.5038920 u, and exact data by Wolniewicz.9

ger by one order of magnitude when compared to those forH+

2 . Our model does not present any particular difficulty. InTable III and Figure 4 we compare our data with the exactones by Wolniewicz and Poll,11 those obtained with calcula-tions using a constant mass, and those by Alijah et al. For thehighest two levels, where g/u coupling becomes appreciable,the quality of all the models deteriorates somewhat. Exceptfor those two levels, the accuracy of our predictions is thesame as for H+

2 . We note that the very weakly bound level,v = 22, which is located only 0.43 cm−1 below dissociation,has been omitted just as v = 19 for H+

2 .Satisfactory results were also obtained for HD, in par-

ticular after a scaling with the factor f = 1.250. These dataare presented in Table IV and Figure 5, where they arecompared with other empirical results and the exact ones of

TABLE II. H2 (J = 0): Exact values of the non-adiabatic corrections,9 deviations according to different models(atomic mass, Moss mass, Alijah Hinze, Schwenke, Schwenke scaled, core mass, scaled core mass) and their rootmean squares. The vibrational Moss mass is μvib = 0.5038920 u. For the notation, see Table I.

v exact �AM �MM �AH �S �SS �CM �SCM

0 − 0.499 − 0.089 − 0.045 − 0.034 0.059 − 0.003 0.146 0.0621 − 1.335 − 0.321 − 0.197 − 0.168 0.155 − 0.010 0.325 0.0832 − 2.091 − 0.507 − 0.313 − 0.267 0.251 − 0.006 0.454 0.0633 − 2.773 − 0.645 − 0.389 − 0.329 0.363 0.025 0.585 0.0624 − 3.382 − 0.733 − 0.425 − 0.352 0.482 0.076 0.707 0.0675 − 3.921 − 0.768 − 0.417 − 0.334 0.611 0.147 0.779 0.0276 − 4.385 − 0.750 − 0.366 − 0.274 0.745 0.235 0.893 0.0577 − 4.765 − 0.678 − 0.271 − 0.174 0.895 0.354 0.955 0.0438 − 5.046 − 0.557 − 0.138 − 0.038 1.046 0.486 1.012 0.0469 − 5.198 − 0.395 0.023 0.122 1.178 0.615 1.003 − 0.001

10 − 5.177 − 0.210 0.193 0.286 1.277 0.731 0.975 − 0.03011 − 4.918 − 0.025 0.345 0.426 1.308 0.802 0.914 − 0.04412 − 4.327 0.129 0.443 0.502 1.217 0.782 0.758 − 0.09613 − 3.234 0.172 0.401 0.417 0.894 0.567 0.478 − 0.18114 − 1.648 0.259 0.363 0.289 0.468 0.303 0.315 − 0.005rms 0.487 0.318 0.299 0.837 0.449 0.740 0.071

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164316-6 Diniz, Alijah, and Mohallem J. Chem. Phys. 137, 164316 (2012)

TABLE III. HD+ (J = 0): Exact values of the non-adiabatic corrections11

and deviations according to different models. The vibrational Moss mass isμvib = 0.671532691 u. All data are in cm−1.

v exact �AM �MM �AH �CM �SCM

0 − 0.086 − 0.026 − 0.007 − 0.002 0.014 0.0151 − 0.219 − 0.099 − 0.044 − 0.030 0.013 0.0152 − 0.342 − 0.161 − 0.074 − 0.051 0.011 0.0153 − 0.456 − 0.212 − 0.097 − 0.065 0.008 0.0134 − 0.560 − 0.252 − 0.113 − 0.074 0.005 0.0115 − 0.655 − 0.283 − 0.122 − 0.077 0.001 0.0096 − 0.741 − 0.305 − 0.125 − 0.075 − 0.002 0.0077 − 0.816 − 0.318 − 0.123 − 0.068 − 0.006 0.0048 − 0.882 − 0.324 − 0.116 − 0.057 − 0.009 0.0029 − 0.937 − 0.321 − 0.105 − 0.042 − 0.012 − 0.001

10 − 0.980 − 0.312 − 0.090 − 0.024 − 0.015 − 0.00311 − 1.010 − 0.298 − 0.073 − 0.004 − 0.018 − 0.00612 − 1.026 − 0.278 − 0.054 0.017 − 0.021 − 0.00813 − 1.024 − 0.254 − 0.034 0.038 − 0.024 − 0.01114 − 1.005 − 0.225 − 0.014 0.057 − 0.026 − 0.01415 − 0.963 − 0.191 0.007 0.074 − 0.027 − 0.01516 − 0.897 − 0.151 0.029 0.086 − 0.023 − 0.01217 − 0.803 − 0.105 0.052 0.092 − 0.014 − 0.00418 − 0.677 − 0.054 0.071 0.090 0.002 0.01019 − 0.515 − 0.003 0.086 0.078 0.024 0.03020 − 0.314 0.045 0.091 0.042 0.051 0.05521 − 0.066 0.024 0.031 0.031 0.024 0.024rms 0.222 0.081 0.060 0.019 0.017

Wolniewicz.9 Kutzelnigg’s approach seems to not have beenapplied to HD.

