Gerade-ungerade mixing in the hydrogen molecule

Transcription

1 Geade-ungeade mixing in the hydogen molecule Kzysztof Pachucki 1, and Jacek Komasa, 1 Faculty of Physics, Univesity of Wasaw, Hoża 69, Wasaw, Poland Faculty of Chemisty, A. Mickiewicz Univesity, Gunwaldzka 6, Poznań, Poland (Dated: Mach 18, 011) In homonuclea molecules, such as H, the ovibational levels close to the dissociation theshold do not have definite symmety with espect to the invesion of electonic vaiables. This effect the geade ungeade mixing esults fom inteactions between magnetic moments of electons and potons. We calculate this mixing on the level of adiabatic appoximation and numeically solve the system of nuclea diffeential equations. It tuns out, that the coections to the dissociation enegy of ovibational levels esulting fom the mixing ae negligible in compaison with the pesent accuacy of expeiments. As a co-poduct, the most accuate to date clamped nuclei potential fo the b 3 Σ + u state has been obtained. PACS numbes: GS, aj, vn, Df I. INTRODUCTION The hydogen molecule, due to its simplicity, can be accuately calculated fom fist pinciples using the quantum electodynamic (QED) theoy. At the cuent accuacy of about cm 1, apat fom nonadiabatic and elativistic effects, the O(α 3 ) and the dominating pat of O(α 4 ) QED coections have to be included. The excellent ageement with ecent expeimental esults fo H 1, ], D 1, 3] and HD 4, 5] indicates a good undestanding of all physically significant effects and is a basis fo a futhe impovement in theoetical desciption of the hydogen molecule. At pesent, the main uncetainty comes fom the highe ode nonadiabatic O(µ 3 n ), elativistic ecoil O(α /µ n ), and QED O(α 4 ) coections. Once these thee tems ae known, the accuacy of ovibational levels could be inceased up to about 10 6 cm 1 ( 30 khz), povided that all othe small effects ae detemined, in paticula those due to the finite poton chage adius p and the geade ungeade mixing. The coection due to p can easily be calculated, but the accuate value of p is pesently an issue. The esult of the ecent detemination of p fom the muonic hydogen Lamb shift 6] is 5% smalle than pevious deteminations fom the hydogen spectum and fom the electon-poton scatteing. This 5% gives uncetainty in H dissociation enegy of about cm 1, thusxs the poton chage adius discepancy has to be esolved to be able to each 10 6 cm 1 accuacy. The coection esulting fom geade ungeade mixing appeas due to the inteactions between the electon and the poton magnetic moments. An expeimental evidence of this phenomenon has been epoted fo iodine 7] and cesium 8] dimes and a detailed theoetical account of hypefine inteactions in diatomic homonuclea molecules has been given in 9]. The enegy shift caused by such inteactions in H can, in pinciple, be as lage as the hypefine splitting in the hydogen atom, which amounts to 140 MHz 0.05 cm 1. Until now howeve, the magnitude of this effect fo the hydogen Electonic addess: kp@fuw.edu.pl Electonic addess: komasa@man.poznan.pl molecule has been unknown and its detemination is the pupose of this wok. At the accuacy level of 10 6 cm 1 the pedicted H spectum is sensitive to uncetainties in fundamental constants. In paticula, the pesent uncetainty in the poton-to-electon mass atio, equal to 0.4 ppb, yields about 14 khz uncetainty fo the gound state dissociation enegy. This means, that the poton-to-electon mass atio can be detemined fom the H spectum moe accuately than it is known pesently, if both theoy and expeiment pass the theshold of 14 khz uncetainty. Expeimentalists 10] have aleady consideed such level of pecision, while fom theoetical point of view, we have not yet investigated in detail the feasibility of such an uncetainty in H. Cetainly the most challenging is the accuate calculation