The hydrogen molecule H 2 in inclined configuration in a weak magnetic field
A. Alijah, J.C. López Vieyra, D.J. Nader, A.V. Turbiner, H. Medel Cobaxin
aa r X i v : . [ phy s i c s . a t o m - ph ] M a y The hydrogen molecule H in inclined configuration in aweak magnetic field Alexander Alijah
Universit´e de Reims Champagne-Ardenne, Groupe de Spectrom´etrie Mol´eculaire etAtmosph´erique (UMR CNRS 7331), U.F.R. Sciences Exactes et Naturelles,Moulin de la Housse B.P. 1039, F-51687 Reims Cedex 2, France
Juan Carlos L´opez Vieyra, Daniel J. Nader, Alexander V. Turbiner
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 70-543, 04510 Ciudad de M´exico, M´exico
H´ector Medel Cobaxin
Instituto Tecnol´ogico de Estudios Superiores de Monterrey,64849 Monterrey, N.L., M´exico
Abstract
Highly accurate variational calculations, based on a few-parameter, physi-cally adequate trial function, are carried out for the hydrogen molecule H ininclined configuration, where the molecular axis forms an angle θ with respectto the direction of a uniform constant magnetic field B , for B = 0 , . , . . θ = 0 ◦ , ◦ , ◦ are studied in detail withemphasis to the ground state 1 g . Diamagnetic and paramagnetic suscepti-bilities are calculated (for θ = 45 ◦ for the first time), they are in agreementwith the experimental data and with other calculations. For B = 0 , . . E vs R are built for each inclination, theyare interpolated by simple, two-point Pad´e approximant P ade [2 / R ) withaccuracy of not less than 4 significant digits. Spectra of rovibrational states Email addresses: [email protected] (Alexander Alijah), [email protected] (Juan Carlos L´opez Vieyra), [email protected] (Daniel J. Nader), [email protected] (Alexander V. Turbiner), [email protected] (H´ector Medel Cobaxin)
Preprint submitted to JQSRT May 15, 2019 re calculated for the first time. It was found that the optimal configurationof the ground state for B ≤ B cr = 0 .
178 a.u. corresponds always to theparallel configuration, θ = 0, thus, it is a Σ g state. The state 1 g remainsbound for any magnetic field, becoming metastable for B > B cr , while for B cr < B <
12 a.u. the ground state corresponds to two isolated hydrogenatoms with parallel spins.
Keywords: variational method, weak magnetic field, critical magnetic field,magnetic susceptibility, ro-vibrational states
1. Introduction
More than fifty years have passed since it was predicted that extremelystrong magnetic fields up to B = 10 − G ( B ∼ × − a.u.), whichare by far beyond those that can be reached in the laboratory, could existin neutron stars remnant of a supernova explosion as effect of the magneticfield flux compression [1] (see also [2, 3, 4]). As for magnetized white dwarfs,the surface magnetic field can reach B ∼ G (see e.g. [5] and referencestherein). Soon afterwards it was recognized that the structure of atomsand molecules might be qualitatively different under strong magnetic fields B & B ( B = 1 a . u . ≡ . × G = 2 . × T) [6, 7, 8] from the field-freecase. The electronic clouds assume a well-pronounced cigar-like form, andmolecules become oriented along the magnetic line. Eventually, the problembecomes quasi-one-dimensional, where longitudinal and transverse motionsof the electrons are almost separated. This gives hope to develop an analyt-ical theory in the domain of very strong magnetic fields. The situation getsmuch more complicated in the domain of intermediate magnetic fields, say,of order of B ∼ − a.u., where quadratic corrections to the linear Zeemaneffect become significant. This domain is ‘slightly’ above the magnetic fieldsreachable in the laboratory. In this case we do not see hope to develop ana-lytic approaches. We will call the fields 0 . . B . intermediate magnetic fields.Due to mainly technical difficulties in solving the Schr¨odinger equationin the presence of intermediate and strong magnetic fields, only a relativelysmall number of simple atomic and molecular systems has been studied.Naturally, the hydrogen atom H and the hydrogen molecular ion H +2 arethe most studied systems, see e.g. [9] and [10], respectively, and referencestherein. The first quantitative study of the H molecule was carried out2y one of the present authors in 1983 [11]. In the majority of studies ofmolecules and molecular ions all non-adiabatic terms in the Hamiltonianare neglected by assuming an infinite nuclear mass (what is usually calledBorn-Oppenheimer (BO) approximation of zeroth order). The fact is that inboth the H and H +2 systems the binding energy grows dramatically with anincrease of the magnetic field strength, it hints at the possible existence ofunusual chemical species in strong magnetic fields. Other simple, traditionalsuch as H +3 [12, 13], and exotic compounds mainly formed by protons and/or α -particles (helium nuclei) and one or two electrons have been studied to acertain degree. For a discussion, see [10] for one-electron systems and [14]for two-electron systems.Recently, a detailed study of the H +2 molecular ion in inclined configu-ration (when the molecular axis and the magnetic line form some non-zeroangle) was carried out for intermediate and strong magnetic fields [10, 15].It was shown that for the ground state the optimal configuration is alwaysparallel, where the molecular axis and magnetic field direction coincide. Thespectra of rovibrational states was exhaustively studied.As for the H molecule, it was found long ago that the minimal energy(ground) state evolves with magnetic field strength being realized by dif-ferent states depending on the strength of the magnetic field, see [16, 17]and references therein. At zero and weak magnetic fields, the H groundstate is realized by the spin-singlet S = 0, Σ g state in parallel configura-tion, but with the magnetic field strength increasing to above the criticalfield strength of B cr = 0 .
