Vibrating H - American Chemical Society

Article pubs.acs.org/JPCA

Vibrating H3+ in a Uniform Magnetic Field Héctor Medel Cobaxin* and Alexander Alijah* Groupe de Spectrométrie Moléculaire et Atmosphérique (UMR CNRS 7331), U.F.R. Sciences Exactes et Naturelles, University of Reims Champagne-Ardenne, Moulin de la Housse B.P. 1039, F-51687 Reims Cedex 2, France S Supporting Information *

ABSTRACT: Potential energy surfaces are obtained for singlet H3+ in magnetic fields of up to 2350 T. The magnetic interaction was treated by first-order perturbation theory and the interaction terms computed ab initio. They were then fitted to a functional form and added to a recent, highly accurate adiabatic potential energy surface. In its most stable orientation, the molecule is arranged such that the magnetic field vector is in the molecular plane. The most stable configuration is no longer D3h as in the field-free case, but C2v, though the stabilization energy is extremely small, of the order of 0.01 cm−1 for a 2350 T field. Finally, we have calculated, for a range of magnetic field strengths and orientations, all the vibrational eigenvalues that are below the barrier to linearity in the field-free case.

1. INTRODUCTION When celebrating stunning 45 years of Oka’s work in the field of astrochemistry, what molecule could be more appropriate to report on in this Festschrift than H3+? It was Oka1 who first observed its rotation−vibrational spectrum in the laboratory more than 30 years ago, in 1980. This discovery has kindled research activities dedicated to H3+ in many areas, such as molecular spectroscopy, theoretical chemistry, and astrochemistry. Soon after the discovery of the laboratory spectrum, H3+ was detected in extraterrestrial surroundings: in the atmospheres of giant planets such as Jupiter,2,3 Saturn,4 and Uranus,5 in interstellar space,6 in the central molecular zone of our galaxy7 and even outside of our own galaxy.8 This list, which is not meant to be complete, shows that H3+ is truly ubiquitous. From an astrochemical point of view we note that H3+ acts as a strong acid, or proton donor, and catalyzes a plethora of astrochemical reactions.9,10 For theorists, H3+ is equally interesting. Its geometrical shape, an equilateral triangle, first suggested by Coulson,11 is not compatible with a classical Lewis structure. H3+ is hence the prototype of a molecule with a three-center−two-electron, 3c−2e, bond. Due to this high symmetry, H3+ has no permanent dipole moment and therefore no pure rotational spectrum. Oka’s search for its rotationvibrational spectrum was favored by the accurate calculations of Carney and Porter.12,13 Up to the present, H3+ has continued to serve as a benchmark molecule for the development of theoretical methods. The most recent theoretical papers on H3+ 14−21 demonstrate the enormous progress made since then. It is almost impossible and beyond the scope of this article, to give justice to all the important contributions to the field of H3+. Instead, we refer our readers to the papers of three discussion meetings on Astronomy, Physics, and Chemistry of H3+, organized by Oka.22−24 These meetings were held at the Royal Society London in the years 2000, 2006, and 2012. © 2013 American Chemical Society

Judging from these papers, no investigation has been reported on the vibrational states of H3+ in the presence of magnetic fields, except from early estimates by Warke and Dutta.25 Turbiner and co-workers,19,26,27 recognizing the importance of such work, studied the geometrical deformation of H3+ in a strong field and its stability and orientation. The present article can be considered a continuation of research along a line initiated by those authors. Knowledge of how the rotation− vibrational states are affected by the interaction of the molecule with such a field would likely facilitate the detection of H3+ in the universe at locations with magnetic fields such as interstellar media. We report here the first results on the deformation of the potential energy surface in a weak to medium magnetic field and its effect on the vibrational states.

2. THEORY Calculations of vibrational states of molecules and ions are usually based on two approximations: the separation of the collective motion, or center-of-mass motion, from the internal, vibrational, motion, and the separation of nuclear vibrational and electronic motions. When a molecular ion is placed in a magnetic field, it is subject to the velocity-dependent Lorentz force. The interaction with the magnetic field introduces fundamentally different boundary conditions to the Schrödinger equation compared to the field free case so that the validity of the usual approximations must be questioned. In the first subsection, we will present this aspect in an intuitive way, Special Issue: Oka Festschrift: Celebrating 45 Years of Astrochemistry Received: December 30, 2012 Revised: March 5, 2013 Published: March 5, 2013 9871

dx.doi.org/10.1021/jp312856s | J. Phys. Chem. A 2013, 117, 9871−9881

The Journal of Physical Chemistry A

Article

system. Baye and Vincke30 pointed to the possibility of an approximate separation, which, if needed, may be followed by a perturbational treatment of the induced coupling terms. Schmelcher and Cederbaum31 investigated the validity of the Born−Oppenheimer approximation in magnetic fields. They showed that this approximation is, in principle, valid, but that one should bear in mind the shielding effect of the electrons on the nuclear charges, which interact with the field. This shielding effect is accounted for by the diagonal adiabatic correction term. A further discussion of these issues can be found in ref 32. The first potential energy surfaces for an ion in a magnetic field were reported for H2+ by Larsen33 and Khersonskij.34 The same system was then also studied by Wille,35 Kappes and Schmelcher,36 Turbiner and Lopez-Vieyra,37 and more recently, by Baye et al.38 We are not aware of potential energy surfaces for further ions in the magnetic field, though Telgren et al.39 reported recently on geometry optimizations of molecules in magnetic fields. Turbiner and co-workers studied stability and equilibrium structures of one and two-electron molecules and ions in the magnetic field by use of specially tailored trial functions for the variational problem; see for example refs 40−42. For a general overview on one and two-electron systems see refs 40 and 43. Of particular interest in our context is their work on H3+.19,26 In the first paper they show that this molecule becomes linear at very large field strengths. Furthermore, for large fields a triplet state becomes the electronic ground state. The latter paper deals with the development of efficient trial functions for the variational solution of the electronic Schrödinger equation. 2.2. Hamiltonian for H3+. Under the weak-field conditions described above in which the collective motion is assumed separable from the internal motion and the Born−Oppenheimer approximation holds, the electronic Hamiltonian of H3+ interacting with a constant magnetic field B = (B1, B2, B3) is written as

