www.elsevier.com/locate/theochem

Journal of Molecular Structure: THEOCHEM 769 (2006) 39–46

Calculation of electronic energies and vibrational levelsof molecular ions from the Hamilton–Jacobi equation:

H2þ, D2

þ, T2þ, HD+ and DT+

J.A. Campos, D.L. Nascimento, M. Wolf, A.L.A. Fonseca *, O.A.C. Nunes

Instituto de Fısica – UnB, Caixa Postal 04455, 70919-970 Brasılia, DF, Brazil

Received 17 March 2006; accepted 24 April 2006Available online 11 July 2006

Abstract

Starting with the Hamilton–Jacobi equation we apply Hylleraas’ method in association with the series established by Wind-Jaffe, toH2þ, D2

þ, T2þ, HD+ and DT+ molecular ions in order to calculate the electronic energies as well as vibrational levels. In the later it is

necessary to have the exact value of the electronic energy as a function of the nuclear distance. The dissociation of these molecular ions isimportant to study proton injection in magnetic mirrors and in Tokomak leading to the controlled thermonuclear fusion.� 2006 Elsevier B.V. All rights reserved.

Keywords: Hamilton–Jacobi equation; Electronic energies; Vibrational levels; Hylleraas’ method; Molecular ions

1. Introduction

In spite of the progress brought about by quantummechanics, classical and semi-classical methods are stillimportant in the study of atoms and molecules. In specialthe Hamilton–Jacobi equation (HJE) has been recentlyused to describe: chemical reaction dynamics [1]; band edgefor periodic potential [2]; bound state of wave functions [3];the modified de Broglie approach for a single particle in anelectromagnetic field [4]; canonical transformation forfermionic systems [5]; Bohmian trajectories [6]; the WKBapproximation [7]; the multidimensional wave functionsand tunnel splitting in the vibrational spectrum of a non-rigid molecule such as H2

þ molecule [8], yielding the cor-rect value for the ground state energy [9]; the He atom,reproducing with good agreement the atomic ground-statebinding energy [10]. Furthermore, the Hylleraas’ methodhas been recently used for many electron atoms [11], open-ing new possibility to this method.

In this work we propose to obtain the exact solution of theHamilton–Jacobi equation through eigenvalue equations.

0166-1280/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2006.04.054

* Corresponding author. Tel.: +55 61 3307 2900; fax: +55 61 3307 2363.E-mail address: alaf@fis.unb.br (A.L.A. Fonseca).

We then apply this equation in association with the seriesestablished by Wind-Jaffe to molecular ions H2

þ, D2þ, T2

þ,HD+ and DT+ in order to calculate the electronic energiesand vibrational levels. For the later task it is necessary tohave the exact value of the electronic energy as a functionof nuclear distance. Such a study with these molecular ionsis important regarding controlled thermonuclear fusion pro-cess of Tokomak plasmas.

2. Method

To begin with, we define pm ¼ ð~p;H þ a/Þ and pmpm +

m2 = 0, and obtain the usual HJE given by:

p2 � ðH þ a/Þ2 þ m2 ¼ 0: ð1Þ

In previous works this approach was applied to the H2þ

molecule, yielding the correct value for the ground stateenergy [9] and reproducing the atomic ground-state bind-ing energy of the He atom [10].

In the current work we shall use the eigenvalue differen-tial equation which stems from the problem of HJE, Eq.(1), with H = E = constant. The relativistic Hamiltonianfor the electron is given by:

40 J.A. Campos et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 39–46

H rel ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2

1 þ m21

q� a

1

raþ 1

rb

� �; ð2Þ

where as usual a = e2(�h = c = 1). The non-relativistic ver-sion of the Hamiltonian equation (2) for the electron andthe system Hamiltonian in the adiabatic approximationare, respectively, given by:

H 1 ¼p2

1

2m1

� a1

raþ 1

rb

� �; ð3aÞ

H ffi H 1 þaR: ð3bÞ

Here ra and rb are the distances from protons a and b to theelectron and R is the distance between the protons. Theelectron problem can be separated only in the confocalelliptic coordinate system [12] defined by

n ¼ ra þ rb

R; g ¼ ra � rb

R; ð4Þ

where the g, f ranges are �1 6 g 6 +1 and �1 6 n <1.The Hamiltonian H1 in this new system of coordinates iswritten as