178 a.u., see below, the ground state changes to aspin-triplet S = 1, Σ u , state which is a repulsive state (!). It correspondsto two hydrogen atoms at large distances with electron spins antiparallel tothe magnetic field, hence, the hydrogen molecule does not exist as a com-pact system. It is worth noting that this value of the critical magnetic fieldwas calculated accurately in present paper and confirms the rough estimate B cr ≃ . B &
12 a.u., the ground state is realized by a spin triplet S = 1, Π u state,see [17] and references therein. A similar behavior is observed in the caseof the linear H +3 molecular ion in strong magnetic fields: the ground stateevolves from the spin-singlet Σ g state for weak magnetic fields B . × G( ≃ . Σ u state for intermediate andstrong fields and, eventually, to a spin-triplet Π u state for magnetic fields B & × G ( ≃
21 a.u.) [12]. In such studies the parallel configurationof the molecular axis and the magnetic field direction is explicitly assumed.3on-aligned configurations, where the molecular axis is not parallel to themagnetic field direction, have received much less attention. This is due tothe fact that such configurations require a much larger computational effortto reach the accuracies obtained in the parallel case. The present authorsare not aware of any studies of inclined configurations for the H moleculefor B . . arbitrarilyoriented i.e. with the molecular axis forming an angle θ with respect to the di-rection of a uniform magnetic field B in lowest spin-singlet state 1 g . The mag-netic field strengths of interest in this work are chosen to be B = 0 , . , . . , . × , . × and 4 . × G), where theground state is realized by the spin-singlet state Σ g at θ = 0 for B < B cr .We use the variational method with trial functions designed following a cri-terion of physical adequacy [18, 10]. Three inclinations θ = 0 ◦ , ◦ and 90 ◦ will be studied in detail and the potential energy curves for each inclinationand each magnetic field will be constructed. The two-dimensional potentialenergy surfaces are obtained by interpolation in the θ coordinate. This al-lows us to calculate for the first time the lowest rovibrational levels of theH molecule in weak and intermediate magnetic fields, where this moleculeexists as a compact object. A study of the magnetic susceptibility of theH molecule is also performed. We will follow in presentation our previouswork on H +2 in weak and intermediate magnetic fields [15]. Atomic units willbe used through the text.
2. The Hamiltonian and generalities
We consider the hydrogen molecule H interacting with an external mag-netic field B . The origin of coordinates is chosen in the midpoint of the lineconnecting the nuclei (molecular axis). The molecular axis in turn forms anangle θ with respect to the magnetic field direction (chosen to coincide withthe z -axis). A convenient gauge which describes a magnetic field orientedparallel to the z -axis, is the linear gaugeˆ A = B [( ξ − y, ξx, , (1)where ξ is a parameter. If ξ = 0 the linear gauge is reduced to the Landaugauge, and if ξ = 1 / ξ is considered as an extra variationalparameter. 4ince the nucleus mass is by far larger than the electron mass, we canneglect all non-adiabatic coupling terms in the Hamiltonian to obtain theorder zero BO approximation. Thus, the electronic Hamiltonian in atomicunits ( ~ = m e = c = 1) is given byˆ H e = − X i =1 ∇ i − iB X i =1 (( ξ − y i ∂ x i + ξx i ∂ y i ) + S · B + 12 B X i =1 (cid:0) ξ x i + ( ξ − y i (cid:1) − X i =1 (cid:18) r ia + 1 r ib (cid:19) + 1 r + 1 R , (2)where ∇ i is the Laplacian operator with respect to the coordinates of the i -th electron r = ( x i , y i , z i ), r ia,ib are the distances between the i -th electronand the nuclei a or b , respectively, r ij is the distance between the electronsand R is the distance between the nuclei. As usual, the contribution to theenergy due to the Coulomb interaction between the nuclei (1 /R ) is treatedclassically. Hence, R is considered an external parameter. In the particularcase θ = 0 ◦ , the component of the angular momentum along the z -axis isconserved and the term linear in B becomes L · B for ξ = . The spinZeeman term S · B with the total electron spin S = S + S is included inthe Hamiltonian. However, for the spin-singlet states with S = 0 this termdoes not contribute to the total energy and can be excluded.Finally, the nuclear motion can be treated as vibrations and rotationsfollowing the BO approximation with the electronic energy taken as the po-tential in the nuclear Hamiltonian.
3. The trial function
Following physical relevance arguments (see, e.g. [18]) we designed aspatial trial function which is a product of Landau orbitals, Coulomb orbitalsand a correlation term in exponential form: ψ ( r , r ) = Y k =1 ( e − α ka r ka − α kb r kb − Bβkx x k − Bβky y k ) e α r (3)where α ka,kb , β kx , β ky with k = 1 , α are variational parame-ters. In (3) the variational parameters α ka , α kb ( k = 1 ,
2) have the meaningof screening (or anti-screening) factors (charges) for the nucleus a, b respec-tively, as it is seen from the k -th electron. The variational parameters β kx ,5 ky account for the screening (or anti-screening) factors for the magnetic fieldseen from k -th electron in x, y direction respectively, and the parameter α “measures” the screening (or anti-screening) of the electron correlation in-teraction. This spatial function reproduces adequately the behavior of theelectrons near the Coulomb singularities and the harmonic oscillator at longdistances arising from the magnetic field. In a certain way the trial func-tion (3) is a generalization of the trial function presented in [19] for the fieldfree case. It reproduces two physical situations: for small internuclear dis-tances the trial function (3) mimics the interaction H +2 + e (if α a = α b and α a = α b ) while for large internuclear distances it mimics the interactionH − H (if α a = α b and α b = α a ).We consider a trial function which is a superposition of three Ans¨atze:a general Ansatz of the type (3), a H − H type Ansatz and a H +2 + e typeAnsatz Ψ = A ψ + A ψ H + H + A ψ H +2 + e , (4)where A , , are linear variational parameters. Each Ansatz has its own setof variational parameters. Without loss of generality A may be set equal tothe unity, therefore the total number of variational parameters is 27 includingthe internuclear distance R and ξ as variational parameters.In the singlet state ( S = 0) the trial function (4) must be