of O(α 4 ) and estimation of O(α 5 ) coections. II. WAVE FUNCTIONS AND HAMILTONIAN In the adiabatic appoximation, the total spatial wave function φ is appoximated by a poduct of electonic and nuclea functions φ(, R) = φ el ( ) χ(r) Y LM ( n) (1) with φ el being the solution to the clamped nuclei Schödinge equation H el φ el = E( R) φ el, () Y LM ( n) a spheical hamonic with n = R/R. The nuclea function χ fulfills the Bon-Oppenheime adial Schödinge equation 1 1 µ n R ] L (L + 1) R + R µ n R + E(R) χ vl (R) = E vl χ vl (R), (3) whee µ n is the atio of the nuclea educed mass to the electon mass, and atomic units ae used thoughout the pape. In the nonelativistic appoximation the symmety of the invesion of electonic coodinates ( 1, ) with espect to the

2 geometical cente of the H molecule is conseved, and the splitting δe of the clamped nuclei enegies between the lowest geade and ungeade states vanishes exponentially fo lage intenuclea distances R 11, 1] (in atomic units) δe(r) = E u (R) E g (R) = R 5/ e R 1+O(1/ R)]. (4) Since these states become asymptotically degeneate, a small petubation may significantly mix them, and geade and ungeade symmety will not be peseved. To descibe this mixing, let us define the following two electonic spatial functions φ g ( 1, ; R) = φ g ( 1,, R) = φ g (, 1, R) (5) fo the geade (X 1 Σ + g ) state, and φ u ( 1, ; R) = φ u ( 1,, R) = φ u (, 1, R) (6) fo ungeade (b 3 Σ + u ) state. Both functions ae assumed to be solutions to the clamped nuclei Schödinge equation () with coesponding enegies E g and E u. In the asymptotic egion they take the Heitle-London fom 13] φh ( 1A ) φ H ( B ) + φ H ( 1B ) φ H ( A ) ] φ g ( 1,, R) = φ u ( 1,, R) = (7) φh ( 1A ) φ H ( B ) φ H ( 1B ) φ H ( A ) ] whee φ H is the gound state atomic hydogen function. The leading elativistic coections, as given by the Beit- Pauli Hamiltonian, violate the invesion symmety of electon coodinates with espect to the geometical cente. As a esult, electonic states do not have a definite symmety and ovibational enegies ae slightly shifted. One expects this shift to be the most significant fo states laying close to the dissociation theshold, whee nuclei ae fa apat fom each othe. Among all the elativistic coections, the dominating one at lage intenuclea distances esults fom the magnetic inteactions between all the fou paticles, electons and potons, epesented by a and b, (e = 4 π α) δh = m α a>b e a e b 4 π + g a g b 4 m a m b s i a s j b 3 ab π 3 g a g b m a m b s a s b δ (3) ( ab ) ( δ ij 3 i ab j ab 3 ab (8) )]. (9) δh causes the geade ungeade mixing and also contibutes to the geade ungeade splitting of the clamped nuclei enegies. In fact, the elativistic coection to this splitting goes like R 3 (fo J 0) and, at lage R, dominates ove the nonelativistic splitting, Eq. (4). To account fo the splitting and the mixing, we will include in the nuclea equation both the diagonal and off-diagonal matix elements between geade and ungeade states. III. MATRIX ELEMENTS FOR THE GERADE UNGERADE MIXING AND SPLITTING Among all the spin inteactions in Eq. (9), the potonpoton and the local (Diac δ) electon-electon inteactions can be neglected. The fist one is vey small and the second one vanishes exponentially fo lage distances. We neglect also the tenso electon-nucleus inteaction, because it is much smalle than the scala inteaction, which is popotional to the Diac δ. As a esult of these appoximations, δh is a sum of the tenso electon-electon spin and the local electonnucleus inteactions. In atomic units they take the fom δh = α si 1 s j ( 1 3 δ ij 3 i 1 j ) π α g p m s a 3 m I X δ (3) ( ax ) p a,x (10) = δh 1 + δh, (11) whee a = 1, labels electons and X = A, B nuclei. Let us intoduce the notation L fo otational angula momentum, S = s 1 + s fo the total electon spin, I = s A + s B