to make this article more easily accessible to the broad readership of this Festschrift. References to rigorous investigations will be given. In the second subsection, the equations relevant for our problem will be derived. 2.1. General Considerations. To start, let us consider a classical charged particle in a magnetic field. It evolves on a helical trajectory on which it moves freely in the direction of the magnetic field vector, whereas its perpendicular motion is bound. The radius of the helix is known as the Larmor radius. In quantum mechanics, the perpendicular motion becomes quantized,28 its discrete harmonic oscillator levels are known as Landau states. For a more complex charged system such as a molecular ion, the situation is much more complicated. As the motion of the center of mass can no longer be separated from the internal motion, except for its component along the field, it is expected intuitively that the collective Landau oscillation interferes with the nuclear vibration. As far as rotation is concerned, there is only one degree of freedom, that of the rotation around a molecular axis parallel to the field. The other two degrees of freedom are transformed into internal degrees, making the potential energy surface dependent on two more variables than usual. For H3+ it depends on the three internuclear distances plus the two angles describing the orientation of the magnetic field, for any given field strength, which is a parameter. Any motion on such a five-dimensional surface is coupled to the infinite number of Landau oscillators, leading to Feshbach resonances. There are two magnetic interaction terms in the Hamiltonian, the linear Zeeman term, which leads just to a constant energy shift, and the quadratic term, which is responsible for the Landau oscillations. Fortunately, the magnetic interaction terms are usually very small, due to the enormous numerical value of the quantum mechanical unit of the magnetic field B0 = 2.35 × 105 T = 2.35 × 109 G. We consider in the present article field strengths of up to B/B0 = 0.01 or B = 2350 T. For comparison, the strength of the earth’s magnetic field is of the order of ∼10−5 T. To make our problem tractable, we will first assume that the collective Landau oscillation is decoupled from the nuclear vibrations. The energy of the Landau states with excitation quantum number N is given by, neglecting the linear Zeeman shift, B EN = (N + 1/2) MB0

/=

1 2

∑

(pĵ + A j)2 +

j = 1,2

∑ κ = 1,2,3

1 − Rκ

∑ j = 1,2 λ = 1,2,3

1 1 + rj ,λ r12

+ B ·Ŝ

(2)

where p̂j = −i∇j is the 3-vector of the canonical momentum of the jth electron, A is the vector potential of the field B, Ŝ = ∑jŝj is the operator of the total electron spin. The index λ runs over protons 1, 2, and 3, rj,λ is the distance between electron j and proton λ, r12 is the interelectronic distance, and Rκ are the internuclear distances. Choosing the vector potential to be in Coulomb gauge, ∇j·Aj = 0 (which implies [Aj,pj] = 0) and defining it as Aj = 1/2B × rj, we obtain, with the definition of the angular momentum lĵ = rj × p̂j, the final expression of the Hamiltonian

(1)

where M is the mass of the particle. Taking M as the mass of the H3+ ion, we obtain for B/B0 = 0.01 the frequency ω = 1.8 × 10−6 Eh = 0.4 cm−1, which is 4 orders of magnitude smaller than the vibrational frequencies. Thus it appears reasonable to make such a separation. The second approximation is to assume that meaningful vibrational frequencies may be calculated for a fixed orientation of the molecule in the magnetic field, which implies that the molecule does not reorient in the magnetic field during a vibration. Given that a typical vibrational period is about one hundredth of a typical rotational period, this is sensible approximation. The problem of the vibrating H3+ is thus reduced to a vibrational problem on a three-dimensional potential energy surface, which depends on the two orientation angles and on the magnetic field strength. The quadratic magnetic interaction term is evaluated by perturbation theory. Conditions for the separability of the center of mass motion in a homogeneous magnetic field have been derived rigorously in a paper by Avron et al.29 In general, no separation of collective and internal motions seems possible for a charged

/=

1 2 +

∑ j = 1,2

pĵ +

∑ κ = 1,2,3

1 − Rκ

1 1 B ·(L̂ + 2Ŝ) + 2 8

∑ j = 1,2 λ = 1,2,3

∑ j = 1,2

1 1 + rj ,λ r12

[B2 rj 2 − (B ·rj)2 ] (3)

where L̂ = ∑jlĵ is the operator of total angular momentum of the electrons. The first term in the second line of the above equation is the linear Zeeman term, the second the quadratic. 2.3. Weak Magnetic Fields and Energy Correction. If we consider a weak magnetic field, we may use perturbation theory to study the behavior of the system in interaction with it. 9872



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To this end, the Hamiltonian of eq 3 is written as sum of the unperturbed Hamiltonian, /0 1 /0 = 2

∑

2

∑

pĵ +

j = 1,2

κ = 1,2,3

1 − Rκ

∑ j = 1,2 λ = 1,2,3

1 1 + rj ,λ r12

ΔE(B) =

= (4)

∑

[B2 rj 2 − (B ·rj)2 ] (5)

j = 1,2

δE(θ ,ϕ) =

∑

[⟨rj 2⟩ − ⟨(n·̂ rj)2 ⟩]

j = 1,2

1 1 ⟨B ·(L̂ + 2Ŝ)⟩ − 2 2

(10)

Note that the sum over the two electrons has disappeared, as the two electrons have equal contributions. With our definition of the coordinate system, where the nuclei are in the x−y plane, the expectation values satisfy the inequalities ⟨x2⟩ > ⟨z2⟩ and ⟨y2⟩ > ⟨z2⟩ and ⟨xz⟩ = ⟨yz⟩ = 0. Hence, the last two terms of eq 10 vanish identically. The potential energy surface for H3+ in a uniform magnetic field can now be written as

α ,β

where Ψ is the normalized electronic zero-order wave function, solution of eq 4. The last term of the eq 6 contains the magnetic susceptibility tensor

V (R1 ,R 2 ,R3 ,θ ,ϕ;B) = V B = 0(R1 ,R 2 ,R3) + B2 δE(θ ,ϕ) (11)

where V (R1,R2,R3) is the field-free potential energy surface and δE(θ,ϕ) is the correction term derived in eq 10. This surface depends on five coordinates, which are the three internuclear distances and the two orientation angles of the molecule in the field. The magnetic field strength enters as a parameter. To be sure, the expectation values inside the magnetic correction term and hence the term itself also depend on the internuclear distances. B=0

2

(⟨rj ⟩δαβ − ⟨rjαrjβ⟩) (7)

For the singlet ground state of the system, the linear term in B is zero, because L̂ = 0 and Ŝ = 0. So the only important term that contributes to the energy correction is the susceptibility. The energy correction depends on both the intensity and direction of the magnetic field B. In the molecule fixed coordinate system, with origin at the nuclear center of mass and the protons placed in the x−y plane, the Cartesian components of the magnetic field vector can be written in spherical coordinates as ⎛ sin θ cos ϕ ⎞ ⎜ ⎟ B = B⎜ sin θ sin ϕ ⎟ = Bn ̂ ⎜ ⎟ ⎝ cos θ ⎠