symmetricwith respect to the exchange of the electrons and in the gerade (g) state thetrial function (4) must be symmetric with respect to the exchange of nuclei.Therefore the operator (1 + ˆ P ab )(1 + ˆ P ) , (5)where ˆ P ab is the operator of symmetrization of nuclei and ˆ P is the operatorof symmetrization of the electrons, must be applied to the trial function (4).The calculation of the variational energy using the trial function (3) in-volves two major parts: (i) 6-dimensional numerical integrations which wereimplemented by an adaptive multidimensional integration C -language rou-tine ( cubature ) [20], and (ii) a minimizer which was implemented with theFortran minimization package MINUIT from CERN-LIB. Our C -Fortran hy-brid program was parallelized using MPI. The 6-dimensional integrationswere carried out using a dynamical partitioning procedure: the domain ofintegration is manually divided into sub-domains following the profile of theintegrand. Then each sub-domain is integrated on separated processors us-ing the routine CUBATURE . In total, we have a division into 960 subregionsfor the numerator and ∼ ∼ for the numericalintegrations for each subregion, the time needed for one evaluation of thevariational energy (two integrations) is 2 × seconds ( ∼
37 min) with 96processors at the cluster
KAREN (ICN-UNAM, Mexico). It was checkedthat this procedure stabilizes the estimated accuracy and is reliable in thefirst three to four decimal digits. However, in order to localize the domain ofminimal parameters, a minimization procedure with much less sample pointswas used in each sub-domain, and a single evaluation of the energy usuallytook ∼ −
20 mins. Once a domain is roughly localized, the number ofsample points is increased by a factor of ∼ . Typically, a minimizationprocess required several hundreds of evaluations. As a general strategy, thevariational energy corresponding to the general Ansatz only is calculated infirst place. Then, either the H − H type Ansatz or the H +2 + e type Ansatz isadded as a first correction, depending on which configuration yields a bettervariational result, and the energy is minimized using the superposition oftwo Ans¨atze. Eventually, the remaining configuration is included in the finaltrial function and a final minimization is carried out. The whole process isvery lengthy and cumbersome due to the absence of a fast minimization pro-cedure. Computations were mainly performed in parallel on 96 processors onthe cluster ROMEO at the University of Reims, France, and on the cluster
KAREN at ICN-UNAM, Mexico.
4. Results
The electronic energies and the equilibrium distances of H in the 1 g state are presented in Table 1 for magnetic fields B = 0 , . , .
175 and0 . B ≤ B cr = 0 .
178 a.u. thelowest energy state of H is realized by the 1 g state in parallel configuration.For B cr = 0 .
178 a.u. the energy of the 1 g state at the equilibrium minimumcoincides with the energy of two hydrogen atoms infinitely separated andhaving both electron spins antiparallel to the magnetic field direction. Thus,for B = 0 . g is, in fact, a m-eta-stable state. We studiedthe geometrical configurations with angles θ = 0 ◦ , ◦ and 90 ◦ between themagnetic field direction and molecular axis in great detail, while some samplecalculations were carried out for the intermediate angles θ = 15 ◦ , ◦ and60 ◦ , ◦ to check the smoothness of the angular dependence of both, theenergy and the equilibrium distance (see below). For all inclinations thepotential energy curve E vs R exhibits a well pronounced minimum at a7nite internuclear distance R . As the magnetic field increases, for any giveninclination the system becomes more strongly bound (the binding energyincreases) and more compact (the internuclear equilibrium distance R eq isreduced), see Table 1. Note that for the field-free case B = 0 our energy isin agreement with one of the most accurate results [21] in ∼ × − a.u.in spite of the very simple form of the trial function that we used. We mustemphasize that for parallel configuration θ = 0 our energies are systematicallybetter than the ones from [16] in 3 decimal digits (d.d.), which leads to amore accurate value of the critical magnetic field strength B cr . For a givenmagnetic field, the total energy increases while the equilibrium distance R eq shows a small decrease with growth of the inclination angle from θ = 0 to90 ◦ , see Figures 1 and 2. Such an increase in E , and decrease in R eq , aremore pronounced as the magnetic field increases. Thus, for all magnetic fieldsstudied, the optimal configuration corresponds to the parallel configurationas it is expected. The angular dependence of the variational energy E ( B, θ )and the equilibrium distance R eq ( B, θ ) for a fixed magnetic field strength B is very simple and is well-described by the hindered rotator model, see Eq.(15) and captions of Figs. 1 and 2. This observation is in agreement with thetest calculations for angles θ = 15 ◦ , ◦ , ◦ and 75 ◦ .
5. Potential Energy Curves
Potential energy curves E vs R of the state 1 g of the H molecule inmagnetic fields B = 0 , . , . θ = 0 , ◦ , ◦ are builtfrom variational results obtained in the domain R ∈ [1 ,
2] a.u. and extendedbeyond following the procedure discussed in [22] for approximating potentialcurves in diatomic molecules (see also references therein). It is evident thatthe asymptotic behavior of the electronic energy of H at small distances R → E ≈ R + E He ( B ) + c R + O ( R ) , (6)where E He ( B ) is the ground state energy of the helium atom in a magneticfield ( B ) (the so-called united atom limit), and the coefficient in front of R depends on the magnetic field and the inclination θ , c = c ( B, θ ); at B = 0this coefficient vanishes c = 0 (see [22] and references therein). As for theasymptotic limit R → ∞ , the expansion of the energy E is given by E ≈ E ( B ) + c R − c R + c R + O (cid:18) R (cid:19) , (7)8 able 1: Total electronic energy and equilibrium distance of H in the state 1 g vs magneticfield B and inclination θ based on trial function (4), see text. Energies E and equilibriumdistances R eq rounded to 5th and 3rd d.d., respectively. ∗ For B = 0 . g state isno longer the ground state. Results marked † are from Ref. [16], those marked ‡ from [21](rounded). The binding energy E bind ≡ E (H) − E (H ) with respect to dissociation toH + H is shown in the last column, where the energies for the H atom in ground state aretaken from [9]. B (a.u.) θ (degrees) E (a.u.) R eq (a.u.) E bind (a.u.)0 . − . .
