Ortho-para transition in molecular hydrogen

Ortho-para transition in molecular hydrogen

Krzysztof Pachucki

*

Institute of Theoretical Physics, University of Warsaw, Hoża 69, 00-681 Warsaw, Poland

Jacek Komasa†

Faculty of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznań, Poland

共Received 22 December 2007; published 14 March 2008兲

The radiative ortho-para transition in molecular hydrogen is studied. This highly forbidden transition is very sensitive to relativistic and subtle nonadiabatic effects. Our result for the transition rate in the ground vibra-tional level ⌫共J=1→J=0兲=6.20共62兲⫻10−14_yr−1 _{is significantly lower in comparison to all the previous} approximate calculations. Experimental detection of such a weak line by observation of, for example, cold interstellar molecular hydrogen is at present unlikely.

DOI:10.1103/PhysRevA.77.030501 PACS number共s兲: 31.15.ac, 31.30.J-, 33.70.Ca, 95.30.Ky

The hydrogen molecule in the ground electronic state can exist in a nuclear triplet state共S=1, ortho-H2兲 with the odd angular momentum L, or in a singlet state 共S=0, para-H2兲 with the even L. The question we raise is, what are the physi-cal mechanisms for possible transitions between these two classes of states? The nonradiative transition, for example, in interstellar molecular hydrogen is mostly induced by colli-sions with atomic H. The corresponding rates were obtained by Sun and Dalgarno in关1兴. The radiative transition which is

much weaker can, in principle, take place at sufficiently low densities and temperatures. The relativistic spin-orbit inter-action共nuclear spin and the electron momenta兲 is the most obvious source of this transition as it mixes slightly the ortho-H2 and para-H2 states 关see Eq. 共1兲兴. This effect has been considered in the original work of Raich and Good in 关2兴, although not in a complete and systematic way. It has

happened that a tiny nonadiabatic correction to the total H2 wave function significantly changes the theoretical predic-tions for this rate. Moreover, the spin-orbit mixing is not the only effect, which makes this transition possible. There are also relativistic corrections to the E1 coupling to the electro-magnetic field which are barely known in the literature. These corrections in the context of the H2 molecule have been derived for the first time by Dodelson in关3兴 using the

Feinberg-Sucher formalism. Here we rederive this result in a much simpler way. Because of the summation over the infi-nite H2spectrum, the calculations of the ortho-para transition amplitude are not completely trivial. The most elaborate cal-culations so far, by Raich and Good关2兴 including Dodelson

corrections关3兴, gave the rate of 1.85共46兲⫻10−13_yr−1_{, which} did not include all important contributions. The purpose of this work is to present a complete theoretical description of the radiative ortho-para conversion of the H2 molecule, in-cluding the results of direct numerical calculations of the transition probability for the lowest rotational levels.

The interaction of an arbitrary molecule with the electro-magnetic field, whose characteristic wavelength is much larger than the size of this molecule, including the relevant spin-orbit interaction is关4兴 ␦H = −

兺

A eAxជA· Eជ−

兺

b ebxជb· Eជ −

兺

A eA 2mA 关gAxA i sA j B_,ij +共gA− 1兲sជA⫻ xជA·⳵tEជ兴 +

兺

A,b eAeb 4␲ 1 2xAb3

冋

gA mAmb s ជA· xជAb⫻ pជb −共gA− 1兲 mA 2 sជA· xជAb⫻ pជA

册

. 共1兲

In the above, A , b are the indices of the nuclei and electrons, respectively, xA, xbare the coordinates of the nuclei and

elec-trons with respect to the mass center, xជAb= xជA− xជb, and gA is

the nuclear g factor. Moreover, the electromagnetic fields

Eជ, Bជ and their derivatives are assumed in Eq.共1兲 to be at the

mass center.