1 2 {⟨x ⟩(1 − sin 2 θ cos2 ϕ) 4

− 2⟨xz⟩ sin θ cos θ cos ϕ − 2⟨yz⟩ sin θ cos θ sin ϕ}

∑ χα ,β Bα Bβ

j = 1,2

(9)

− 2⟨xy⟩ sin 2 θ cos ϕ sin ϕ

(6)

∑

B 8

+ ⟨y 2 ⟩(1 − sin 2 θ sin 2 ϕ) + ⟨z 2⟩ sin 2 θ

1 = E(0) + ⟨B ·(L̂ + 2Ŝ)⟩ 2 1 + ∑ [B2⟨rj 2⟩ − ⟨(B·rj)2 ⟩] 8 j = 1,2

χαβ

j = 1,2

where

E(B) = ⟨Ψ|/|Ψ⟩ = ⟨Ψ|/o + >|Ψ⟩ = E(0) + ⟨>⟩

1 =− 4

[B2 ⟨rj 2⟩ − ⟨(B ·rj)2 ⟩]

= B2 δE(θ ,ϕ)

The energy of the system up to first order in perturbation theory is given by

= E(0) +

∑ 2

and the perturbation > 1 1 > = B ·(L̂ + 2Ŝ) + 2 8

1 8

3. RESULTS AND DISCUSSION In this section we show how the potential energy surface and the vibrational frequencies are altered if the molecule interacts with a not too strong uniform magnetic field. 3.1. Cuts through the Five-Dimensional Potential Energy Surface. To understand better the five-dimensional surface, we consider here some well-defined three-dimensional cuts by searching for those values of the orientation angles that make the correction term, eq 10, stationary. Applying the gradient operator to the function δE(θ,ϕ) and setting the result equal to zero

(8)

where θ is the angle between the molecular z-axis and the field and ϕ is the angle formed by the projection of the field on the plane with respect the x-axis (Figure 1). This yields for the firstorder energy correction

⎛∂ ⎞ ⎜ ⎟ ∂θ ⎜ ⎟δE(θ ,ϕ) = 0 ∇θ ,ϕ δE(θ ,ϕ) = ⎜ 1 ∂ ⎟ ⎜ ⎟ ⎝ sin θ ∂ϕ ⎠

(12)

yields the following two equations for the critical angles: [⟨x 2⟩ + ⟨y 2 ⟩ − 2⟨z 2⟩ + (⟨x 2⟩ − ⟨y 2 ⟩) cos 2ϕ + 2⟨xy⟩ sin 2ϕ] sin 2θ =0 [(⟨x 2⟩ − ⟨y 2 ⟩) sin 2ϕ − 2⟨xy⟩ cos 2ϕ] sin θ = 0 Figure 1. Coordinate system for magnetic field B.

(13) 9873



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Table 1. Expectation Values of ⟨x2⟩, ⟨y2⟩, ⟨z2⟩, and ⟨xy⟩ for the Classical 69 Geometrical Configurations point

R1

R2

R3

⟨x2⟩

⟨y2⟩

⟨z2⟩

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

1.1781 1.2807 1.3923 1.5146 1.6500 1.8016 1.9738 2.1731 2.4096 2.7005 1.0245 1.1544 1.2991 1.4625 1.8701 2.1366 2.4743 1.0459 1.3257 1.0097 1.1379 1.4415 1.6257 0.9744 1.0990 1.2372 1.5689 1.7743 2.0193 1.0612 1.1951 1.3448 1.7106 1.9425 2.2265 1.1150 1.2550 1.4124 1.5921 2.0526 2.3657 2.7825 1.2124 1.3643 1.5368 1.7366 2.2657 2.6456 3.1901 1.3179 1.4838 1.6748 1.8996 2.5222 3.0051 1.6158 1.8295 2.0867 2.8437 3.5086 2.3062

1.1781 1.2807 1.3923 1.5146 1.6500 1.8016 1.9738 2.1731 2.4096 2.7005 2.1366 1.9964 1.8701 1.7553 1.5528 1.4625 1.3782 1.2498 1.1103 1.4415 1.3585 1.2073 1.1379 1.6675 1.5689 1.4775 1.3123 1.2372 1.1662 1.8213 1.7106 1.6088 1.4269 1.3448 1.2678 2.3657 2.1995 2.0526 1.9209 1.6926 1.5921 1.4991 2.6456 2.4415 2.2657 2.1114 1.8497 1.7366 1.6328 3.0051 2.7407 2.5221 2.3356 2.0290 1.8996 3.1331 2.8437 2.6081 2.2379 2.0867 2.9557

1.1781 1.2807 1.3922 1.5146 1.6500 1.8016 1.9738 2.1731 2.4096 2.7005 2.1366 1.9964 1.8701 1.7553 1.5528 1.4625 1.3782 1.2498 1.1103 1.4415 1.3585 1.2073 1.1379 1.6675 1.5689 1.4775 1.3123 1.2372 1.1662 1.8213 1.7106 1.6088 1.4269 1.3448 1.2678 2.3657 2.1995 2.0526 1.9209 1.6926 1.5921 1.4991 2.6456 2.4415 2.2657 2.1114 1.8497 1.7366 1.6328 3.0051 2.7407 2.5221 2.3356 2.0290 1.8996 3.1331 2.8437 2.6081 2.2379 2.0867 2.9556

0.51662 0.56533 0.62100 0.68531 0.76055 0.84999 0.95848 1.09362 1.26809 1.50519 0.62395 0.64964 0.67966 0.71574 0.81842 0.89714 1.01230 0.49551 0.54246 0.51927 0.54019 0.59632 0.63567 0.54482 0.56593 0.59084 0.65859 0.70714 0.77266 0.59380 0.61892 0.64878 0.73154 0.79259 0.87784 0.68350 0.71399 0.75005 0.79413 0.92411 1.02911 1.19323 0.75295 0.78917 0.83276 0.88735 1.05642 1.20320 1.45728 0.83572 0.87853 0.93157 1.00038 1.22869 1.44888 0.98754 1.05218 1.14091 1.46650 1.83763 1.32127