H 1 ¼H n þ H g

n2 � g2; ð5Þ

where Hn and Hg are given by:

H n ¼1

2m1

ðn2 � 1Þp2n þ

k2u

n2 � 1

" #� 2an;

H g ¼1

2m1

ð1� g2Þp2g þ

k2u

1� g2

" #: ð6Þ

In this work we will consider, instead of the direct solutionof the HJE problem with H1 = E1, the alternative varia-tional problem d �dV(H1 � E1)w2 = 0, where dV is the vol-ume element in the new coordinates, Here w(n,g,u) is anauxiliary action function defined by the relationspn ¼ 1

wowon and pg ¼ 1

wowog. Also, (H1 � E1)w2 in the volume

integral, is a quadratic form. By executing the indicatedvariation we get the corresponding eigenvalue differentialequation:

o

onðn2 � 1Þ ow

on

� �þ o

ogð1� g2Þ ow

og

� �

þ 1

n2 � 1þ 1

1� g2

� �o2w

on2

þ 1

2R2ðn2 � g2ÞE1 þ 2aRn

� �w ¼ 0; ð7Þ

13ð1� kÞð1þ kÞb2 þ k2b2 � k 1

15ð2þ kÞð1þ kÞb2

13ð2� kÞð1� kÞb2 11

21b2 � 2

21b2k2 � k� 6 1

63ð4þ

0 135ð4� kÞð3� kÞb2 39

77b2 þ 2

7

0 0

. . . . . .

������������

which has a solution that can be separated as a product offunctions of only one variable [12]:

wðn; g;uÞ ¼ eikuuY ðgÞQðnÞ; ð8Þyielding the following set of ordinary differential equa-tions:

d

dnðn2 � 1Þ dQ

dn

� �þ kþ 2aRnþ 1

2R2n2E1 �

k2u

n2 � 1

!Q ¼ 0;

ð9Þ

d

dgð1� g2Þ dY

dg

� �� kþ 1

2R2g2E1 þ

k2u

1� g2

!Y ¼ 0: ð10Þ

A solution to Eq. (10) is immediately found by Hylleraas’method [12], writing

Y ðgÞ ¼X1‘¼k

c‘Pku

‘ ðgÞ; ð11Þ

where P ku

‘ ðgÞ are Legendre polynomials. For ku = 0,1,2,3, . . . we will have molecular orbitals r,p,d,c, . . . Onthe other hand for ‘ = 0,2,4, . . . we obtain levels s,d,g,j, l, . . . (even atomic orbitals) and for ‘ = 1,3, 5, . . . levelsp, f, i, k, . . . (odd atomic orbitals).

With the recurrence rule determined by Legendre’sequation

d

dgð1� g2Þ dY

dg

� �þ nðnþ 1Þ �

k2u

1� g2

!Y ¼ 0; ð12Þ

we obtain the following recurrence relation for coeffi-cients c‘:

A‘�2c‘�2 þ B‘c‘ þ C‘þ2c‘þ2 ¼ 0: ð13Þ

Here,

A‘�2 ¼ð‘� kuÞð‘� 1� kuÞb2

ð2‘� 3Þð2‘� 1Þ ; ð14Þ

B‘ ¼ð‘þ 1� kuÞð‘þ 1þ kuÞb2

ð2‘þ 3Þð2‘þ 1Þ þ ð‘� kuÞðlþ kuÞb2

ð2‘þ 1Þð2‘� 1Þ� k� ‘ð‘þ 1Þ; ð15Þ

C‘þ2 ¼ð‘þ 2þ kuÞð‘þ 1þ kuÞb2

ð2‘þ 3Þð2‘þ 5Þ ; ð16Þ

and b2 ¼ 12R2E1. Eqs. (14)–(16) can be considered as an infi-

nite set of linear homogeneous equations, requiring thedeterminant of matrix coefficient to vanish:

0 0 . . .

kÞð3þ kÞb2 0 . . .

7b2k2 � k� 20 . . . . . .

0 . . . . . .

. . . . . . . . .