fo the total nuclea spin, J = L + S and F = J + I. We will use the basis L, S, J, I, F, m F in the evaluation of matix elements. Not all the values of angula momenta ae allowed, due to the Pauli exclusion pinciple. Fo the geade state of H molecule (S = 0), I = 0 fo even L and I = 1 fo odd L, fo the ungeade state (S = 1), I = 1 fo even, and I = 0 fo odd L. Let us conside now the fist component δh 1 in Eq. (10) and ewite it in tems of the total electon spin δh 1 = α S i S j ( 1 3 δ ij 3 i 1 j ) 1 1. (1) Its expectation value in the ungeade state is δh 1uu φ u δh 1 φ u = β(r) S i S j (δ ij 3 n i n j ), (13) whee n i = R i /R and β(r) = α 4 b(r), (14) b(r) = φ u 3( 1 n) φu. (15) Fo asymptotic intenuclea distances the electon-electon distance 1 can be eplaced by R, thus b(r) /R 3, and δh 1uu becomes much lage than the nonelativistic splitting δe. Matix elements of δh 1uu in the basis L, S, J, m J ae diagonal in S and J, and do not vanish only fo S = 1 and L = 0, ±. Hence, we ae left with only fou types of ma-

3 3 tix elements: L, 1, L + 1 δh 1uu L, 1, L + 1 = L (L ) β (16) (L + 3)(L 1) L, 1, L δh 1uu L, 1, L = (L(L + 1) 3) β (17) (L + 3)(L 1) L, 1, L 1 δh 1uu L, 1, L 1 = (L + 1)(L + 3) β (18) (L + 3)(L 1) L 1, 1, L δh 1uu L + 1, 1, L = L (L + 1) 3 β, (19) L + 1 whee we have intoduced the double-baket notation J, M Q J, M = δ M M J Q J (0) fo a scala opeato Q. The second tem in Eq. (10), δh, is at fist ewitten as δh = π α 3 g p m m p { S I δ (3) ( 1A ) + δ (3) ( 1B ) (1) +δ (3) ( A ) + δ (3) ( B ) ] + ( s 1 s ) ( I A I B ) δ (3) ( 1A ) δ (3) ( 1B ) δ (3) ( A ) + δ (3) ( B ) ]}, whee A and B efe to the two nuclei. Although othe tems containing ( s 1 ± s ) ( I A I B ) might also be pesent in the above, they wee omitted because thei matix elements between φ g and φ u states vanish. The diagonal matix element of δh is δh uu φ u δh φ u = γ(r) S I, () γ(r) = π α 3 and the off-diagonal is g p m m p c(r), (3) c(r) = φ u δ (3) ( 1A ) + δ (3) ( 1B ) +δ (3) ( A ) + δ (3) ( B ) φ u (4) δh gu φ g δh φ u = γ (R) ( s 1 s ) ( I A I B )(5), γ (R) = π α 3 g p m m p c (R), (6) c (R) = φ g δ (3) ( 1A ) δ (3) ( 1B ) δ (3) ( A ) + δ (3) ( B ) φ u. (7) In the asymptotic egion, the matix elements of the electonnucleus Diac δ function appoach the atomic hydogen value, thus c( ) = c ( ) = π. (8) Nonvanishing matix elements in the angula momentum basis L, S, J, I, F, M F ae L, 1, L + 1, 1, L δh uu L, 1, L + 1, 1, L = γ L + L + 1 (9) L, 1, L, 1, L δh uu L, 1, L, 1, L = 1 γ L(L + 1) (30) L, 1, L 1, 1, L δh uu L, 1, L 1, 1, L = γ L 1 L (31) L, 1, L + 1, 1, L δh uu L, 1, L, 1, L = L L + 3 γ L + 1 L + 1 (3) L, 1, L, 1, L δh uu L, 1, L 1, 1, L = γ L + 1 L 1 L L + 1 (33) L, 1, J, 1, L δh gu L, 0, L, 0, L = γ J + 1 L + 1 (34) L, 1, J, 0, J δh gu L, 0, L, 1, J = γ. (35) All these matix elements depend implicitly on R and ae included in the clamped nuclei potential in the nuclea Schödinge equation. IV. NUCLEAR EQUATIONS FOR THE GERADE UNGERADE MIXING To take into account the states mixing, we employ the matix fom of the equation L (L + 1) R + µ n R R µ n R ] +E g (R) + δe spin (R) ẼvL χ vl (R) = 0, (36) whee δe spin (R) is a matix fomed fom matix elements of δh, Eq. (10), in a petinent basis. The dimension of the matix δe spin (R) is detemined by the numbe of close lying levels and it depends on the otational quantum numbe L. The pincipal question we ask is what is the value of the diffeence δe gu = E vl ẼvL, (37) which we shall call the geade ungeade mixing coection to the dissociation enegy of a ovibational level (v, L). In what follows we conside thee sepaate cases depending on the quantum numbe L. We note in passing, that the L- mixing in Eq. (19) can potentially play a ole only fo L = 0, because in this case the diagonal spin-spin inteaction epesented by β is absent in δe spin (R).