0.51662 0.56533 0.62100 0.68531 0.76055 0.84999 0.95848 1.09362 1.26809 1.50519 0.97294 0.91414 0.86074 0.81034 0.70838 0.64946 0.57605 0.53960 0.49282 0.62021 0.59274 0.53684 0.50578 0.72171 0.68716 0.65401 0.58659 0.54872 0.50420 0.80860 0.76607 0.72555 0.64326 0.59645 0.54005 1.12276 1.04572 0.97692 0.91284 0.78399 0.70799 0.60850 1.32078 1.21562 1.12388 1.03993 0.87279 0.77131 0.62764 1.59833 1.44620 1.31769 1.20301 0.97830 0.83568 1.78352 1.58946 1.42280 1.10482 0.88655 1.74196

0.41267 0.44033 0.47046 0.50340 0.53963 0.57969 0.62429 0.67429 0.73073 0.79479 0.53150 0.53498 0.53750 0.53908 0.53902 0.53706 0.53340 0.41243 0.41243 0.43916 0.44004 0.44003 0.43911 0.46722 0.46900 0.47009 0.47007 0.46889 0.46679 0.49949 0.50163 0.50295 0.50292 0.50140 0.49865 0.57028 0.57428 0.57717 0.57903 0.57893 0.57637 0.57136 0.61368 0.61816 0.62138 0.62350 0.62332 0.61988 0.61269 0.66275 0.66769 0.67105 0.67336 0.67302 0.66817 0.72436 0.72740 0.72967 0.72899 0.72156 0.79366

9874

⟨xy⟩ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0



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Table 1. continued point

R1

R2

R3

⟨x2⟩

⟨y2⟩

⟨z2⟩

⟨xy⟩

62 63 64 65 66 67 68 69

3.2754 1.3923 1.6500 1.9738 1.3923 1.6500 1.9738 1.6500

2.4881 1.7159 2.0595 2.5321 1.9151 2.3276 2.9426 2.6679

2.4881 1.1346 1.3409 1.5873 1.0230 1.2106 1.4308 1.0925

1.82728 0.42854 0.49380 0.56930 0.37931 0.42314 0.45637 0.34740

1.25534 0.56565 0.79683 0.86761 0.52048 0.62818 0.75665 0.51676

0.79219 0.41162 0.46664 0.52885 0.37445 0.41682 0.45167 0.34750

0.0 −0.00011113 −0.00027178 −0.00072298 −0.00016093 −0.00037844 −0.00081781 −0.00011536

Their solutions are θc = 0, π, and any ϕ, and

of inertia of the unperturbed molecule can be chosen to coincide with these specific directions. 3.2. Ab Initio Calculations and Fit of Data. The electronic expectation values were calculated with the MOLPRO package44 at the MRCI/cc-pV6Z level. A molecular fixed coordinate system was defined with origin at the center of mass and the molecule in the plane z = 0. The first two atoms were placed such that their distance vector is parallel to the xaxis, with identical, negative y-coordinates. The third atom was then placed according to the two remaining bond lengths and has a positive value of the y-coordinate. With this, for symmetric configurations the nuclei are laid on the principal axes of inertia and the y-axis coincides with the 2-fold symmetry axis, C2. See Figure 1 for this coordinate system and the orientation of the magnetic field vector. Initial computations were done for the classical 69 configurations around the minimum proposed by Meyer et al.45 These data are collected in Table 1. As the table shows, the expectation values can be of significant size; hence higher order polynomials had to be employed to obtain a good fit, which, in turn, requires a sufficient amount of data points. Therefore, additional configurations were generated, using the mapping procedure of Meyer et al.. It is based on the usual symmetry coordinates

θc = π /2 ϕc = ±

⎛ ⟨x 2⟩ − ⟨y 2 ⟩ 1 arccos⎜⎜ 2 2 2 2 2 ⎝ (⟨x ⟩ − ⟨y ⟩) + 4⟨xy⟩

⎞ ⎟ ⎟ ⎠

(14)

where the sign of ϕc depends on the sign of the expectation value ⟨xy⟩. It can be shown that the latter set of angles minimizes the correction term. Inserting these angles into eq 10 gives the expression for the minimum energy correction, δEmin =

1 2 [⟨x ⟩ + ⟨y 2 ⟩ + 2⟨z 2⟩ 8 −

(⟨x 2⟩ − ⟨y 2 ⟩)2 + 4⟨xy⟩2 ]

(15)

Two important observations can be made: First, θc = π/2, which implies that the field vector is in the molecular plane. Second, eq 15 shows that the equilateral triangular configuration is no longer stable. For such configurations, the square root term vanishes identically. To understand better the meaning of this correction, let us consider the special case of C2v configurations. Here, ⟨xy⟩ = 0 (when the symmetry axis is parallel to one of the coordinate axes) and the critical angle ϕc of eq 15 assumes a simple expression, ϕc = 0, π/2. The energy correction becomes δE x ,y ,z

⎧⟨y 2 ⟩ + ⟨z 2⟩ for θ = π /2, ϕ = 0 ⎪ 1⎪ = ⎨⟨x 2⟩ + ⟨z 2⟩ for θ = π /2, ϕ = π /2 4⎪ ⎪⟨x 2⟩ + ⟨y 2 ⟩ for θ = 0 ⎩

Sa =

1 (R1̃ + R̃ 2 + R3̃ ) 3

Sx =

1 (2R1̃ − R̃ 2 − R3̃ ) 6

Sy =

1 (R̃ 2 − R3̃ ) 2

(16) (17)

(19)

Morse functions of the deformation coordinates, R̃ κ = [1 − exp(−β(Rκ/R0 − 1))]/β, β = 1.3, R0 = 1.65 a0 are then employed to generate the grid in terms of the internuclear distances, Rκ. Just as these authors, we have separated our configurations in two different domains, one corresponding to symmetric configurations (isosceles triangles) and the other to nonsymmetric configurations (scalene triangles). A total of 1615 different triangular configurations (952 isosceles and 663 scalene) were calculated, the data are not reported here. Our ranges and step sizes for the symmetric configurations were, using Meyer’s notation,

(18)

which are the x-, y-, and z-directions in the nuclear center of mass coordinate system (Figure 1). The z-orientation of the magnetic field does not correspond to a minimum as this has to be in-plane. The minimum will either be in the x- or in yorientation. Simplifying eq 15 shows that the minimum corresponds to x-orientation if ⟨x2⟩ > ⟨y2⟩, which is attained for triangles for which the two equal sides are smaller than the third side. Otherwise, for triangles with the two equal sides larger than the third side, which have ⟨x2⟩ < ⟨y2⟩, the minimum is at y-orientation. For equilateral triangular configurations, all in plane orientations are equivalent. If the molecule was to vibrate on the minimum energy surface, it would have to reorient in the magnetic field as it passes through the D3h configuration. Such a process is not likely, as we have discussed in the General Considerations. To study the interaction of H3+ with a magnetic field, we have chosen the three fixed orientations of eq 18. The principal axes