������������: ð17Þ

Table 1Comparison of potential energies E (in atomic units) in the Born–Oppenheimer approximation for H2

þ

R-1sr [15] This work R-2sr [15] This work

0.2500 �2.101445 �2.10144289336 0.5000 �1.534926 �1.534926005122.0000 �0.602635 �0.60263420472 2.0000 �0.139135 �0.139135124269.0000 �0.501194 �0.50119544445 9.0000 �0.101301 �0.10130134922

16.0000 �0.500035 �0.50003636946 20.0000 �0.118499 �0.1184992898618.0000 �0.500024 �0.50002195710 28.0000 �0.121519 �0.1215200369920.0000 �0.500015 �0.50001423235 32.0000 �0.122299 �0.1222991048616.0000 �0.500035 �0.50003636946 36.0000 �0.122843 �0.12284384379

R-5sr This work R-5gr This work

1.2500 �0.726684 �0.72668375634 0.75000 1.253281 �1.2532813062.5000 �0.332282 �0.33228169690 3.0000 0.252481 �0.252481140

10.0000 �0.048572 �0.04857154518 9.0000 0.021066 �0.02106594330.0000 �0.004412 �0.00441218625 12.0000 �0.019524 �0.01952430240.0000 �0.009862 �0.00986170848 15.0000 �0.050391 �0.05039083845.0000 �0.011550 �0.01154980516 18.0000 �0.068461 �0.068461345

100.0000 �0.017744 �0.01774429533 21.0000 �0.076403 �0.076403283

R-5fp This work R-9lp This work

1.0000 �0.919847 �0.91984741444 1.2500 �0.775305 �0.775305249653.0000 �0.251966 �0.25196595168 5.0000 �0.175254 �0.17525414254

10.0000 �0.013609 �0.01360871984 10.0000 �0.075088 �0.0750877773014.0000 �0.012707 �0.01270670587 15.0000 �0.041467 �0.0414671459724.0000 �0.031043 �0.03104344590 20.0000 �0.024375 �0.0243754501726.0000 �0.031843 �0.03184329969 22.5000 �0.018548 �0.0185476796128.0000 �0.032281 �0.03228063171 25.0000 �0.013785 �0.01378523464

R-6id This work R-8kd This work

1.0000 �0.944427 �0.94442718081 1.2500 �0.768744 �0.768744212163.0000 �0.277623 �0.27762258413 5.0000 �0.168657 �0.16865706577

10.0000 �0.042781 �0.04278052559 10.0000 �0.068374 �0.0683741749514.0000 �0.012872 �0.01287170766 15.0000 �0.034557 �0.0345569095216.0000 �0.003295 �0.00329498334 20.0000 �0.017194 �0.0171935228718.0000 �0.004203 �0.00420319471 22.5000 �0.011211 �0.0112106226020.0000 �0.010171 �0.01017064385 25.0000 �0.006293 �0.00629251333

R-5g/ This work R-5gc This work

1.0000 �0.920032 �0.920032422 1.2500 �0.720200 �0.720200183.0000 �0.253631 �0.253631360 5.0000 �0.122848 �0.12284811

10.0000 �0.023511 �0.023511160 10.0000 �0.028784 �0.0287844914.0000 �0.002007 �0.002007385 15.0000 �0.001644 �0.0016439816.0000 �0.009272 �0.009271688 20.0000 �0.009484 �0.0094835118.0000 �0.014527 �0.014527047 22.5000 �0.012567 �0.0125668320.0000 �0.018404 �0.018403945 25.0000 �0.014730 �0.01472984

Table 2aComparison of equilibrium distances R and electronic energies E1 for 1sr

R-1sr This work [23]

0.6 �1.6714847144446982 �1.67148471450.8 �1.5544800944452937 �1.55448009151.0 �1.4517863133784486 �1.517863133781 �1.45178631302.0 �1.1026342144949465 �1.026342144949 �1.10263421503.0 �0.9108961973823069 �0.910896197382 �0.91089619745.0 �0.7244202951676077 �0.7244202951676 �0.7244202952