4 4 A. Case: L even and 0 We span the nuclea wave function in the following basis The δe spin (R) matix is now obtained using Eqs. (16)-(18) and (9)-(35) and assumes the fom L, 0, L, 0, L, M L, 1, L + 1, 1, L, M L, 1, L, 1, L, M L, 1, L 1, 1, L, M. (38) δe spin = 0 γ L+3 L (L ) δe + β (L+3)(L 1) γ L+ L+1 γ L+3 γ γ L L+1 γ L 1 L+3 γ γ L+1 L L+3 δe β (L(L+1) 3) (L+3)(L 1) γ 1 L(L+1) L 1 0 γ L+1 L δe + β γ L 1 0 γ L+1 L 1 L (L+1)(L+3) (L+3)(L 1) γ L 1 L (39) The basis is B. Case: L odd L, 0, L, 1, J, M L, 1, J, 0, J, M. (40) δe spin (R) matices fo thee diffeent values of J = L + 1, L, L 1 ead ( ) 0 γ δe L+1 spin = γ (41) δe + β δe L spin = δe L 1 ( 0 γ γ γ spin = ( 0 γ L (L ) (L+3)(L 1) δe β (L(L+1) 3) (L+3)(L 1) δe + β (L+1)(L+3) (L+3)(L 1) C. Case: L = 0 ) ) (4) (43) This is the only case whee we include the L-mixing and thus the basis is and the δe spin (R) becomes 0, 0, 0, 0, 0, 0 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0 (44) 0 3γ 0 δe spin = 3γ δe γ β 0. (45) β δe + 3/(µ n R ) The nuclea equation (36) with the matices δe spin (R) pesented above has been solved numeically as descibed in the following section. V. NUMERICAL PROCEDURES AND RESULTS Vey accuate clamped nuclei potential fo the X 1 Σ + g state has been epoted ecently in 1]. Fo the whole enegy cuve, an accuacy of the ode of has been eached. It is the most accuate esult to date fo H itself but also fo any molecula system with two o moe electons. Inceasing the accuacy to this level has been possible thanks to the discovey of analytic fomulas fo two-cente two-electon integals with exponential functions 14]. In this wok, we epot on an analogous calculation fo the b 3 Σ + u state. In ode to achieve the highest numeical accuacy, diffeent basis sets ae used, depending on the intenuclea distance R. Fo R < 1 boh, the James-Coolidge basis functions 15, 16] of the fom ψ {n} ( 1, ) = (1 ± ˆP 1 )(1 ± î) e α ( 1A+ 1B ) β ( A + B ) n1 1 ( 1A 1B ) n ( A B ) n3 ( 1A + 1B ) n4 ( A + B ) n5 (46) have been employed. The antisymmety pojecto (1 ± ˆP 1 ) ensues singlet (+) o tiplet ( ) state, while the spatial pojecto (1 ± î) the geade (+) o ungeade ( ) symmety. Since in the actual numeical calculations, one can use only a finite numbe of basis functions, one has to somehow select the most appopiate finite subset of functions in Eq. (46). We assume theefoe, that the finite basis consists of all functions with nonnegative integes n i such that 5 n i Ω (47) i=1 with Ω = 3,... 18, and the final esult is obtained by a numeical extapolation to Ω. Fo R < 1. boh we used the James-Coolidge basis with two diffeent nonlinea paametes α β, wheeas fo 1. R 1 boh with α = β. The nonlinea paametes wee optimized sepaately fo each intenuclea distance R, and then the exponential convegence to a complete basis set as Ω has been obseved. To descibe the molecule at 1 R 0 boh, the gene-