Sa ∈ [−0.70, 0.80] ΔSa = 0.05 Sx ∈ [−0.65, 0.50] ΔSx = 0.05 Sy = 0

(20)

whereas for the nonsymmetric configurations 9875



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Sa ∈ [−0.40, 0.40] ΔSa = 0.02 Sx = 0 Sy ∈ [0.20, 0.50]

ΔSx = 0.02

(21)

The electronic expectation values so generated were fitted to polynomials of the form PN (Q 1 ,Q 2 ,Q 3) =

∑

cijkQ 1iQ 2jQ 3k (22)

i+j+k≤N

where the functions Qi, Q 1 = R1̃

Q 2 = R̃ 2 + R3̃

Q 3 = R̃ 2 − R3̃

(23)

reflect the reduced symmetry of the molecule when interacting with the field. An expectation value might be symmetric or antisymmetric with respect to a permutation of particles 1 and 2. Hence Q3 can take either even (for x2, y2, z2) or odd (for xy) powers. The functions R̃ κ were simply taken as displacement coordinates, R̃ κ = Rκ − Rref, from the reference configuration. Each set of the expectation values, consisting of 1615 data points, was fitted to a polynomial of degree 10, with 161 coefficients, for the symmetric functions, or of degree 8, with 70 coefficients, for the antisymmetric function. To obtain the potential energy surface for a given orientation of the magnetic field, the magnetic correction term, eq 9 is then added to the very accurate, global potential energy surface by Pavanello et al.,20,21 which includes diagonal adiabatic and relativistic corrections. 3.3. Three-Dimensional Minimum Surface. In Figure 2 the energy correction is shown for the 62 symmetric

Figure 3. Minimum energy path with reorientation of the molecule in the magnetic field of B/B0 = 0.1 or B = 23500 T. The two other coordinates remain practically constant, ρ = 2.165 a0 and θhyp = 0.14°.

θhyp is the angle of latitude, where θhyp = 0° corresponds to the pole of the hemisphere and θhyp = 90° to the equator. The pole is the locus of the equilateral triangle, whereas all linear configurations are found on the equator. ϕhyp is the angle of longitude. ϕhyp = 30° ± n × 60°, n = 0, 1, 2, ... indicate C2v configurations. Note that the 3-fold symmetry of the field-free surface, which allows reduction of the range of the angle ϕhyp to a 60° interval, is lowered as the molecule changes along the path its orientation with respect to the magnetic field. The minimum energy path extends from ϕhyp = −90° to ϕhyp = 90°, linking two different C2v configurations, the first with characteristic angle ∠HHH < 60°, the second with ∠HHH > 60°. The lower of the two configurations, at ϕhyp = −90°, corresponds to y-orientation, the upper to x-orientation. A barrier is located at ϕhyp = 12°, which is a nonsymmetric configuration. Except for ϕhyp = ±30°, all intermediate configurations along the path are nonsymmetric. This may be understood from Figure 4, which shows the evolution of the three side lengths. The difference between the two minima is only 42.5 nEh or 0.009 cm−1, whereas the barrier height, from

Figure 2. Energy correction δE = ΔE/B 2 (hartree), for C 2v configurations, as function of the sides of the isosceles triangle Rk (bohr). The contour lines clearly show the ridge at D3h configurations, R1 = R2.

configurations of Meyer et al. as function of the sides of the triangle R1, R2 = R3. We note that the configurations with R1 < R2 are y-oriented, whereas those with R1 > R2 are x-oriented, just as discussed in the text following eq 18. Configurations at identity, R1 = R2 = R3, correspond to a peak in the energy correction. This peak is caused by an intersection of the x- and y-directed surfaces. Figure 3 shows the minimum energy path, parametrized using Johnson’s hyperspherical coordinates,46 as a function of the hyperspherical angle ϕhyp. The computer code of Mendes Ferreira et al.47 was employed, where details can be found. In the hyperspherical mapping, all configurations are projected on a hemisphere with (hyper)radius ρ, which controls the overall size of the triangle. The two remaining coordinates are the hyperangles θhyp and ϕhyp; their interpretation is the following:

Figure 4. Variation of the three side lengths along the minimum energy path for a magnetic field of B/B0 = 0.1 or B = 23500 T. The mean value of the three side lengths is constant and equal to the side length of the most stable equilateral triangle in y-orientation. 9876



Article

Figure 5. Variation of the three side lengths R1, R2 = R3 of the minimum energy triangle as a function of the magnetic field B/B0, for x-orientation (left) and y-orientation (right).

right to left, is 38.4 nEh or 0.008 cm−1. This is at the limit of what can be resolved numerically. An interesting question is how the minimum configurations depend on the magnetic field strength. In Figure 5, we have plotted the three side lengths of the most stable triangles for xand y-orientations, and a range of magnetic field strengths. We observe first of all that the distances are decreased significantly and second that the difference between the two characteristic side lengths of the C2v configurations augments rapidly with increasing field. This should affect the relative stability of the xand y-orientations. We have seen that at weak fields the yorientation is more stable, but at larger fields it must be the xorientation. The rationale is provided by Figure 4: In the yorientation, the molecular triangle becomes sharper, which destabilizes the molecule due to increased Coulomb repulsion. In the x-orientation, however, the triangle widens up and one may speculate that eventually it will be transformed into a linear configuration. This would explain the result obtained by Turbiner et al.26 that for strong fields H3+ becomes linear. Unfortunately, our perturbative approach does not permit a quantitative analysis extending to very strong fields. 3.4. Three-Dimensional Cuts through the Surface at Fixed Orientations. The shape of the potential energy surface is considerably changed by the magnetic interaction terms. The equilateral triangular structure is no longer the most stable configuration. Though it is preserved if the field is oriented perpendicular to the molecular plane, such an orientation is not stable. Furthermore, we observe shrinking of the equilibrium distances, an effect well-known from other molecules.48 Figure 6 shows cuts through the potential energy surface along the three axes, x, y and z, for a field of B = 47000 T. This high field strength was chosen to make the differences due to orientation clearly visible. In the plots, the two equal distances were held constant at their equilibrium value Req. At equilateral configurations, the x- and y-orientations are equivalent and the correction terms coincide. For smaller distances of the third bond compared to Req, the y-curve is lower, whereas for larger distances it is the x-curve. When the free R approaches its maximum value of 2Req, the y-curve approaches the z-curve, as these give the two equivalent orientations of the field vector perpendicular to the linear molecule axis. The x-curve arrives at the more stable linear configuration parallel to the field axis. We note that our internal coordinate system was defined such that in the linear configurations the three nuclei are located on the x-axis. The result that for linear configurations the parallel