For ortho-H₂共the first excited rotational state兲 the nuclear spin S = 1 couples to the orbital angular momentum J = 1 of the nuclei, giving the total angular momentum characterized by quantum numbers F = 0 , 1 , 2. In the para-H2 共the ground rotational state兲 the total angular momentum is F=0, there-fore the one photon ortho-para transition from any other F = 0 level is strictly forbidden, while the F = 1 level decays by the E1 transition and the F = 2 level decays by the M2 tran-sition.

Let us first consider the M2 transition from the F = 2 level of ortho-H₂, to the F = 0 level of para-H₂. This transition comes from the following interaction with the electromag-netic field, which is obtained from Eq.共1兲:

␦H = − egp 4mp 共sA j − sB j_兲Ri B_,ij, 共2兲

with R=xA− xB and the proton g factor gp

= 5.585 694 713共46兲 关5兴. From this Hamiltonian one obtains

the transition rate⌫共M2兲⬅⌫2, *krp@fuw.edu.pl

†

komasa@man.poznan.pl

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⌫2= 2␣␻ 1 5

兺

m_F

冟

冓

1_⌺ 0

冏

gp 4mp 共kជ·Rជ兲共sជA− sជB兲 ⫻ kជ

冏

3⌸2,mF

冔

冟

2 ⬇ 1 120␣␻ 5

冉

gpR0 mp

冊

2 , 共3兲

whereR0 is the average distance between protons, and the last equation holds for the nuclear H2 wave function, which is strongly peaked aroundR₀, as is the case for the lowest rovibrational levels.

The calculation of the rate for the E1 transition from the

F = 1 level of ortho-H2 to the F = 0 level of para-H2 is more complicated as it includes corrections to the wave function coming from the spin-orbit interaction. Since it is the ⌬F = 1 transition, the operators in the interaction Hamiltonian in Eq.共1兲 can be simplified, namely, xA

i

sA j_→⑀ijk_共xជ

A⫻sជA兲k/2 and

this Hamiltonian becomes

␦H = HLS+ e共xជ1C+ xជ2C兲 · Eជ + e 4mp

冉

gp 2 − 1

冊

Rជ ⫻ 共sជA− sជB兲 ·⳵tEជ, 共4兲 HLS= − i 2HជLS·共sជA− sជB兲, 共5兲 HជLS= gp␣ 2mpm hជ1− 共gp− 1兲␣ 2mp 2 hជ2⫻ ⵜជR, 共6兲 hជ₁=

冉

xជ1A x_1A3 − xជ1B x_1B3

冊

⫻ ⵜជ1+

冉

x ជ2A x_2A3 − x ជ2B x_2B3

冊

⫻ ⵜជ2, 共7兲 hជ₂=xជ1A x_1A3 + xជ1B x_1B3 + x ជ2A x_2A3 + xជ2B x_2B3 . 共8兲

The resulting transition rate⌫共E1兲⬅⌫1 is ⌫1= 4 3␣␻ 1 3

兺

m_F

冟

冓

1_⌺ 0

冏

Qជ ⫻ 共sជA− sជB兲 2

冏

3_⌸ 1,mF

冔

冟

2 =2 9␣␻兩具⌺兩Q k_兩⌸k_典兩2_{, 共9兲} where Qk₌ 1 2mp

冉

gp 2 − 1

冊

␻ 2_Rk₋␻ 2⑀ ikj_共x 1C i + x_2Ci 兲 1 E_⌸− HHLS j −␻ 2⑀ ikj HLS j 1 E_⌺− H共x1C i + x_2Ci 兲, 共10兲

and H is the four-body nonrelativistic Hamiltonian. In order to simplify the evaluation of Qk we perform the adiabatic expansion, namely, the expansion of the resolvent in the ki-netic energy of the nuclei