10.0 �0.6005787289439264 �0.6005787289439 �0.600578728915.0 �0.5667156051634333 �0.5667156051634 �0.566715605220.0 �0.5500142593309156 �0.5500142593309 �0.550014259325.0 �0.5400058008027918 �0.5400058008028 �0.540005800830.0 �0.5333361240769541 �0.5333361240769

J.A. Campos et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 39–46 41

42 J.A. Campos et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 39–46

Eq. (9) on the other hand has been solved by Jaffe’smethod [13] starting from the trial solution

QðnÞ ¼ ðn2 þ 1Þk=2ðnþ 1Þryðn� 1Þðnþ 1Þ

� �e�bn; ð18Þ

where

r ¼ Rb� 1� k; ð19Þ

y ¼X1r¼0

grn� 1

nþ 1

� �r

; ð20Þ

whose recurrence relations are given by

Ar;r�1gr�1 þ Br;rgr þ Cr;rþ1grþ1 ¼ 0; ð21ÞAr;r�1 ¼ ðr � 1� rÞðr � 1� r� kÞ; ð22Þ

Table 2bComparison of equilibrium distances R and electronic energies E1 for 2pr

R-2pr This work [23]

0.6 �0.5243103718434056 �0.52430500000.8 �0.5427459207023233 �0.54274000001.0 �0.5648136251036241 �0.564813625101502.0 �0.6675343922023829 �0.66753439220240 �0.66753439223.0 �0.7014183333732244 �0.70141833337323 �0.70141833345.0 �0.6772916132282817 �0.67729161322830 �0.6772916132

10.0 �0.5999010686027201 �0.59990106860270 �0.599901068615.0 �0.5667087290356798 �0.56670872903567 �0.566708729020.0 �0.5500141976532722 �0.55001419765330 �0.550014197725.0 �0.5400058002843242 �0.54000580028430 �0.550014197730.0 �0.5333361240727691 �0.53333612407273

Table 2cComparison of distances R and electronic energies E1 for 3dr

R-2pp This work [24]

5 �0.1213848123677511 �0.1213858 �0.1345106312691379 �0.1345119 �0.1339140316900742 �0.133914

10 �0.1327162901456616 �0.13271612 �0.1299510306633994 �0.12995115 �0.1268786937434897 �0.12687918 �0.1254627975188663 �0.12546319 �0.1252358006344060 �0.12523620 �0.1250835494886558 �0.12508324 �0.1248668171805402 �0.12486730 �0.1248779527422754 �0.124878

Table 2dComparison of distances R and electronic energies E1 for 2pp

R-3dr This work [24]

5 �0.1060130778364009 �0.1060138 �0.1735116416597216 �0.1735129 �0.1750006907546308 �0.175001

10 �0.1731174404887226 �0.17311712 �0.1653295969212709 �0.16533015 �0.1524813909493626 �0.15248118 �0.1424735611882319 �0.14247419 �0.1399314273887813 �0.13993120 �0.1377660804134798 �0.13776624 �0.132170711938498630 �0.1288576424142943

Br;r ¼ k� b2 þ 2brþ ðrþ kÞðk þ 1Þ þ 2rðr� 2bÞ � 2r2;

ð23ÞCr;rþ1 ¼ ðr þ 1Þðr þ k þ 1Þ: ð24ÞThe infinite system represented by Eq. (21) was then solvedby the same method as the one used for Eq. (13), i.e., bysolving the determinant:

B00 C01 0 0 0 . . .

A10 B11 C12 0 0 . . .

0 A21 B22 C23 0 . . .

0 0 A32 B33 C34 . . .

0 0 0 A43 B44 . . .

. . . . . . . . . . . . . . . . . .

�����������������

�����������������

¼ 0: ð25Þ

However, this time only a numerical solution is possible[14] and the method of Newton–Raphson was used. Eqs.(21) and (13) together allow us to find the electronic energyE = E1 + 1/R and the parameter k.