Figure 6. Cuts through the potential energy surfaces corresponding to three orientations x, y, and z of the magnetic field B/B0 = 0.2. Two internuclear distances were kept at their respective equilibrium values for the three orientations, which are Req = 1.62806, 1.63605, and 1.62580 a0. The third distance is varied. The strong field was chosen to make the differences visible.

orientation is preferred has been obtained before by Turbiner et al.26 3.5. Calculation of Vibrational States. The vibrational calculations were performed with the hyperspherical harmonics code.49,50 Three orientations of the magnetic field were considered, x-, y-, and z-orientation. For symmetric configurations, the molecule is thus oriented in the field along one of its principal axes. As we have discussed in the Theory, because vibration is much faster than rotation, it is reasonable to assume that the vibrating molecule keeps its orientation. In Tables 2−4 we present data for the lowest vibrational states, up to approximately 10 000 cm−1 in the field-free case, and follow them as the magnetic field strength is increased. In the first line of these tables, the absolute energy of the hypothetical lowest vibrational state, (0,0°), is given for reference. The energy values of the excited states are presented relative to (0,0°). Note that for (fermionic) H3+ totally symmetric states are not permitted by the Pauli principle. The lowest rovibrational state of H3+ has J = 1. Still the vibrational A′1 data are required to estimate the bandheads of a rovibrational transition. The vibrational states are classified by three quantum numbers: that of the totally symmetric vibration, v1, the 9877


9878

10211.45 10594.36

A2′ (1,33) A′1 (2,2°)

A1 B2 A1 A1 A1 B2 A1 B2 A1 A1 B2 A1 B2 A1 A1 B2 A1 B2 A1 A1 B2 A1 A1 B2 A1 A1 B2 B2 A1

(0,1,0) (0,0,1) (1,0,0) (0,2,0) (0,0,2) (0,1,1) (1,1,0) (1,0,1) (2,0,0) (0,1,2) (0,2,1) (0,3,0) (0,0,3) (1,2,0) (1,0,2) (1,1,1) (2,1,0) (2,0,1) (0,4,0) (0,2,2) (0,3,1) (3,0,0) (1,1,2) (1,2,1) (1,3,0) (0,0,4) (0,1,3) (1,0,3) (2,2,0)

A1 (0,0,0)

Γ(S2) (v1,v2,v3)

2521.71 2522.35 3178.80 4779.64 4999.63 4999.75 5555.71 5556.08 6263.27 7008.49 7008.45 7288.30 7497.04 7773.58 7872.97 7874.94 8491.56 8491.12 9005.47 9115.49 9118.02 9253.95 9662.34 9658.01 9974.30 10002.90 10007.63 10220.23 10602.50

−1.3234459

B = 0.001

2522.07 2524.62 3179.19 4782.46 5003.11 5003.51 5558.24 5559.73 6264.43 7013.51 7013.43 7293.95 7506.87 7783.50 7878.45 7885.77 8498.76 8497.25 9013.50 9119.93 9128.78 9256.62 9682.13 9667.18 9985.09 10016.64 10032.16 10244.12 10623.86

−1.3234423

B = 0.002

B = 0.004

B = 0.005

Zero Point Energy (Eh) −1.3234364 −1.3234281 −1.3234176 Relative Vibrational Energies (cm−1) 2522.64 2523.41 2524.37 2528.35 2533.49 2539.96 3179.80 3180.63 3181.62 4786.95 4792.87 4799.93 5008.85 5016.79 5026.83 5009.54 5017.58 5027.32 5562.23 5567.47 5573.72 5565.60 5573.45 5583.01 6266.24 6268.56 6271.26 7021.12 7030.62 7041.41 7021.18 7031.18 7042.95 7302.89 7314.65 7328.79 7522.29 7542.37 7566.22 7798.80 7818.20 7840.43 7887.23 7898.98 7913.48 7902.18 7922.84 7946.69 8509.40 8522.42 8537.06 8506.66 8518.60 8532.45 9025.04 9038.95 9054.48 9126.96 9136.26 9147.56 9144.11 9162.59 9183.28 9260.50 9265.18 9270.36 9709.38 9740.99 9775.22 9680.88 9697.68 9716.46 9999.41 10016.09 10034.72 10038.44 10065.92 10096.70 10066.15 10105.34 10146.98 10279.11 10321.92 10369.94 10652.77 10683.28 10711.01

B = 0.003

2525.50 2547.67 3182.75 4807.85 5038.87 5038.50 5580.80 5594.04 6274.23 7053.04 7056.08 7344.89 7593.08 7864.24 7930.72 7972.98 8552.77 8547.73 9071.12 9160.59 9205.53 9275.83 9811.09 9736.41 10055.02 10129.11 10189.34 10420.87 10736.98

−1.3234049

B = 0.006

2526.76 2556.53 3184.00 4816.37 5052.78 5050.86 5588.54 5606.31 6277.38 7065.17 7070.19 7362.57 7622.27 7888.34 7950.89 8001.23 8569.20 8564.06 9088.51 9175.14 9228.85 9281.47 9847.95 9757.00 10076.72 10162.13 10231.38 10472.64 10762.61

−1.3233900

B = 0.007

2528.15 2566.44 3185.32 4825.25 5068.39 5064.18 5596.82 5619.61 6280.63 7077.58 7084.96 7381.49 7653.17 7911.63 7974.27 8031.14 8586.11 8581.17 9106.42 9191.03 9252.90 9287.20 9885.33 9777.87 10099.61 10195.18 10272.45 10523.42 10788.14

−1.3233730

B = 0.008

2529.64 2577.32 3186.70 4834.32 5085.55 5078.28 5605.55 5633.77 6283.95 7090.12 7100.14 7401.37 7685.25 7933.44 8000.79 8062.55 8603.35 8598.86 9124.67 9208.07 9277.41 9292.97 9922.79 9798.79 10123.61 10227.98 10312.22 10571.76 10813.53

−1.3233541

B = 0.009

2531.23 2589.07 3188.11 4843.43 5104.06 5093.00 5614.66 5648.65 6287.29 7102.71 7115.54 7421.96 7718.00 7953.72 8029.94 8095.39 8620.81 8616.98 9143.18 9226.12 9302.21 9298.74 9959.93 9819.63 10148.69 10260.42 10350.58 10616.77 10838.70

−1.3233332

B = 0.010

a The zero point energy, which increases with increasing field strength, is given in absolute values. The energy values of the remaining states are given relative to the zero-point energy. Their units are wavenumbers, cm−1. Quantum numbers are according to S3 permutational symmetry for zero field, the correlation to S2 quantum numbers for non-zero field is given for convenience. Note that states with symmetry A1′ or A1 do not exist for a fermionic system such as H3+.