B. Rovibrational states J > 0

To illustrate the validity of our scaled core mass model(SCM) also for rotationally excited states, we have computedthe non-adiabatic corrections for J = 5 and J = 10 forour four molecules H2, H+

2 , and their isotopologues. Asalready for J = 0 the SCM model outperforms the other

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0 2 4 6 8 10 12 14 16 18 20 22

no

nad

iab

atic

-co

rrec

tio

ns

/ cm

-1

v

exactΔSCM

ΔMM

FIG. 4. HD+ (J = 0): Non-adiabatic corrections to the vibrational states.Comparison of our scaled results (SCM) with those obtained with the con-stant MM, here μvib = 0.671532691 u, and the exact data by Wolniewicz andPoll.11

TABLE IV. HD (J = 0): Exact values of the non-adiabatic corrections9

and deviations according to different models. The vibrational Moss mass isμvib = 0.6716882 u. All data are in cm−1.

v exact �AM �MM �AH �CM �SCM

0 − 0.371 − 0.054 − 0.022 0.020 0.116 0.0521 − 1.001 − 0.208 − 0.117 0.112 0.263 0.0782 − 1.574 − 0.339 − 0.195 0.185 0.386 0.0903 − 2.098 − 0.442 − 0.251 0.237 0.494 0.0934 − 2.574 − 0.517 − 0.284 0.268 0.587 0.0905 − 3.002 − 0.563 − 0.294 0.276 0.665 0.0816 − 3.382 − 0.580 − 0.281 0.261 0.732 0.0697 − 3.711 − 0.566 − 0.244 0.221 0.786 0.0558 − 3.986 − 0.522 − 0.182 0.159 0.830 0.0419 − 4.196 − 0.450 − 0.100 0.076 0.858 0.024

10 − 4.330 − 0.353 0.000 − 0.023 0.871 0.00711 − 4.365 − 0.240 0.108 − 0.131 0.859 − 0.01712 − 4.278 − 0.115 0.217 − 0.235 0.820 − 0.04513 − 4.023 0.002 0.305 − 0.318 0.738 − 0.08414 − 3.551 0.098 0.358 − 0.361 0.610 − 0.12615 − 2.796 0.159 0.358 − 0.341 0.442 − 0.14716 − 1.683 0.189 0.301 − 0.241 0.275 − 0.07717 − 0.224 0.023 0.038 − 0.024 0.024 − 0.026rms 0.360 0.231 0.221 0.632 0.076

models, having almost the same accuracy as the Jaquetand Kutzelnigg approach, we do not need to refer to thesemodels any more. It suffices to demonstrate that the rms dataof the high J values are comparable to those obtained forJ = 0. The only model we retained is the Moss massmodel, since it represents the best that can be achievedwith different, but constant, vibrational and rotationalmasses, and is a standard procedure. Within the MM model,we employed the optimized masses by Moss17 for H+

2(μvib = 0.5037518 u, μrot = 0.5036853 u) and by Bunkeret al.19 for H2 (μvib = 0.5038920 u, μrot = 0.5036382 u).The reduced masses for HD+ were calculated from thereduced masses of H+

2 and D+2 , and are (μvib

= 0.6715326916 u, μrot = 0.67144835 u). Similarly, thereduced masses for HD were calculated from the reduced

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

0 2 4 6 8 10 12 14 16 18

no

nad

iab

atic

-co

rrec

tio

ns

/ cm

-1

v

exactΔSCM

ΔMM

FIG. 5. HD (J = 0): Non-adiabatic corrections to the vibrational states. Com-parison of our scaled results (SCM) with those obtained with the constantMM, here μvib = 0.6716882 u, and the exact data by Wolniewicz.9

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164316-7 Diniz, Alijah, and Mohallem J. Chem. Phys. 137, 164316 (2012)

TABLE V. Root mean square values, in cm−1, of six models: core mass (CM), scaled core mass (SCM), coremass and rotational mass (CRM), scaled core mass and rotational mass (SCRM), scaled core mass and rotationalmass with scaling factor from J = 0 (SCRM (J = 0)), scaled core mass and rotational mass with unique scalingfactor (SCRMu), The scaling factor is given in parentheses.