1 E_⌸− H= 1 EBO− HBO − 1 EBO− HBO 共␦E_⌸−␦HM兲 ⫻ 1 EBO− HBO + ¯ , 共11兲 similarly for _E1

⌺−H, and the expansion of the wave function

␾⌺=␺共xជ1C,xជ2C;R兲␭0共R兲/

冑

4␲+␦␾⌺, 共12兲

␾_⌸k

=␺共xជ1C,xជ2C;R兲␭1

k_共Rជ_兲/

冑

4␲+␦␾_⌸, 共13兲 with␭₁k=␭1Rk/R and with normalization

冕

dR R2_␭ 0

2_{共R兲 =}

冕

_{dR R}2_␭ 1

2_{共R兲 = 1. 共14兲} While the exact nonadiabatic correction to the wave function is unknown, we need only the first order m/mp part of the

correction, which explicitly depends on the nuclear state关6兴

␦␾⌺= − 2 mp 1 EBO− HBO ⵜRl ␺ⵜl␭0/

冑

4␲, 共15兲 ␦␾⌸k = − 2 mp 1 EBO− HBO ⵜ_Rl _␺ⵜl_␭ 1 k_/

冑

4␲. 共16兲 We introduce now the perturbed electronic wave functions

␺1 i = 1 EBO− HBO 共x1C i + x_2Ci 兲␺, 共17a兲 ␺2 j = 1 EBO− HBO h₁j␺, 共17b兲 ␺3 l = 1 EBO− HBO ⵜ_Rl _␺ , 共17c兲

to simplify the matrix elements of Qkin Eq.共10兲,

具⌺兩Qk_兩⌸k_{典 =} 1 2mp

冉

gp 2 − 1

冊

␻ 2_R 0− ␻共gp− 1兲 mp2R0 +␻2 gp␣ 4mpm X₁−␻ gp␣ 4mp 2_mR 0 共X2+ 2X3兲, 共18兲 where we used the commutator

i关p₁k+ p₂k,HBO− EBO兴 =␣h2 k , 共19兲 and X1=⑀ikjnk具␺1 i_兩␺ 2 j_典 R0, 共20a兲 X2=共␦kl− nknl兲⑀ikj关具⳵Rl ␺1 i_兩␺ 2 j_{典 − 具␺} 1 i_兩 ⳵Rl ␺2 j_典兴 R0, 共20b兲

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X3=共␦kl− nknl兲⑀ikj关具␺3 l_兩共x 1C i + x_2Ci 兲兩␺₂j典 + 具␺l₃兩h₁j兩␺₁i典兴_R 0, 共20c兲 with nជ⬅Rជ₀/R₀. One notes that derivatives of nonlinear and linear parameters in␺关see Eq. 共24兲兴 with respect to R do not

contribute to the above matrix elements, which significantly simplifies the numerical computations.

Results for Xican be expressed in terms of dimensionless

factors Fi, X1= − 9 m2␣4R2F1共m␣R兲, 共21a兲 X2= 9 m2␣4R3F2共m␣R兲, 共21b兲 X₃= −2m ␣ F3共m␣R兲, 共21c兲

which are chosen in such a way that Fi共m␣R兲 vanish at R

= 0 and approach 1 forR→⬁. However, in the case of F₂ this large R limit is only a rough approximation, since we have not been able to perform this limit analytically. We can now return to Eq.共18兲 and obtain a compact formula for the

matrix element in the transition rate⌫1 of Eq.共9兲, 具⌺兩Qk_兩⌸k_{典 =}␻ 2_R 0 2mp

冋

冉

gp 2 − 1

冊

− 9 2 gpF1 共m␣R0兲3

册

− ␻ mp 2_R 0

冋

共gp− 1兲 − gpF3+ 9 4 gpF2 共m␣R0兲3

册

. 共22兲 Numerical evaluation of the⌫2rate according to formula 共3兲 is straightforward. To obtain the ortho-para energy

spac-ing ␻ and the average internuclear distance R₀, we em-ployed the accurate Kołos–Le Roy–Schwartz interaction po-tential 关7兴, which includes the adiabatic and relativistic

energy corrections. With this potential we solved numerically the radial Schrödinger equation to obtain the energies and wave functions corresponding to the lowest ortho and para levels. The numerical values used here are ␻= 2␲ ⫻118.49 cm−1 _{and R}

0= 1.449 a.u., and the resulting M2 transition rate with physical constants from Ref.关5兴 is

⌫2= 1.07共1兲 ⫻ 10−14yr−1. 共23兲 The accurate evaluation of⌫₁and the corresponding elec-tronic matrix elements Fiin Eqs.共21兲 is a challenging task.