3. Results

In our calculations we worked with matrices of order 22in Eqs. (17) and (25) and found all states described in theliterature with a reasonably good agreement as shown inTable 1, where the electronic energies for some states arepresented. The comparison of equilibrium distances R

and electronic energies E1 for 1sr, 2pr, 3dr and 2pp statesare shown in Tables 2a–2d, respectively. In Tables 3a and3b we have the equilibrium distances Re and equilibriumenergies Ve; Table 4 shows a comparison between valuesfound in this work and those found in the literature forequilibrium distance Re, equilibrium energy Ve, and vibra-tional frequencies xe of H2

þ ion. Tables 5a–5c show vibra-tional frequencies xe for various states of ions H2

þ, D2þ,

T2þ, HD+ and DT+. These results were determined using

a potential energy curve obtained with Maple’s linear leastsquare fitting of electronic energy points in the neighbor-hood of the equilibrium point Re. To accomplish the linearleast square fitting we used 21 points separated by 0.05 a.u.to fit a degree-7 polynomial. The vibrational frequenciescan be determined from our polynomial by calculating

the force constant e, given by e ¼ ðd2EdR2 ÞRe

, and using the def-

inition xe ¼ffiffiel

q. The ions reduced masses used in this work

were: lH þ2¼ 918:5477060, lHDþ ¼ 1224:416554, lD þ

2¼

1835:684280, lDTþ ¼ 2201:331125, lT þ2¼ 2748:874857.

We have also found here that the results are in goodagreement with those found in the literature and the readercan look at Table 3a, 3b and 4 for this information. We cansee for instance in the 2pp case, if compared to reference[18] where the obtained frequency has taken the adiabaticcorrections to the electronic energy, that we have a0.00218% error.

Table 3aEquilibrium distance Re and equilibrium energies Ve (in atomic units) for R orbital in H2

þ

R Re Ve

1s 1.9974641068391440421 �.602634336541191675612p 12.546083224827289151 �0.50006079056395475843d 8.8341646124899194457 �0.1750490359090718154d 17.849217053120482328 �0.05882062666225727234f 20.921041131893873245 �0.130655086618111802605g 23.900267127886817853 �0.07824535362239550386i 40.520590341736538754 �0.06063995570395146472017g 49.306611518158863312 �0.0207367835112425434807i 56.081465707432845642 �0.03267896081467709757j 47.361115152053330012 �0.043596961882422362378j 59.675815112201862826 �0.025486461370839886908k 68.170533135867631476 �0.0351997154414115152569k 84.548563429776531563 �0.021809880020849795529l 79.234081509632920844 �0.02762761613215976231010o 103.94118756496270819 �0.02304148343489525171

Table 3bEquilibrium distance Re and equilibrium energies Ve (in atomic units) for P, U and K orbitals in H2

þ

P Re Ve

2p 7.9307152212765269113 �0.13451381661558800904f 18.607803077507718920 �0.07124680574143678575f 31.455255621100588576 �0.03250735500088449595g 35.656842238682074557 �0.05826796664354467216i 39.107498805660154573 �0.04160045012618739587i 52.069214228638131513 �0.02348082157653059437j 59.768368831110534332 �0.03420356812816731038j 80.072993323045967497 �0.02084069028025225888k 68.325481935784413434 �0.02682656672295463659l 92.618497778697960824 �0.0225520566436224370U4f 32.474124862908914799 �0.03125685627114609306i 48.641098319838044811 �0.02331786680144528117j 76.633149154887131835 �0.0208353401141923669D3d 17.969639580 �0.057033503397

Table 4Comparison between this work and others concerning equilibrium distances Re, equilibrium energies (Ve) and vibrational frequencies xe for H2

þ ion

State Re Ve xe

1sr 1.9974641068391440421 �0.60263433654119167561 2322.49590693486301181.997193320 [16] �0.6026346191065 [16] 2324.38 [16]

2pr 12.546083224827289151 �0.5000607905639547584 28.4394335345259112.5 [19]

3dr 8.8341646124899194457 �0.175049035909071815 437.359044880591538558.83416450 [17] �0.1750490358955 [17] 437.478 [17]

437.156 [18]4dr 17.849217053120482328 �0.0588206266622572723 89.066162492637150579

17.849217047 [19] �0.0588206266622 [19] 89.089026 [19]5gr 23.900267127886817853 �0.0782453536223955038 133.5885612258425

23.9002671274 [19] �0.0782453536223 [19] 133.62285 [19]6ir 40.520590341736538754 �0.060639955703951464720 43.84979118064506