5 5 alized Heitle-London functions 13] ψ k ( 1, ) = (1 ± ˆP 1 )(1 ± î) (48) e ( 1A+ B ) n 1k 1 n k 1A n 3k 1B n 4k A n 5k B with Ω up to 16 have been applied. These functions ae the most appopiate fo lage intenuclea distances, and we have checked that at R = 1 boh the accuacy achieved with genealized Heitle-London functions is close to that with the symmetic James-Coolidge basis. The egion of R > 0 boh is found to be numeically insignificant, as δe vanishes exponentially at lage R, see Eq. (4). All the numeical esults fo R < 0 boh, afte extapolation to a complete basis set, ae listed along with the estimated eo in Table I. This is the most accuate clamped nuclei enegy cuve fo the b 3 Σ + u state among all obtained so fa. Numeical calculations wee pefomed in the quadupole pecision, which nevetheless was not always sufficient. Fo some values of R we obseved numeical instabilities fo highest values of Ω, these esults had to be dopped, and thus numeical extapolation includes much lage uncetainties. Fo small values of R we obseved much slowe numeical convegence. It is elated to the fact, that ia is then close to ib, and the one paamete selection of the finite basis set Eq. (47) is not the most effective. To evaluate the matix elements in the functions β, γ, and γ (see Eqs. (14), (4), and (7)), we employed exponentially coelated Gaussian (ECG) functions 17, 18] of the fom ψ k ( 1, ) = (1 ± ˆP 1 )(1 ± î) (49) exp A k,ij ( i s k,i )( j s k,j ), i,j=1 whee the matices A k and vectos s k contain nonlinea paametes, 5 pe basis function, to be vaiationally optimized. Fo both, X 1 Σ + g and b 3 Σ + u states, 600-tem bases wee optimized with espect to E g o E u, fo R spead ove the ange (0, 1) boh. Fo each R, the geade {ψ g,k } 600 k=1 and ungeade {ψ u,k } 600 k=1 basis sets wee meged togethe to fom 100-tem expansions 600 k=1 (c k ψ g,k + c k+600 ψ u,k ) fo φ g/u yielding the E g/u accuate to a faction of micohatee. Next, using these φ g/u, we evaluated the electonic matix elements of δh, Eqs. (15), (4), and (7). Thei numeical values ae pesented in Table II. The egula adial Schödinge equation (3) as well as the coupled set of adial diffeential equations (36) have been solved using Discete Vaiable Repesentation (DVR) method 19]. The discete spectum consists of 301 eigenvalues, each coesponding to a bound ovibational level (v, L) accommodated by the E g potential of H. The geade ungeade mixing coections δe gu to the dissociation enegy of all the levels ae listed in Table III. Fo a vast pat of the levels the coections ae of the ode of 10 8 cm 1 o even smalle. Only fo the highest vibational quantum numbes v 1, values two odes of magnitude lage can be found. The lagest coection of ca cm 1 appeas fo the v = 14, L = level. In all cases, the coections incease the dissociation enegy, i.e. lowe the enegy level. Among the components of the Beit-Pauli Hamiltonian, which ae due to the magnetic inteaction between all the paticles, Eq. (9), the poton-poton inteaction and the electonelecton contact inteaction have been a pioi discaded, as expected to be vey small. It tuns out, that the elativistic coections to geade ungeade splitting do not play any ole, eithe. The main contibution to the oveall mixing effect comes fom the off-diagonal matix elements δh gu expessed though the γ (R), Eq. (7). VI. CONCLUSION We have shown