40 0 . − . ‡ . . − . .
397 0 . − . † . † − . .
396 0 . − . .
394 0 . .
175 0 − . .
390 0 . − . .
387 0 . − . .
384 0 . . ∗ − . .
385 0 . − . † . † − . .
382 0 . − . .
379 0 . E ( B ) is the energy of two (infinitely separated) hydrogen atoms intheir ground state in the magnetic field of strength B (however, with oppositeelectron spin projections so that S · B = 0), the term ∝ /R corresponds tothe quadrupole-quadrupole interaction (repulsive for 0 , ◦ and attractive for45 ◦ ) between two separated hydrogen atoms in the magnetic field (which isthe leading order interaction at R → ∞ ). The term ∝ /R corresponds tothe induced dipole-dipole interaction (in second order perturbation theoryin 1 /R for B = 0) between two separated hydrogen atoms (see [11] and[8]). The coefficients c , , can depend on the magnetic field strength andinclination c , , = c , , ( B, θ ). In absence of a magnetic field c , = 0. Ingeneral, the quadrupole-quadrupole interaction energy (in a.u.) is given by E Q = 34 Q zz ( B ) P (cos θ ) R , (8)where Q zz is the quadrupole moment of the hydrogen atom in a magnetic fieldof strength B (see [11]), P is 4th Legendre polynomial. Thus, the coefficient9 E ( a . u . ) θ (deg) B=0B=0.1 a.u.B=0.175 a.u.B=0.2 a.u.
Figure 1: Total energy E of H , g state vs inclination θ for B = 0 and B = 0 . , . . E ( B, θ ) = E ( B, ◦ ) + A sin ( θ ), where A = ( E ( B, ◦ ) − E ( B, ◦ )), see Eq. (15). c is known. For weak magnetic fields B we use the approximation thequadrupole moment in perturbation theory (see [23]) Q zz = − B + 61532 B + . . . . (9)Now we interpolate both asymptotic expansions (6) and (7) via the two-point Pad´e approximant P ade [ N/N + 4]( R ) with N = 2 as the minimaldegree, which guarantees that the expansions (6) and (7) are described func-tionally correct, E ( R ) = 1 R a + a R + a R ( b + b R + b R + b R + b R + b R + b R ) + E ( B ) , (10)where the constraints b = a , b = a ( E ( B ) − E He ( B )) + a , R eq ( a . u . ) θ (deg) B=0B=0.1 a.u.B=0.175 a.u.B=0.2 a.u.
Figure 2: Equilibrium distance R eq of H in the 1 g state vs inclination angle θ for B =0 , . , . , . R eq ( B, θ ) = R eq ( B, ◦ ) + C sin ( θ ),where C = R eq ( B, ◦ ) − R eq ( B, ◦ ) (c.f. Eq (15)). are imposed in order to reproduce the first two leading terms in (6) exactlyplus the condition c = Q zz ( B ), it implies the relation a = c b . Without loss of generality we can set a = 1. Therefore, we have six freeparameters a , b , b , b , b , b to fit the variational energies at internucleardistances near the equilibrium, R ∈ [1 ,
2] a.u. for B = 0 , . , . θ = 0 , ◦ , ◦ . The value of the parameters is presented inTable 2. The potential energy curves are shown in Fig 3. In general, thecurves (10) reproduce four d.d. in energy at R ∈ [1 ,
2] a.u.
6. Magnetic Susceptibility
The trial function (4), in spite of its simplicity, incorporates accuratelythe many physics features of the H molecule in a magnetic field. In orderto verify this assertion for weak magnetic fields we calculated the magnetic11 E T ( a . u . ) R (a.u.)
B=0, θ =0 o B=0.1, θ =0 o B=0.1, θ =45 o B=0.1, θ =90 o B=0.2, θ =0 o B=0.2, θ =45 o B=0.2, θ =90 o -1.16-1.155 1.3 1.4 1.5 -1.16-1.155 1.3 1.4 1.5 -1.16-1.155 1.3 1.4 1.5 -1.16-1.155 1.3 1.4 1.5 -1.17-1.165 1.3 1.4 1.5 -1.17-1.165 1.3 1.4 1.5 -1.17-1.165 1.3 1.4 1.5 Figure 3: Potential Energy curves of the 1 g ground state of the H molecule for B =0 , . , . θ = 0 , ◦ , ◦ . The insets show amplified energy curves for B = 0 . B = 0 . θ = 0 ◦ , while the highest energy curve corresponds to the perpendicularconfiguration θ = 90 ◦ . For B = 0 . corresponds to theasymptotic energy of the repulsive 3 u triplet state (the energy of two hydrogen atomsinfinitely separated with spins antiparallel to the magnetic field direction) and lies belowthe minimum for the 1 g state. susceptibility (magnetizability). To make this calculation we follow the recipeproposed in our work on H +2 [15].It is well known that the response of the molecule to an external magneticfield falls into two parts: diamagnetic and paramagnetic. Correspondingly,there are two contributions to the magnetic susceptibility: a paramagnetic, χ p , originating from the linear Zeeman term in the Hamiltonian (2) whentreated within second order perturbation theory in powers B , and a diamag-netic, χ d , coming from the quadratic Zeeman term ∼ B in the first orderof perturbation theory. Thus, the total magnetic susceptibility is the sum ofthe two terms χ = χ d + χ p .In general, the magnetic susceptibility tensor χ αβ is defined by the coef-12 able 2: Fitted parameters (rounded to 5 d.d.) in the Pad´e approximant (10) for theH potential energy curves E vs R for B = 0 , . , . B (a.u.) θ a b b b b b ◦ -1.28814 1.09317 -0.94343 0.79009 -0.28161 0.047580 . ◦ -1.29188 0.97020 -0.64470 0.51708 -0.17165 0.0313645 ◦ -1.29133 1.16648 -1.12137 0.94652 -0.34039 0.0564490 ◦ -1.29330 1.30768 -1.38159 1.13784 -0.40315 0.063950 . ◦ -1.30393 1.04298 -0.80653 0.65880 -0.22496 0.0388645 ◦ -1.30284 1.09918 -0.99912 0.90103 -0.35196 0.0640890 ◦ -1.30207 1.13486 -1.06132 0.93558 -0.35573 0.06400ficients in the operator H ′ = − X α,β χ αβ B α B β , with B α , B β being the components of the magnetic field. As for the diamag-netic susceptibility it is given in first order PT in B as χ dαβ = − X i =1 (cid:0) h r i i δ αβ − h r i,α r i,β i (cid:1) , (11)where both the expectation value h r i i of the position vector squared of the i -th electron and the 2nd order tensor h r i,α r i,β i , α, β = x, y, z are taken withrespect to the field-free wavefunction at equilibrium distance R eq . If themagnetic field direction is chosen along the z -axis, B = B ˆ z the tensor χ dαβ appears in diagonal form and contains a single non-zero component, χ dzz ≡ χ d , χ d = − X i =1 ( h x i i + h y i i ) , (12)where the symmetric gauge is assumed to be taken. On the other hand,the paramagnetic contribution to the susceptibility is much more difficult tocalculate, since it occurs in second order PT. In general, the paramagneticsusceptibility is much smaller than the diamagnetic one. In principle, thiscontribution to the susceptibility can be easily evaluated as the difference χ p = χ − χ d , where χ is the total magnetic susceptibility at a given inclination.13s for the ground state, the total magnetic susceptibility can be calculated ina straightforward way as the coefficient in front of the B term in the energyexpansion E ( B, θ ) = E (0) − χ ( θ ) | B | + . . . , (13)at R = R eq .The results for the susceptibilities are presented in Table 3 for inclina-tions θ = 0 ◦ , ◦ , ◦ , they are compared with the experimental data fromRamsey [24], and with other calculations, when available. In general, all sus-ceptibilities grow with the inclination angle. For θ = 0 our χ d are larger thanthe values obtained in the past in [25, 26]. They are closer to experimentaldata being different from experimental data in one portion × − in spite ofthe fact that our trial function is much simpler than the ones used in [25]and [26]. As for θ = 45 ◦ , the susceptibilities are calculated for the first timeto the best of our knowledge, while for θ = 90 ◦ our χ d agrees in 2 d.d. with[26] and differs from experimental data in 2 × − . Concerning χ p , it issuperior to all nine values calculated previously and collected in Table XIIof [27], however, it still differs from experimental data in ∼
7. Rovibrational levels
The lowest rovibrational states of H and D were calculated for thefield strengths B = 0 . B and B = 0 . B , where B = 2 . × Gauss= 2 . × T, as described in [15]. To keep the present paper self-contained,the method is briefly summarized below. Starting point is the nuclear Hamil-tonian expressed in spherical coordinates,ˆ H nuc = − M s R ∂ ∂R R + 2 M s R ˆ L R − M s B ˆ L z + 18 M s B R sin θ + ˜ V ( R, θ ) . (14)Here, M s denotes the total mass of the nuclei, ˆ L z is the projection of angularmomentum along z -axis and θ the angle between the molecular and the z -axis. The two-dimensional potential, ˜ V ( R, θ ), is parametrized as a hinderedrotator, where only the lowest expansion term is maintained, to yield˜ V ( R, θ ) = ˜ V ( R,
0) + X n V ,n ( R )2 [1 − cos(2 nθ )]14 able 3: Diamagnetic χ d , paramagnetic χ p and total χ susceptibilities of H in the state1 g for different inclinations θ at R = R eq . The paramagnetic susceptibility χ p obtained asthe difference χ p = χ − χ d (see text) is included for completeness. The expectation valuesof the squared components of the position vector of each electron h x , i , h y , i and h z , i (in a.u.) are also included for B = 0 at the equilibrium distance R eq = 1 .
40 a.u., they wereobtained using the trial function (4). exp
Experimental results from [24], see also Table Iin [28]. Results marked as a are from [25], b from [26], c from [27]. θ h x , i h y , i h z , i χ d χ p χ ◦ . . . − . . − . . a . a . a − . a . a . b . b . b − . b . b − . exp ◦ . . . − . . − . ◦ . . . − . . − . . a . a . a − . a . b . b . b − . b (0 . − . c − . exp . exp ≈ ˜ V ( R,
0) + V ( R ) sin θ (15) V ( R ) = ˜ V ( R, − ˜ V ( R,