40.5205903415 [19] �0.06063995570396 [19] 43.861048 [19]2pp 7.9307152212765269113 �0.13451381661558800909 266.03882222547398582

7.93071496 [20] �0.1345138166227 [20] 266.111 [20]266.033 [18]

4fp 18.607803077507718920 �0.071246805741436785701 129.24926235103118.607803076 [19] �0.07124680574143 [19] 129.28244 [19]

5gp 35.656842238682074557 �0.0582679666435446721 35.471218932710135.656842239 [19] �0.05826796664355 [19] 35.480325 [19]

3dd 17.969639580 �0.05703350339780 71.21368923592817.969598578 [19] �0.05703350663903 [19] 71.248226 [19]

J.A. Campos et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 39–46 43

44 J.A. Campos et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 39–46

Proceeding further, by using Eqs. (11) and (18) into Eq.(8) we find the wave function according to:

Wðg; n;/Þ ¼ N eikuuX1‘¼k

c‘Pku

‘ ðgÞðn2 þ 1Þk=2ðnþ 1Þr

�X1r¼0

grn� 1

nþ 1

� �r

e�bn; ð26Þ

where N is the normalization constant which is to be deter-mined by the condition

Z 1

�1

Z 1

�1

Z 2p

0

N 2jWðg; n;/Þj2 dV ¼ 1: ð27Þ

Table 5aVibrational frequencies (in cm�1) for R orbitals

R H2þ HD+ D2

þ

1s 2322.495906934863 2011.598112453338 1642.88372p 28.43943353452591 24.63242697068786 20.11743d 437.3590448805915 378.8125638969608 309.37834d 89.06616249263715 77.14344030431744 63.00344f 87.18517199911855 75.51424607624305 61.67295g 133.5885612258425 115.7059079436297 94.49766i 43.84979118064506 37.97989779317727 31.01847g 21.82795495120848 18.90598508589706 15.44067i 17.49197420636991 15.15043457839904 12.37347j 55.46070253874628 48.03653010072109 39.23178j 29.08031494221747 25.18751764971524 20.57078k 23.96613581862956 20.75794124398759 16.95319k 12.85539814462230 11.13452754226259 9.09369l 27.61201313774619 23.91576809374537 19.532110o 14.29591165568505 12.38220864738956 10.1126

Table 5bVibrational frequencies (in cm�1) for P orbitals

P H2þ HD+ D2

þ

2p 266.038822225473 230.425892691608 188.194f 129.249262351031 111.947483483170 91.425f 39.0050795111449 33.7837169428054 27.595g 35.4712189327101 30.7229118632242 25.096i 56.5865837106851 49.0116967021159 40.027i 28.1484778595531 24.3804196862680 19.917j 22.4327782709127 19.4298445444442 15.868j 9.90157555652576 8.57611444668090 7.008k 28.3797995455048 24.5807758054932 20.079l 13.9512674832548 12.0836998040979 9.86

Table 5cVibrational frequencies (in cm�1) for U orbitals

U H2þ HD+ D2

þ

4f 25.851837027262 22.391215593626 18.286i 27.405795851935 23.737155807862 19.387j 9.7389995392890 8.4353014495827 6.88

The elliptic confocal coordinates (g,n,/) relate to Carte-sian coordinates (x,y,z) through transformation equationsgiven by

x ¼ R2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� g2Þðn2 þ 1Þ

qcosð/Þ;

y ¼ R2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� g2Þðn2 þ 1Þ

qsinð/Þ; ð28Þ

z ¼ R2

gn;

dV ¼ R3

8ðn2 � g2Þdndgd/:

The expansion coefficients are determined from Eqs. (21)and (13) by the continuous fraction method:

DT+ T2þ

02159861 1500.248249264250 1342.5433917835414421724039 18.37084416073252 16.43971618801903910150413 282.5181045196970 252.82003459636998962344988 57.53351553786470 51.485639879477621736431608 56.31846379703805 50.398313374599587785492885 86.29337279018133 77.222284679210860010080820 28.32537713109983 25.347836870187582632521393 14.10006842328006 12.617880870485284671154754 11.29918188485224 10.111421212794720931719438 35.82560539214463 32.059647328004868995471611 18.78483034129126 16.810184483030492952871458 15.48125583178332 13.8538789988017632429453592 8.304121657427494 7.4311992440007024480696152 17.83636054852561 15.961417051901462073552531 9.234641218973485 8.263903357397635