Molecule J CM SCM CRM SCRM SCRM (J = 0) SCRMu

H2 0 0.740 0.071 0.740 0.071 0.071 0.071(1.24) (1.24) (1.24) (1.24)

5 0.833 0.090 0.743 0.063 0.063 0.063(1.28) (1.24) (1.24) (1.24)

10 1.055 0.176 0.720 0.054 0.054 0.054(1.39) (1.24) (1.24) (1.24)

HD 0 0.632 0.076 0.632 0.076 0.076 0.080(1.25) (1.25) (1.25) (1.24)

5 0.721 0.109 0.665 0.093 0.096 0.105(1.29) (1.26) (1.25) (1.24)

10 0.946 0.213 0.734 0.138 0.159 0.174(1.39) (1.28) (1.25) (1.24)

H+2 0 0.082 0.033 0.082 0.033 0.033 0.033

(1.08) (1.08) (1.08) (1.08)5 0.110 0.038 0.089 0.032 0.034 0.034

(1.12) (1.09) (1.08) (1.08)10 0.191 0.065 0.108 0.038 0.047 0.047

(1.21) (1.11) (1.08) (1.08)

HD+ 0 0.019 0.017 0.019 0.017 0.017 0.072(0.99) (0.99) (0.99) (1.08)

5 0.030 0.026 0.022 0.022 0.025 0.062(1.02) (1.00) (0.99) (1.08)

10 0.116 0.064 0.068 0.047 0.075 0.049(1.14) (1.06) (0.99) (1.08)

masses of H2 and D2 reported by Bunker et al.19 and are(μvib = 0.6716882 u, μrot = 0.6714063 u). Note that Bunkeret al. kept the rotational reduced masses for H2 and D2 equalto the nuclear reduced mass.

In our method, we also require different reduced vibra-tional and rotational masses for the core. Initially, a reducedvibrational core mass was obtained via Eq. (7) for each vi-brational state, while the reduced rotational core mass wasleft constant and equal to the nuclear reduced mass. Alreadyat this stage, our results, see the SCM rms data in Table V,are superior to those obtained with all the other models, notshown here. We noticed a slight dependence of the scalingfactor on the rotational quantum number J. Furthermore, ourpredicted non-adiabatic shifts deteriorate with increasing J.Jaquet and Kutzelnigg,40 in their studies on H+

2 , also notedloss of accuracy when J was increased, which they attributedto a shortcoming of the rotational mass. Analysing our owndata, we arrived at the same conclusion. A more sensible re-duced rotational core mass should be close to the reduced nu-clear mass near equilibrium and tend to the average mass ofthe atomic fragment at dissociation. We cured our results intu-itively by adding a simple correction function, Eq. (6), to therotational core mass, that switches smoothly between thesetwo values. We call this the core plus rotational mass (CRM)approach.

The functional form was inspired from the plots of therotational masses obtained fully ab initio by Pachucki andKomasa28 and from the LCAO approximation by Jaquet

and Kutzelnigg.40 The two parameters in this function, thesmoothness parameter, α, and the turning point, RT, were cho-sen as α = 1.5 a−1

0 and RT = 3.0 a0 for H2. With this simpleparametrization of the rotational core mass, our results im-proved significantly. As an example, we show in Table VIand Fig. 6 the new results for the hydrogen molecule andJ = 10. The rms is reduced dramatically by one order ofmagnitude to just rms = 0.054 cm−1 from its constant mass

TABLE VI. H2 (J = 10): Exact non-adiabatic corrections9 and deviationsof different models. The Moss masses used are μvib = 0.5038920 u, μrot

= 0.5036382 u.

v exact �MM �CM �SCM �CRM �SCRM

0 − 0.863 0.359 0.530 0.403 0.205 0.0491 − 1.648 0.231 0.692 0.330 0.353 0.0452 − 2.359 0.142 0.832 0.253 0.504 0.0633 − 2.998 0.096 0.953 0.178 0.617 0.0504 − 3.566 0.094 1.060 0.111 0.730 0.0555 − 4.060 0.137 1.154 0.052 0.795 0.0196 − 4.469 0.224 1.233 0.007 0.870 0.0137 − 4.775 0.348 1.294 − 0.026 0.935 0.0228 − 4.944 0.497 1.324 − 0.049 0.959 0.0109 − 4.924 0.649 1.304 − 0.067 0.938 − 0.011

10 − 4.631 0.768 1.212 − 0.084 0.852 − 0.04711 − 3.931 0.807 1.016 − 0.089 0.670 − 0.10612 − 2.551 0.713 0.716 0.020 0.415 − 0.093rms 0.466 1.055 0.177 0.720 0.053

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164316-8 Diniz, Alijah, and Mohallem J. Chem. Phys. 137, 164316 (2012)