9969.98 9998.30

A′1 (1,33) E′ (0,44)

9252.98 9654.73

7286.34 7493.65 7770.11 7871.10

A1′ (0,33) A2′ (0,33) A′1 (1,2°) E′ (1,22)

A′1 (3,0°) E′ (1,31)

6262.86 7006.72

A1′ (2,0°) E′ (0,31)

9002.48 9113.96

5554.84

E′ (1,11)

A1′ (0,4°) E′ (0,42)

3178.67 4778.68 4998.47

A′1 (1,0°) A1′ (0,2°) E′ (0,22)

8488.97

2521.59

E′ (0,11)

E′ (2,11)

−1.3234471

B = 0.000

A′1 (0,0°)

Γ(S3) (v1,v|l|2 )

Table 2. Energies of the Lowest Vibrational States with the Magnetic Field in x-Orientation and Field Strengths of up to B/B0 = 0.01 or 2350 Ta

The Journal of Physical Chemistry A Article


7286.34 7493.65 7770.11 7871.10

A1′ (0,33) A2′ (0,33) A′1 (1,2°) E′ (1,22)

9879

10211.45 10594.36

A2′ (1,33) A′1 (2,2°)

A1 B2

A1 B2

A1 B2

A1 B2 A1 B2

A1 B2

A1 B2 A1 B2

A1 B2

2521.60 2522.33 3178.77 4779.29 4999.50 4999.29 5555.17 5556.09 6263.17 7007.19 7007.95 7287.75 7495.32 7772.61 7872.27 7873.41 8489.87 8491.25 9003.87 9114.58 9115.55 9253.68 9655.69 9656.48 9972.73 10002.01 10003.29 10213.95 10599.40

−1.3234460

B = 0.001

2521.61 2524.52 3179.07 4781.07 5002.61 5001.70 5556.12 5559.78 6264.05 7008.53 7011.54 7291.88 7500.30 7779.76 7875.89 7880.04 8492.45 8497.87 9007.88 9116.46 9120.19 9255.66 9658.51 9661.47 9979.26 10013.78 10017.71 10221.08 10612.70

−1.3234427

B = 0.002

2521.63 2528.12 3179.55 4783.86 5007.80 5005.60 5557.63 5565.77 6265.44 7010.60 7017.27 7298.51 7508.41 7790.47 7882.37 7890.32 8496.38 8508.31 9014.10 9119.65 9127.53 9258.67 9663.04 9669.12 9987.03 10033.47 10040.09 10232.04 10628.64

−1.3234373

B = 0.003

B = 0.005

Zero Point Energy (Eh) −1.3234298 −1.3234202 Relative Vibrational Energies (cm−1) 2521.65 2521.66 2533.08 2539.32 3180.18 3180.95 4787.45 4791.56 5015.08 5024.42 5010.81 5017.15 5559.62 5562.01 5573.86 5583.82 6267.25 6269.37 7013.21 7016.16 7024.82 7033.82 7307.29 7317.85 7519.47 7533.22 7803.17 7816.12 7892.42 7906.82 7903.37 7918.40 8501.30 8506.91 8521.92 8538.05 9021.94 9030.81 9124.21 9130.27 9137.13 9148.50 9262.39 9266.59 9669.08 9676.36 9678.80 9690.02 9995.40 10004.52 10058.82 10087.30 10068.58 10101.31 10246.06 10262.59 10641.09 10650.21

B = 0.004

2521.66 2546.75 3181.83 4795.94 5035.79 5024.43 5564.71 5595.41 6271.71 7019.30 7043.94 7329.83 7549.40 7827.99 7926.02 7934.74 8513.00 8556.17 9040.12 9137.98 9161.23 9271.08 9684.64 9702.37 10014.47 10117.13 10136.56 10281.24 10658.22

−1.3234086

B = 0.006

2521.63 2555.28 3182.78 4800.37 5049.09 5032.48 5567.68 5608.41 6274.21 7022.52 7054.85 7342.91 7567.74 7838.29 7949.66 7951.90 8519.45 8575.82 9049.37 9147.47 9174.96 9275.73 9693.66 9715.55 10025.25 10147.24 10172.86 10301.73 10666.05

−1.3233951

B = 0.007

2521.59 2564.81 3183.80 4804.67 5064.19 5041.15 5570.88 5622.61 6276.79 7025.74 7066.28 7356.81 7587.97 7847.24 7976.81 7969.54 8526.19 8596.65 9058.21 9158.79 9189.40 9280.46 9703.19 9729.36 10036.80 10177.06 10208.99 10323.90 10674.01

−1.3233796

B = 0.008

2521.52 2575.25 3184.84 4808.72 5080.89 5050.32 5574.29 5637.83 6279.42 7028.90 7078.00 7371.30 7609.81 7855.27 8006.40 7987.44 8533.19 8618.36 9066.45 9171.88 9204.34 9285.20 9713.01 9743.63 10049.03 10206.29 10243.99 10347.59 10682.19

−1.3233624

B = 0.009

2521.43 2586.50 3185.91 4812.44 5099.00 5059.88 5577.90 5653.91 6282.06 7032.00 7089.86 7386.21 7632.97 7862.76 8037.50 8005.46 8540.46 8640.73 9074.06 9186.59 9219.60 9289.91 9722.97 9758.20 10061.88 10234.84 10277.20 10372.63 10690.60

−1.3233435

B = 0.010

a The zero point energy, which increases with increasing field strength, is given in absolute values. The energy values of the remaining states are given relative to the zero-point energy. Their units are wavenumbers, cm−1. Quantum numbers are according to S3 permutational symmetry for zero field, the correlation to S2 quantum numbers for non-zero field is given for convenience. Note that states with symmetry A1′ or A1 do not exist for a fermionic system such as H3+.