We have represented the electronic ground state wave func-tion as well as the first order perturbed funcfunc-tions defined by Eqs.共17兲, in the form of properly symmetrized linear

com-binations␺=兺kckPˆg,u␾kof Gaussian geminals

␾k=⌶kexp共−␣kx1A2 −␤kx1B2 −␨kx2A2 −␩kx2B2 −␥kx122兲. 共24兲 The projection operators

Pˆ_g,u=1

4共1 + Pˆ12兲共1 ⫾ ıˆ兲 共25兲 ensure the proper symmetry with respect to the exchange of the electrons and with respect to the inversion operation, yielding singlet gerade or ungerade functions. Required⌺+, ⌺−_{, or⌸ symmetry of the electronic wave function was} im-posed by the Cartesian prefactor⌶k. The linear and the

non-linear parameters were optimized variationally with the goal function being the ground state energy in the case of the unperturbed wave function ␺ or pertinent Hylleraas func-tional J关␺k i_{兴 = 具} ␺k i_兩E BO− HBO兩␺k i_{典 + 2具} ␺兩Oˆ兩␺k i_{典, 共26兲}

in the case of the perturbed functions␺_ki. TableIshows ex-TABLE II. Numerical values of the optimum goal functions Eq. 共26兲 and the expectation values comprising the Xifactors Eqs.共27兲

in atomic units. R0= 1.449 R=12.0 Asymptotic E_BO −1.174 073 569 −1.000 002 546 −1.0 J关␺1 x_兴 −3.3582 −4.5241 −4.5 J关␺1y兴 −2.3684 −4.4885 −4.5 J关␺2 x_兴 −7.87⫻10−3 −2.22⫻10−7 0.0 J关␺2z兴 −0.4925 −5.06⫻10−5 0.0 J关␺3 y_{兴 −2.90⫻10}−2 _{−0.2500 −0.25} 具␺1y兩␺2z典 0.7824 0.0312 X₁ −1.5649 −0.0623 F1 0.3651 0.9975 1.0 具⳵Ry␺1 x_兩␺ 2 z_{典 0.7467 0.0014} 具␺1 z_兩⳵ R y_␺ 2 x_典 0.0034 0.0002 X₂ 3.0003 0.0062 F2 1.0142 1.1989 具␺3y兩共x1C x _{+ x} 2C x _兲兩␺ 2 z_{典 0.0707 −0.0029} 具␺3 y_兩共x 1C z + x_2Cz 兲兩␺₂x典 −0.0010 −0.0001 具␺3y兩h1z兩␺1x典 −0.2579 −0.5026 具␺3 y_兩h 1 x_兩␺ 1 z_{典 0.1066 0.4980} X3 −0.3984 −2.0037 F₃ 0.1992 1.0019 1.0

TABLE I. The definitions of the functions used in the computations.

⌶k Pˆ , Eq. 共25兲 Oˆ, Eq. 共26兲

␺ 1 Gerade ␺1 x x1, x2 Ungerade x1+ x2 ␺1y y1, y2 Ungerade y1+ y2 ␺1 z _z 1, z2 Ungerade z1+ z2 ␺2x y1z2− y2z1 Ungerade h1x ␺2 z _y 1, y2 Ungerade h1 z ␺3y y1, y2 Gerade ⳵Ry

ORTHO-PARA TRANSITI4ON IN MOLECULAR HYDROGEN PHYSICAL REVIEW A 77, 030501共R兲 共2008兲

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plicitly the elements defining particular functions with the assumption that the molecule is placed along the Cartesian X axis. The unperturbed wave function has been expanded in a 600-term basis set which enables the electronic ground state energy to be obtained with an error of only 3⫻10−9 _{a.u. A} 1200-term expansion has been employed to represent the perturbed functions. Values of theJ functionals correspond-ing to the optimum parameters are displayed in TableII.