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

0 2 4 6 8 10 12

no

n-a

dia

bat

ic c

orr

ecti

on

s / c

m-1

v

exactΔSCRM

ΔSCM

ΔMM

FIG. 6. H2 (J = 10): Non adiabatic corrections. Comparison of the SCM,scaled core mass and rotational mass (SCRM), and MM models with theexact values.9

value of rms = 0.176 cm−1. With this, the J = 10 data nowhave the same accuracy as the J = 0 data and, even more,the same scaling factor. The same procedure can be appliedto H+

2 after a straightforward adjustment of the two param-eters, RT and α. Since for H+

2 the equilibrium distance isReq(H+

2 ) = 2.0 a0 whereas for H2 this value is much smaller,Req(H2) = 1.4 a0, one would expect a corresponding change inthe turning point, RT, while the variation of the rotational masswith the distance should be smoother. The simple transfor-mations RT (H+

2 ) = RT (H2)Req(H+2 )/Req(H2) = 4.286 a0 and

α(H+2 ) = α(H2)/[Req(H+

2 )/Req(H2)] = 1.050 a−10 of the pa-

rameters for the two molecules give very satisfactory results.We note in particular that further adjustment of the scalingfactor does improve the results only marginally. Since thestandard error of the scaling factors is about 3 %, this im-provement is not significant and the scaling factors can safelybe truncated to two digits after the decimal point.

Comparable accuracy was obtained for the rovibrationalstates of the deuterated species HD and HD+. The verysimilar error bounds of all isotopologues suggests simul-taneous adjustment of the scale factors of the hydrogenicand deuterated species without significant loss in accuracy.Table V gives a summary of these results. Further details onthe rovibrational calculations are provided as supplementalmaterial:53 Tables VII–XIII contain the non-adiabatic correc-tions for the four molecules and for J = 5 and J = 10, whileFigures 7– 9 show the data graphically for J = 10.

The effective reduced vibrational and rotational massesaccurately account for non-adiabatic shifts of rovibrationalenergy levels. In our demonstration we used potential en-ergy curves obtained with standard methods which are read-ily available in any ab initio quantum chemistry program. Infact the non-adiabatic corrections, evaluated as difference be-tween calculations with two different masses, are known tobe largely independent of the quality of the potential energycurve. To compute not only shifts but the absolute energies ofthe rovibrational levels at high accuracy, our masses shouldbe used with more sophisticated potentials, obtained with spe-

cialised computational codes. The curves would also have toinclude relativistic and radiative corrections.

In view of future applications to polyatomic moleculeswe note here an advantage over the formalism of Jaquet andKutzelnigg.39, 40 Those authors obtain correction formulae di-rectly for the reduced vibrational and rotational masses. Asthe precise form of the reduced masses of a general poly-atomic molecule depends on the particular choice of the inter-nal coordinate system, their correction functions would needto be rederived. In our approach, we do not correct reducedmasses, but rather the nuclear masses. To apply it to a poly-atomic, we would choose as internal coordinates the 3N − 6bond lengths and parametrize each of them according to theformulae suggested here. The reduced masses, which are thenobtained through the usual transformations, become auto-matically coordinate dependent. Scaling of the so-obtainednon-adiabatic corrections may be done easily, as a singlerovibrational state is sufficient to determine the correspond-ing scaling factor. But even without scaling, we would ex-pect better results than can be obtained with constant reducedvibrational and rotational masses. Thus, our method is fullypredictive.

IV. CONCLUSIONS

Sophisticated ab initio methods have been developed byother researchers for the calculation of non-adiabatic cor-rections to the rovibrational states of the simplest of di-atomic molecules, H+

2 , H2, and their isotopologues. Whilethose methods manifest a triumph of ab initio theory, theyare highly specialised for the systems studied, and hence theirgeneralization to triatomic or even polyatomic molecules iscomplicated. We have sought here an approach of a differ-ent kind, empirical rather than ab initio, but neverthelessvery accurate and easily transferable. It is based on parti-tioning of the molecular Hamiltonian into one part that de-scribes an effective nuclear core, consisting of the proper nu-cleus and its attached fraction of the electrons, and anotherpart that describes the valence electrons. We have presenteda recipe of how to evaluate effective, R-dependent vibrationalmasses using the Mulliken density, and how to parametrizeefficiently the effective rotational core masses. The method isconceptually appealing, easy to implement, and does not addany additional cost to standard ab initio calculations whereMulliken population analyses are performed routinely. Themethod can be transferred easily to more complicated tri-atomic or polyatomic molecules, the most important being H+

3and isotopologues, hydrogenic clusters such as H+

3 · · · (H2)n

or H+2 · · · (H2)n or dihydrides such as water. Work along these

lines is in progress.

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