0) is the barrier height for a given value of R .The rovibrational wave function can be expanded in terms of vibrationaland rotational basis functions asΨ( R, θ, φ ) = X v,L c v,L ξ v ( R ; θ ′ ) R Y ML ( θ, φ ) (16)where ξ v ( R ; θ ′ ) are solutions of the vibrational part of Eq. (14) at the refer-ence orientation θ ′ , chosen as θ ′ = 0. These are obtained numerically usingthe renormalized Numerov algorithm. The Y ML ( θ, φ ) in the above equationare spherical harmonics.In this basis, the matrix elements of the Hamiltonian in Eq. (14) are givenby D v ′ L ′ M | ˆ H nuc | vLM E = E v δ L ′ L δ v ′ v + 2 M s (cid:28) v ′ | R | v (cid:29) L ( L + 1) δ L ′ L − BMM s δ L ′ L δ v ′ v (cid:20) B M s h v ′ | R | v i + 23 h v ′ | V ( R ) | v i (cid:21) δ L ′ L − (cid:20) B M s h v ′ | R | v i + 23 h v ′ | V ( R ) | v i (cid:21) × ( − M p (2 L ′ + 1)(2 L + 1) × (cid:18) L L ′ (cid:19) (cid:18) L L ′ M − M (cid:19) (17)The terms in parentheses are Wigner 3 j -symbols. The matrix Eq. (17) isdiagonal in M as expected, since M is an exact quantum number. L -functionsare coupled in steps of 2, conserving z -parity, π = ( − L + M . Diagonalizationof the Hamiltonian matrix, Eq. (17), yields the eigenvalues and eigenvectorsof the rovibrational problem.We have computed the lowest rovibrational states for H and D . Al-lowed rovibrational states must obey the permutational symmetry of the twoidentical nuclei. In the case of H , with two fermions, the symmetry of thevibrational and rotational parts of the rovibrational wavefunction must beopposite, while in the case of D , with two bosons, it must be the sameif we consider ortho nuclear spins. For a rovibrational state of given vibra-tional quantum number, v , and projection of the angular momentum on themagnetic field axis, M , the z -parities are thus π = ( − M + v +1 = (cid:26) − ( − M for v even( − M for v odd (18)for H , and π = ( − M + v = (cid:26) ( − M for v even − ( − M for v odd (19)for D .The results of our calculations for the lowest vibrational states, v =0 , , , M ≤ and in Tables 8–11 for D , for the magnetic field strengths B = 0 . B and B = 0 . B . Wenote that at B = 0 . B the molecule is meta-stable. As in our previouswork on H +2 , two models have been considered: the approximate model 1, inwhich off-diagonal terms in v are omitted when setting up the rovibrationalmatrix, Eq. (17), and model 2, in which they are included. The closeness ofthe two sets of results demonstrate that a simple expansion, with just onevibrational function, yields a good approximation of the final rovibrational16avefunction, at least for the lowest vibrational states. Therefore, in thefull expansion of model 2, the coefficients c v,L allow easy identification of thevibrational quantum number of each computed eigenstate.All states are located above the rotational barrier, which is at E barrier = − . E h for B = 0 . B and E barrier = − . E h for B = 0 . B , re-spectively, and hence, L , which is an exact quantum number in the field-freecase, can still be considered a “good” quantum number. It is interesting toanalyse the orientation with respect to the magnetic field axis of the lowestrovibrational state. The lowest state of H , at the field strength of B =0 . B , is located 0 . E h , or 2222 cm − , above the barrier. Yet only oneof the basis functions of the expansion in Eq. (16) contributes effectively toits eigenvector, with coefficient c , = 0 . R, θ, φ ) ∼ [ ξ v ( R ; θ ′ = 0) /R ] Y ( θ, φ ) ∼ [ ξ v ( R ; θ ′ = 0) /R ] cos θ ,which shows that molecule essentially vibrates in the direction of the mag-netic field.In general, within each L -layer, the rotational energy of a vibrational stateincreases with | M | . Figures 5 and 7 show some exceptions for the states v = 1 , and v = 0 , , where the M = 0 state corresponding to L = 2 is above | M | = 1. A similar effect has been observed in the case of H +2 and D +2 . It is due to strong coupling of the L = 0 and L = 2 basis functions,a kind of Fermi resonance of the zero-order states with well-defined L . Theeffect scales as B and is not visible for the lower field strength, B = 0 . B .No strong effect can be seen for the states v = 0 , and v = 1 , ,which have L = 1 , . . . , where the L = 1 and L = 3 layers are sufficientlyseparated in energy. 17 able 4: Rotational energy levels of H in presence of a uniform magnetic field B forthe vibrational state v = 0. The pure vibrational state ( L = 0 in the field-free case) isforbidden but shown here nevertheless as it corresponds to the origin of the rotationalband. In the simple model 1, terms off-diagonal in v are neglected. In model 2, the fullmatrix 17 is diagonalized. Values in parentheses are from Ref. [22]. L Energy/ E h M π
Energy/ E h B = 0 . B = 0 . B = 0 . L = 5 -1.15627 3 1 -1.15184 -1.15200 -1.13952 -1.13965(-1.15660) -2 -1 -1.15177 -1.15193 -1.13944 -1.139582 -1 -1.15188 -1.15204 -1.13966 -1.13979-1 1 -1.15183 -1.15200 -1.13961 -1.139751 1 -1.15189 -1.15206 -1.13972 -1.139860 -1 -1.15187 -1.15204 -1.13970 -1.13984-3 1 -1.15638 -1.15641 -1.14360 -1.143613 1 -1.15655 -1.15657 -1.14393 -1.14393-2 -1 -1.15659 -1.15662 -1.14422 -1.14424 L = 3 -1.16099 2 -1 -1.15670 -1.15672 -1.14444 -1.14445(-1.16130) -1 1 -1.15672 -1.15674 -1.14450 -1.144521 1 -1.15677 -1.15679 -1.14461 -1.144620 -1 -1.15677 -1.15680 -1.14466 -1.14468-1 1 -1.15924 -1.15924 -1.14679 -1.14680 L = 1 -1.16367 1 1 -1.15930 -1.15930 -1.14690 -1.14690(-1.16400) 0 -1 -1.15959 -1.15959 -1.14783 -1.14783 L = 0 -1.16421 0 -1 -1.15994 -1.15995 -1.14795 -1.14796(-1.16455) 18 able 5: Rotational energy levels of H in presence of a uniform magnetic field B for thevibrational state v = 1. See Caption of Table 4 for explications. L Energy/ E h M π