The general formulas 共20兲, in the particular case of the

molecule oriented along the X axis, can be explicitly written as follows: X1= − 2具␺1 y_兩␺ 2 z_典 R0, 共27a兲 X2= 4关具⳵Ry␺1 x_兩 ␺2 z_{典 − 具} ⳵Ry␺1 z_兩 ␺2 x_典兴 R0, 共27b兲 X3= 2关具␺3 y_兩共x 1C x + x_2Cx 兲兩␺₂z典 − 具␺₃y兩共x_1Cz + x_2Cz 兲兩␺₂x典 +具␺₃y兩h₁z兩␺₁x典 − 具␺₃y兩h₁x兩␺₁z典兴_R 0. 共27c兲

TableIIcontains all the expectation values appearing in Eqs. 共27兲 as well as the final Xi and Fi values computed at R0 = 1.449 bohr. To check the correctness of our codes we per-formed additional calculations at large internuclear distance 共R=12.0 bohr兲 and compared the resulting goal functions and the expectation values with analytically derived asymptotic values. This comparison is presented in TableII. Using Eqs.共9兲, 共22兲, and TableIIone obtains the numeri-cal value for the E1 transition rate

⌫1= 1.68共17兲 ⫻ 10−13yr−1, 共28兲 and finally the rate averaged over the total angular momen-tum F,

⌫ = 共5⌫2+ 3⌫1兲/9 = 6.20共62兲 ⫻ 10−14yr−1. 共29兲 Our result for the averaged transition rate is in disagree-ment with the result of Dodelson 关3兴, ⌫=1.85共46兲

⫻10−13_yr−1_{, which is in turn based on the former work of} Raich and Good关2兴 and included direct coupling of nuclear

spin to the radiation field. We confirm in this work the exis-tence of these additional couplings, which here are expressed by the third term in Eq.共1兲. In our opinion, the difference

from our result is due to the omission of the M2 transition, omission of the nonadiabatic contributions corresponding to

X2and X3in Eq.共20兲, less accurate␻, and a lower accuracy of the numerical calculation of the matrix elements in Ref. 关2兴. In particular, without X3 the overall rate ⌫ would be about 24% larger.

The possibility of the experimental detection of the ortho-para H2 line is questionable. Much stronger E2 lines have already been observed at the Infrared Space Observatory 共ISO兲 and served for estimation of the temperature of inter-stellar hydrogen clouds and of the ratio of abundance ortho-H2 to para-H2, which sometimes differs significantly from the equilibrium one关8兴. The much weaker E1 line has

not been observed yet. In fact there is a potential opportunity related with the Herschel Space Observatory to be launched in 2008 关9兴. Its spectral range covers the ortho-para line at

84.4␮m, but its resolution is, probably, not high enough at this wavelength.

K.P. wishes to acknowledge interesting discussions with Krzysztof Meissner, and thanks the Laboratoire Kastler Brossel in Paris for kind hospitality during his stay, when this work was written.

关1兴 Y. Sun and A. Dalgarno, Astrophys. J. 427, 1053 共1994兲. 关2兴 J. C. Raich and R. H. Good, Jr., Astrophys. J. 139, 1004

共1964兲.

关3兴 S. Dodelson, J. Phys. B 19, 2871 共1986兲. 关4兴 K. Pachucki, Phys. Rev. A 76, 022106 共2007兲.

关5兴 P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 共2005兲.

关6兴 K. Pachucki and J. Komasa, e-print arXiv:0801.0561. 关7兴 C. Schwartz and R. J. Le Roy, J. Mol. Spectrosc. 121, 420

共1987兲.

关8兴 D. A. Neufeld, G. J. Melnick, and M. Harwit, Astrophys. J.

506, L75共1998兲.

关9兴 http://herschel.esac.esa.int

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