DT+ T2þ

0146587971 171.851444856511 153.78656283965781510674149 83.4903428599535 74.713906941296913551619046 25.1958688384673 22.547299905811115781191635 22.9131228796433 20.504514317730881334669308 36.5528838566781 32.710474880728016284243959 18.1828973313374 16.271526174176484653460110 14.4907623848027 12.9675054052975416179806721 6.39605923492897 5.7237107682153352604147350 18.3323227634037 16.4052441281773882685315968 9.01201356449038 8.06467813817065

DT+ T2þ

7034046503 16.699350523980 14.9439285825146271129875 17.703151649822 15.84221096266391590166703 6.2910410153043 5.62973197714326

Fig. 1. Wave function for 1sr orbital for R = 2.0 a.u.

J.A. Campos et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 39–46 45

c‘c‘�2

¼ �A‘�2

B‘ þ C‘þ2c‘þ2

c‘

;

gr

gr�1

¼ �Ar;r�1

Br;r þ Cr;rþ1grþ1

gr

;

‘!1 c‘þ2

c‘¼ 0;

r!1 grþ1

gr¼ 0: ð29Þ

By making use of Maple program the expansion coeffi-cients and normalization constants for any wave functionare shown in Table 6. As an illustration we show in Figs.1 and 2, the wave functions for the orbitals 1sr and 2pp,respectively. We also found that the expansion coefficientsagree with those determined by Wallis and Hurbert [21] forthe ground state and also with those obtained by Bateset al. [22].

4. Conclusions

The dissociation of molecular ions H2þ, D2

þ, T2þ, HD+

and DT+ is very important to study proton injection into

Table 6The expansion coefficients and normalization constants for any wave function

1sr R (2.0) a.u. 2pr R (12

N 1.25090139237 N 76.86c0 1 c1 1c2 0.26064890765 c3 0.99c4 0.01001020708 c5 0.37c6 0.00016042570 c7 0.07c8 0.00000141695 c9 0.01c10 0.00000000793 c11 0.00c12 0.00000000003 c13 0.00g0 1 g0 1g1 0.01677993608 g1 0.02g2 0.00040377408 g2 0.00g3 0.00003488830 g3 0.00g4 0.00000463367 g4 0.00

3dr R (8.85) a.u. 4fr R (20

N 0.30590942183 N 2.86c0 1 c1 1c2 �3.5336628784 c3 �2.14c4 �1.0934314345 c5 �1.74c6 �0.1078759246 c7 �0.47c8 �0.0054005251 c9 �0.06c10 �0.0001652173 c11 �0.00c12 �0.0000034238 c13 �0.00g0 1 g0 1g1 0.21691016710 g1 0.24g2 0.00306701280 g2 0.00g3 0.00000845793 g3 0.00g4 0.00000021623 g4 0.00

5gr R (23.9) a.u.

N 0.27883544680 c8 0.73c0 1 c10 0.08c2 �3.6724451607 c12 0.00c4 6.24031924405 g0 1c6 3.63162443154 g1 0.49

magnetic mirrors and Tokomak leading to the controlledthermonuclear fusion. In order to accomplish that, it isimportant to know the exact excited states of theses ions.To determine the exact value of the atomic energy andvibrational levels we started in the present paper with aneigenvalue differential equation obtained from HJE. Thesolution of this eigenvalue equation was determined byusing the method developed by Hylleraas in associationwith the series established by Wind-Jaffe in FORTRAN

s

.55) a.u. 2pp R (7.95) a.u.

975252204836 N 1.045173077998c1 1

5531402569822 c3 0.13640414590767968402279690068 c5 0.00823779270859288142861610494 c7 0.00027377948481560137291985417 c9 0.00000575622761360900889734348 c11 0.00000008349444040058178804911 c13 0.0000000008861886

g0 16888201986717 g1 0.11219656275341550009346553226 g2 �0.000705253759010000130633029 g3 �0.000004181223170000004499769 g4 �0.00000015904138

.9) a. u 4fp R (18.6) a.u.