Energy/ E h B = 0 . B = 0 . B = 0 . L = 4 -1.14010 -2 1 -1.13549 -1.13556 -1.12272 -1.12277(-1.14050) 2 1 -1.13560 -1.13567 -1.12294 -1.12299-1 -1 -1.13558 -1.13566 -1.12297 -1.123021 -1 -1.13564 -1.13571 -1.12308 -1.123130 1 -1.13563 -1.13570 -1.12308 -1.12313-2 1 -1.13888 -1.13888 -1.12575 -1.125752 1 -1.13898 -1.13899 -1.12597 -1.12597 L = 2 -1.14364 -1 -1 -1.13918 -1.13918 -1.12666 -1.12666(-1.14405) 1 -1 -1.13923 -1.13924 -1.12677 -1.126770 1 -1.13924 -1.13924 -1.12651 -1.12652 L = 0 -1.14517 0 1 -1.14071 -1.14071 -1.12840 -1.12840(-1.14555) 19 able 6: Rotational energy levels of H in presence of a uniform magnetic field B forthe vibrational state v = 2. The pure vibrational state ( L = 0 in the field-free case) isforbidden but shown here nevertheless as it corresponds to the origin of the rotationalband. See Caption of Table 4 for explications. L Energy/ E h M π
Energy/ E h B = 0 . B = 0 . B = 0 . L = 5 -1.12004 3 1 -1.11524 -1.11540 -1.10198 -1.10208(-1.12055) -2 -1 -1.11518 -1.11535 -1.10196 -1.102072 -1 -1.11529 -1.11546 -1.10217 -1.10229-1 1 -1.11526 -1.11542 -1.10216 -1.102281 1 -1.11531 -1.11548 -1.10226 -1.102390 -1 -1.11530 -1.11547 -1.10226 -1.10238-3 1 -1.11927 -1.11929 -1.10540 -1.105403 1 -1.11944 -1.11946 -1.10572 -1.10573-2 -1 -1.11952 -1.11955 -1.10623 -1.10624 L = 3 -1.12431 2 -1 -1.11963 -1.11966 -1.10645 -1.10646(-1.12475) -1 1 -1.11966 -1.11969 -1.10649 -1.106511 1 -1.11972 -1.11974 -1.10660 -1.106610 -1 -1.11973 -1.11976 -1.10672 -1.10674-1 1 -1.12189 -1.12189 -1.10851 -1.10852 L = 1 -1.12673 1 1 -1.12194 -1.12194 -1.10862 -1.10863(-1.12720) 0 -1 -1.12232 -1.12232 -1.10986 -1.10986 L = 0 -1.12722 0 -1 -1.12258 -1.12258 -1.10991 -1.10992(-1.12765) 20 able 7: Rotational energy levels of H in presence of a uniform magnetic field B for thevibrational state v = 3. See Caption of Table 4 for explications. L Energy/ E h M π
Energy/ E h B = 0 . B = 0 . B = 0 . L = 4 -1.10550 -2 1 -1.10078 -1.10084 -1.08702 -1.08705(-1.10630) 2 1 -1.10089 -1.10095 -1.08723 -1.08727-1 -1 -1.10088 -1.10095 -1.08732 -1.087361 -1 -1.10094 -1.10100 -1.08742 -1.087470 1 -1.10093 -1.10100 -1.08743 -1.08748-2 1 -1.10379 -1.10379 -1.08954 -1.089552 1 -1.10390 -1.10390 -1.08976 -1.08977 L = 2 -1.10893 -1 -1 -1.10412 -1.10412 -1.09074 -1.09074(-1.10945) 1 -1 -1.10417 -1.10417 -1.09085 -1.090850 1 -1.10417 -1.10418 -1.09031 -1.09033 L = 0 -1.11034 0 1 -1.10550 -1.10550 -1.09251 -1.09251(-1.11085) 21 able 8: Rotational energy levels of D in presence of a uniform magnetic field B for thevibrational state v = 0. See Caption of Table 4 for explications. L Energy/ E h M π
Energy/ E h B = 0 . B = 0 . B = 0 . L = 4 -1.158877 -2 1 -1.16012 -1.16013 -1.14790 -1.147912 1 -1.16017 -1.16019 -1.14800 -1.14802-1 -1 -1.16019 -1.16020 -1.14808 -1.148091 -1 -1.16021 -1.16023 -1.14813 -1.148150 1 -1.16022 -1.16024 -1.14814 -1.14816-2 1 -1.16185 -1.16185 -1.14936 -1.149362 1 -1.16190 -1.16190 -1.14947 -1.14947 L = 2 -1.162594 -1 -1 -1.16209 -1.16209 -1.15012 -1.150121 -1 -1.16212 -1.16212 -1.15017 -1.150170 1 -1.16211 -1.16211 -1.14981 -1.14982 L = 0 -1.164212 0 1 -1.16291 -1.16291 -1.15118 -1.1511922 able 9: Rotational energy levels of D in presence of a uniform magnetic field B forthe vibrational state v = 1. The pure vibrational state ( L = 0 in the field-free case) isforbidden but shown here nevertheless as it corresponds to the origin of the rotationalband. See Caption of Table 4 for explications. L Energy/ E h M π
Energy/ E h B = 0 . B = 0 . B = 0 . L = 5 -1.137626 3 1 -1.14507 -1.14511 -1.13255 -1.13257-2 -1 -1.14507 -1.14511 -1.13260 -1.132632 -1 -1.14513 -1.14516 -1.13271 -1.13274-1 1 -1.14513 -1.14517 -1.13274 -1.132771 1 -1.14515 -1.14519 -1.13279 -1.132820 -1 -1.14515 -1.14519 -1.13280 -1.13283-3 1 -1.14716 -1.14716 -1.13423 -1.134233 1 -1.14724 -1.14725 -1.13439 -1.13439-2 -1 -1.14738 -1.14738 -1.13490 -1.13490 L = 3 -1.142112 2 -1 -1.14743 -1.14744 -1.13501 -1.13501-1 1 -1.14748 -1.14749 -1.13496 -1.134971 1 -1.14751 -1.14752 -1.13502 -1.135020 -1 -1.14753 -1.14754 -1.13521 -1.13522-1 1 -1.14862 -1.14862 -1.13616 -1.13616 L = 1 -1.144658 1 1 -1.14864 -1.14864 -1.13621 -1.136220 -1 -1.14899 -1.14899 -1.13715 -1.13715 L = 0 -1.145172 0 -1 -1.14908 -1.14908 -1.13716 -1.1371623 able 10: Rotational energy levels of D in presence of a uniform magnetic field B for thevibrational state v = 2. See Caption of Table 4 for explications. L Energy/ E h M π