9969.98 9998.30

A′1 (1,33) E′ (0,44)

9252.98 9654.73

6262.86 7006.72

A′1 (2,0°) E′ (0,31)

A′1 (3,0°) E′ (1,31)

5554.84

E′ (1,11)

9002.48 9113.96

3178.67 4778.68 4998.47

A′1 (1,0°) A1′ (0,2°) E′ (0,22)

A′1 (0,4°) E′ (0,42)

2521.59

E′ (0,11)

8488.97

−1.3234471

A′1 (0,0°)

E′ (2,11)

B = 0.000

Γ (v1,v|l|2 )

Table 3. Energies of the Lowest Vibrational States with the Magnetic Field in y-Orientation and Field Strengths of up to B/B0 = 0.01a

The Journal of Physical Chemistry A Article



Article

Table 4. Energies of the Lowest Vibrational States with the Magnetic Field in z-Orientation and Field Strengths of up to B/B0 = 0.01a (v1,v|l|2 )

a

B = 0.000

B = 0.001

B = 0.002

A′1 (0,0°)

−1.32345

−1.32341

−1.32329

E′ (0,11) A1′ (1,0°) A′1 (0,2°) E′ (0,22) E′ (1,11) A1′ (2,0°) E′ (0,31) A′1 (0,33) A2′ (0,33) A1′ (1,2°) E′ (1,22) E′ (2,11) A1′ (0,4°) E′ (0,42) A′1 (3,0°) E′ (1,31) A1′ (1,33) E′ (0,44) A′2 (1,33) A1′ (2,2°)

2521.59 3178.67 4778.68 4998.47 5554.84 6262.86 7006.72 7286.34 7493.65 7770.11 7871.10 8488.97 9002.48 9113.96 9252.98 9654.73 9969.97 9998.30 10211.45 10594.35

2521.86 3179.09 4779.21 4999.03 5555.54 6263.71 7007.54 7287.23 7494.51 7771.06 7872.11 8490.11 9003.58 9115.09 9254.27 9655.95 9971.32 9999.52 10212.78 10595.78

2522.68 3180.33 4780.81 5000.71 5557.64 6266.25 7009.99 7289.89 7497.09 7773.91 7875.13 8493.56 9006.90 9118.51 9258.17 9659.58 9975.35 10003.20 10216.77 10600.04

B = 0.003

B = 0.004

B = 0.005

B = 0.006

Zero Point Energy (Eh) −1.32309 −1.32281 −1.32244 −1.32200 Relative Vibrational Energies (cm−1) 2524.03 2525.91 2528.34 2531.29 3182.41 3185.31 3189.03 3193.58 4783.48 4787.21 4791.99 4797.83 5003.50 5007.40 5012.42 5018.53 5561.12 5566.00 5572.26 5579.90 6270.49 6276.41 6284.01 6293.29 7014.08 7019.79 7027.11 7036.05 7294.31 7300.50 7308.44 7318.12 7501.39 7507.40 7515.12 7524.53 7778.66 7785.30 7793.82 7804.22 7880.15 7887.17 7896.18 7907.16 8499.29 8507.30 8517.58 8530.12 9012.41 9020.12 9030.01 9042.07 9124.19 9132.14 9142.34 9154.78 9264.66 9273.74 9285.39 9299.59 9665.63 9674.09 9684.95 9698.19 9982.07 9991.45 10003.50 10018.19 10009.32 10017.87 10028.86 10042.25 10223.41 10232.69 10244.60 10259.13 10607.13 10617.05 10629.79 10645.31

B = 0.007

B = 0.008

B = 0.009

B = 0.010

−1.32148

−1.32088

−1.32020

−1.31945

2534.78 3198.94 4804.71 5025.74 5588.90 6304.22 7046.59 7329.53 7535.62 7816.48 7920.12 8544.90 9056.29 9169.44 9316.33 9713.79 10035.50 10058.04 10276.25 10663.61

2538.79 3205.10 4812.63 5034.04 5599.26 6316.80 7058.71 7342.66 7548.39 7830.59 7935.02 8561.91 9072.64 9186.29 9335.59 9731.74 10055.41 10076.21 10295.94 10684.66

2543.32 3212.07 4821.58 5043.41 5610.96 6331.00 7072.40 7357.48 7562.81 7846.53 7951.85 8581.12 9091.11 9205.32 9357.35 9752.01 10077.89 10096.72 10318.17 10708.44

2548.37 3219.82 4831.54 5053.84 5623.99 6346.82 7087.65 7373.98 7578.86 7864.28 7970.59 8602.50 9111.66 9226.50 9381.57 9774.58 10102.91 10119.57 10342.92 10734.90

In z-orientation, the field does not alter the symmetry of the vibrational states.

degenerate vibration, v2 and the associated vibrational angular momentum, l, which may take the values l = −v2, −v2 + 2, ..., v2. These vibrational quantum numbers hold exactly as long as the minimum configuration has D3h symmetry, i.e., for the zoriented molecule, and become “good” quantum numbers for small distortions. For larger distortions, different quantum numbers are appropriate, as the degenerate mode is split into its onedimensional components, the bending mode, v2, and the antisymmetric stretching mode, v3, which are well-known from C2v molecules. These modes transform as A1 and B2 in the twoparticle permutation group, S2. By correlation, A1′ in S3 becomes A1 in S2 whereas A2′ becomes B2. As expected, the vibrational frequencies depend not only on the magnetic field strength but also on the orientation of the molecule with respect to the field axis. In a real environment with an ensemble of H3+ molecules, their orientations follow a Boltzmann distribution. If rotation is excited, this would also have an effect on the orientation, because it is quantized with respect to the magnetic field axis. For a given band, the tables can provide the corresponding energy interval. For example, the transition frequency of the fundamental band (0,11) ← (0,0°), calculated at 2521.59 cm−1 with no field present, attains for a field strength of B/B0 = 0.01 or B = 2350 T, a value between 2521.43 cm−1 (A1 component) and 2589.07 cm−1 (B2 component).

In future work we will study the problems of orientation of the molecules in the field and rotational excitation.

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ASSOCIATED CONTENT

S Supporting Information *

Numerical data of the expectation values of all 1615 points. This information is available free of charge via the Internet at http://pubs.acs.org

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: H.M.C., [email protected]; A.A., [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Alexander Turbiner for suggesting this problem, and we thank him and Vladimir Tyuterev for clarifying discussions. H.M.C. is grateful to Consejo Nacional de Ciencia y Tecnologia,́ Mexico, for a postdoctoral grant (CONACyT grant no 171494). This work was also supported by the Computer Center of the Université de Reims ChampagneArdenne and by the COST Action “COnvergent Distributed Environment for Computational Spectroscopy (CODECS)”.

■

4. CONCLUSIONS We have analyzed the problem of vibrating H3+ in an external magnetic field and presented frequencies for the lowest 21 vibrational states and a range of field strengths up to 2350 T and different orientations. These data, which are the first of their kind, can give a clue as to how much vibrational transition frequencies are altered due to the presence of a magnetic field.

REFERENCES

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