that geade ungeade mixing gives coections to the most of ovibational levels of hydogen molecule smalle than 10 6 cm 1. Since the pesent accuacy of theoetical pedictions fo the dissociation enegy in the gound electonic state of the hydogen molecule is cm 1 1, 4], these coections appea to be negligible. The mixing coections become moe significant only fo highly excited ovibational states, whee they appoach 10 5 cm 1. It means, that futhe impovement in the pecision of the dissociation enegies can be obtained by the calculation of the highe ode nonadiabatic and QED effects, assuming that the geade ungeade symmety is conseved. Acknowledgments KP and JK acknowledge the suppot of the Polish Ministy of Science and Highe Education via Gants No. N N and N N , coespondingly. Pat of the computations has been pefomed within a computing gant fom Poznań Supecomputing and Netwoking Cente. 1] K. Piszczatowski, G. Lach, M. Pzybytek, J. Komasa, K. Pachucki, and B. Jezioski, J. Chem. Theoy Comput. 5, 3039 (009). ] J. Liu, E. J. Salumbides, U. Hollenstein, J. C. J. Koelemeij, K. S. E. Eikema, W. Ubachs, and F. Mekt, J. Chem. Phys. 130, (009). 3] J. Liu, D. Speche, C. Jungen, W. Ubachs, and F. Mekt, J. Chem. Phys. 13, (010). 4] K. Pachucki and J. Komasa, Phys. Chem. Chem. Phys. 1, 9188 (010). 5] D. Speche, J. Liu, C. Jungen, W. Ubachs, and F. Mekt, J. Chem. Phys. 133, (010). 6] R. Pohl et al., Natue 466, 13 (010). 7] J. P. Pique, F. Hatmann, R. Bacis, S. Chuassy, and J. B. Kof-

6 6 fend, Phys. Rev. Lett. 5, 67 (1984). 8] H. Weickenmeie, U. Dieme, W. Demtöde, and M. Boye, Chem. Phys. Lett. 14, 470 (1986). 9] M. Boye, J. Vigué, and J. Lehmann, J. Phys. Fance 39, 591 (1978). 10] D. Speche, C. Jungen, W. Ubachs, and F. Mekt, Faaday Discuss. 150 (011). 11] C. Heing and M. Flicke, Phys. Rev. 134, A36 (1964). 1] K. Pachucki, Phys. Rev. A 8, (010). 13] W. Heitle and F. London, Z. Physic 44, 455 (197). 14] K. Pachucki, Phys. Rev. A 80, 0350 (009). 15] H. M. James and A. S. Coolidge, J. Chem. Phys. 1, 85 (1933). 16] J. S. Sims and S. A. Hagstom, J. Chem. Phys. 14, (006). 17] K. Singe, Poc. R. Soc. Lond. A 58, 41 (1960). 18] J. Rychlewski and J. Komasa, in Explicitly Coelated Wave Functions in Chemisty and Physics. Theoy and Applications, edited by J. Rychlewski (Kluwe, Dodecht, 003), p ] G. C. Goenenboom and D. T. Colbet, J. Chem. Phys. 99, 9681 (1993). TABLE I: The clamped nuclei enegy E u(r) of the b 3 Σ + u state of H in atomic units. R/au E u(r) R/au E u(r) (54) (1) (8) (1) (18) (9) (49) (14) () (6) (19) (4) (6) (4) (33) (3) (36) (1) (35) (7) (3) (7) (1) () (33) (3) (7) () (18) (3) (68) (3) (7) (3) (85) (1) (53) (3) (3) () (9) (17) (1) (16) (3) (30) (3) (47) (11) (5) (49) (7) () (98) (3) (3) (1) (30) (8) (4) (4) (0) (3)

7 7 TABLE II: The matix elements fo δe spin in atomic units. Fo c and c the elative uncetainty is bette than 10 3, wheeas fo b all displayed digits ae significant. R/au b(r) c(r) c (R) /π /π

8 8 TABLE III: The geade ungeade mixing coections δe gu to the dissociation enegy of all the bound ovibational levels of H. The enties ae given in units of 10 8 cm 1. v\l