4720736085051 N 0.7526253092671360c1 1

139427639295 c3 �0.81003972943248699494895721 c5 �0.220933669140730423703673021 c7 �0.024283483396169803392693726 c9 �0.001530394361820614544227304 c11 �0.000063585965311038429624035 c13 �0.000001885356423

g0 16211344377862 g1 0.35308450158384999159324136563 g2 0.01862373659171080013978715878 g3 �0.0000000487632950000052530255 g4 �0.000000000000087

632009623 g2 0.04955917793097788589 g3 0.00082405285570397987 g4 0.00000005

057629575

Fig. 2. Wave function for 2pp orbital for R = 7.95 a.u.

46 J.A. Campos et al. / Journal of Molecular Structure: THEOCHEM 769 (2006) 39–46

and the electronic energy as given in Ref. [15]. For the sym-bolic part of the calculation we used Maple codes obtainingfor all molecular orbitals. We found that our results are ingood agreement with those found in the literature for H2

þ

ion which makes us feel confident that those results thatcould not be found in any other work are also reliable.In other words, we are able to determine any state andits wave function, which shows that the present proposedmethod is quite reliable.

Acknowledgements

ALAF and OACN wish to thank the CNPq (Brazilianagency) for Research Grants and also to FINATEC(Brasilia Founding Agency).

References

[1] B.K. Dey, M.R. Janicki, P.W. Ayers, J. Chem. Phys. 121 (2004) 6667.[2] S.S. Rajani, A.K. Kapoor, P.K. Panigrahi, Mod. Phys. Lett. A 19

(2004) 2047.[3] S.S. Rajani, A.K. Kapoor, P.K. Panigrahi, Mod. Phys. Lett. A 19

(2004) 1457.[4] J. Baker-Jarvis, P.P. Kabos, Phys. Rev. A 68 (2003) 042110.[5] C. Ramirez, P.A. Ritto, Rev. Mex. Fis. 49 (2003) 415.[6] N.C. Dias, J.N. Prata, Phys. Lett. A 302 (2002) 261.[7] E.M. Rabei, K.I. Nawafleh, H.B. Ghassib, Phys. Rev. A 66 (2002)

024101.[8] V.A. Benderskii, E.V. Vetoshkin, Russ. J. Phys. Chem. 74 (2000)

1468.[9] D.L. Nascimento, A.L.A. Fonseca, Int. J. Quantum Chem. 95 (2003)

274.[10] D.L. Nascimento, A.L.A. Fonseca, O. Portilho, Int. J. Quantum

Chem. 103 (2004) 505.[11] M. Belen-Ruiz, Int. J. Quantum Chem. 101 (2005) 246, ibid 261.[12] E.A. Hylleraas, Z. Phys. 71 (1931) 739 (English translation in H.

Hettema, Quantum Chemistry: Classic Scientific Papers, WordScientific Series, vol. 8, Singapore, 2000, p. 175, Chapter II).

[13] G. Jaffe, Z. Phys. 87 (1934) 535.[14] H. Wind, J. Chem. Phys. 42 (1965) 2371.[15] T.E. Sharp, Atomic Data 2 (1971) 119.[16] C.L. Beckel, B.H. Hansen, J.M. Peek, J. Chem. Phys. 53 (1970) 3681.[17] M. Shafi, C.L. Beckel, J. Chem. Phys. 59 (1973) 5294.[18] D.M. Bishop, S.K. Shih, C.L. Beckel, F.M. Wu, J.M. Peek, J. Chem.

Phys. 63 (1975) 4836.[19] C.L. Beckel, J.M. Peek, J. Chem. Phys. 104 (1996) 2801.[20] C.L. Beckel, M. Shafi, J.M. Peek, J. Chem. Phys. 59 (1973) 5288.[21] R.F. Wallis, H.M. Hurbert, J. Chem. Phys. 22 (1953) 774.[22] D.R. Bates, K. Ledsham, A.L. Stewart, Philos. Trans. R. Soc. Lond.

Ser. A 246 (1953) 215.[23] J.M. Peek, J. Chem. Phys. 43 (1965) 3004.[24] M. Kroll, J. Mol. Spectrosc. 35 (1970) 436.