Energy/ E h B = 0 . B = 0 . B = 0 . L = 4 -1.122394 -2 1 -1.13318 -1.13320 -1.12030 -1.120322 1 -1.13324 -1.13325 -1.12041 -1.12042-1 -1 -1.13326 -1.13328 -1.12054 -1.120551 -1 -1.13329 -1.13331 -1.12059 -1.120600 1 -1.13330 -1.13332 -1.12058 -1.12060-2 1 -1.13474 -1.13474 -1.12161 -1.121622 1 -1.13480 -1.13480 -1.12172 -1.12173 L = 2 -1.125757 -1 -1 -1.13504 -1.13504 -1.12254 -1.122541 -1 -1.13507 -1.13507 -1.12259 -1.122590 1 -1.13503 -1.13503 -1.12204 -1.12206 L = 0 -1.127217 0 1 -1.13582 -1.13582 -1.12367 -1.1236724 able 11: Rotational energy levels of D in presence of a uniform magnetic field B forthe vibrational state v = 3. The pure vibrational state ( L = 0 in the field-free case) isforbidden but shown here nevertheless as it corresponds to the origin of the rotationalband. See Caption of Table 4 for explications. L Energy/ E h M π
Energy/ E h B = 0 . B = 0 . B = 0 . L = 5 -1.102915 3 1 -1.11933 -1.11936 -1.10609 -1.10611-2 -1 -1.11934 -1.11938 -1.10619 -1.106212 -1 -1.11939 -1.11943 -1.10630 -1.10632-1 1 -1.11940 -1.11944 -1.10634 -1.106361 1 -1.11942 -1.11946 -1.10639 -1.106420 -1 -1.11943 -1.11947 -1.10641 -1.10644-3 1 -1.12123 -1.12124 -1.10752 -1.107523 1 -1.12131 -1.12132 -1.10768 -1.10769-2 -1 -1.12148 -1.12148 -1.10836 -1.10836 L = 3 -1.107489 2 -1 -1.12153 -1.12154 -1.10847 -1.10847-1 1 -1.12158 -1.12159 -1.10828 -1.108291 1 -1.12161 -1.12162 -1.10833 -1.108350 -1 -1.12164 -1.12164 -1.10867 -1.10868-1 1 -1.12262 -1.12262 -1.10960 -1.10960 L = 1 -1.109872 1 1 -1.12264 -1.12264 -1.10966 -1.109660 -1 -1.12303 -1.12303 -1.11073 -1.11073 L = 0 -1.110336 0 -1 -1.12310 -1.12310 -1.10896 -1.1089825 v v v E n e r g y / E h M Figure 4: Rotational structure, up to L = 5, of the four lowest vibrational states of H inthe presence of an external magnetic field B = 0 . B . -1.15-1.14-1.13-1.12-1.11-1.10-1.09-1.08 -6 -4 -2 0 2 4 6v v v v E n e r g y / E h M Figure 5: Rotational structure, up to L = 5, of the four lowest vibrational states of H inthe presence of an external magnetic field of B = 0 . B . v v v E n e r g y / E h M Figure 6: Rotational structure, up to L = 5, of the four lowest vibrational states of D inthe presence of an external magnetic field B = 0 . B . -1.15-1.14-1.13-1.12-1.11-1.10 -6 -4 -2 0 2 4 6v v v v E n e r g y / E h M Figure 7: Rotational structure, up to L = 5, of the four lowest vibrational states of D inthe presence of an external magnetic field of B = 0 . B . . Conclusions We have investigated the problem of the hydrogen molecule vibrating androtating in the presence of an external magnetic field for the field strengths of B = 0 . , . , .
175 a . u . and B = 0 . . u . (4 . × T). It was shown that for
B > B cr = 0 .
178 a.u. H exists in the form of two isolated hydrogen atomswith anti-parallel electron spins to the magnetic field direction. For magneticfields larger than 12 a.u. the molecule gets bound in parallel configurationwith Π u as the ground state, see e.g. [16, 17].Highly accurate variational calculations, based on a few-parameter phys-ically adequate trial function, are carried out for inclined configurations,where the molecular axis forms an angle θ with respect to the direction ofa uniform constant magnetic field. We calculated diamagnetic and param-agnetic susceptibilities (for θ = 45 ◦ for the first time), they closely describeexperimental data and agree very well with other calculations or are superior.The two-dimensional potential energy surfaces were built for magnetic fieldsfor B = 0 . . molecule withrespect to the magnetic field is the most stable one even though the moleculebecomes metastable for B = 0 . B > B cr . This holdstrue also if the vibrational zero-point energy is taken into account. Thoughthe rovibrational ground state is located well above the barrier to perpendic-ular orientation, the vibrating molecule remains in its parallel orientation.The lowest rovibrational states have then been calculated for the first time.Their energy values are reported for the four lowest vibrational states androtational excitation up to M = 5, for both the H and D isotopologues.
9. Acknowledgements
The authors thank the high-performance computer centre ROMEO ofthe University of Reims Champagne-Ardenne, CRIANN of the Region ofNormandy, France and cluster KAREN (ICN-UNAM, Mexico) for generousallowance of super-computer time. The research by J.C.L.V., D.J.N., A.V.T.is partially supported by CONACyT grant A1-S-17364 and DGAPA grantIN108815 (Mexico). This work was also supported by the Programme Na-tional de Plan´etologie (PNP) of CNRS/INSU, co-funded by CNES. Two ofthe authors A.A. and A.V.T. have the honor and the privilege to know closelyVladimir Tyuterev to whom this paper is dedicated.28 eferences [1] L. Woltjer, X-rays and type I supernova remnants., Astrophys. J. 140(1964) 1309–1313. doi:10.1086/148028.[2] F. Pacini, Energy emission from a neutron star, Nature 216 (1967) 567–568. doi:10.1038/216567a0.[3] T. Gold, Rotating neutron stars as the origin of the pulsating radiosources, Nature 218 (1968) 731–732. doi:10.1038/218731a0.[4] P. Goldreich, W. H. 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