Diamagnetic corrections to low-excited electronic states of light molecular systems

Compact astrophysical objects like white dwarfs and neutron stars generate very strong magnetic fields. In their atmosphere, accretion disks, and the interstellar medium surrounding them, the predominant matter consists of hydrogen, helium and few-particle atoms and molecules. In this work, we have studied the effects of strong magnetic fields on light atomic and molecular systems like hydrogen molecular ions and antiprotonic helium. The full energy of the investigated systems is calculated in the Born-Oppenheimer approximation for a few low-lying electronic states. We investigate the case where the leading effect comes from the quadratic magnetic field term in the interaction Hamiltonian. The electronic energies and wavefunctions are computed in spheroidal coordinates by a finite difference method on a nonequidistant grid. The diamagnetic contributions to the total energy for an arbitrary value of the magnetic field strength are also calculated in perturbation theory. Then they are compared with the exact results, and where possible - with previous calculations. The electronic energy terms could be used for precise computations of the spectral lines of the hydrogen molecular ion and antiprotonic helium. This will allow searching for objects, like neutron stars and white dwarfs with strong magnetic fields, containing such molecular systems in their atmospheres or accretion discs. When the effects from the approximations are taken into account, the tabulated results presented in this work could also find an application in studies for much weaker magnetic field strengths existing in laboratory environments.


Introduction
The light atomic and molecular systems consist of only a few elementary particles, which allows their physical characteristics to be studied with high precision [1,2,3].Their energy and transition spectra have been investigated in detail [1,4,5].The interaction of those systems with external fields is also studied by many authors [6,7,8].The Stark and linear Zeeman effects in many few-particle systems have been calculated with great precision [3,6].However, more subtle effects, that have been difficult to measure with experimental methods in the past, have not been explored as thoroughly.The quadratic Zeeman effect has a very small impact on the energy spectrum of the atoms and molecules when compared to the linear one for magnetic fields that exist on Earth.At these field strengths, it is weaker than the hyperfine interactions and the relativistic effects, but the development of new spectroscopic techniques makes it possible to be observed [9,10].
In astrophysical objects like neutron stars and white dwarfs, exist very high magnetic fields of the order of or greater than 1 atomic unit (2.35051756758(71) × 10 5 T).In a recent review of magnetic white dwarfs [11], the authors show that these objects could have magnetic fields in the range from 10 −1 to 10 5 T. Similar and stronger fields are present in neutron stars.In these intense fields, the quadratic magnetic interaction contains much of the energy of the atomic or molecular system and to a great extent determines its behavior [8,11,12].In these exotic environments, some chemical composites could not form, but other bonds are stronger and even new types of bonds in molecules become possible [12].Neutron stars and white dwarfs have thin atmospheres that consist of the most abundant elements in the Universe.They also accrete matter from the interstellar medium, mostly hydrogen and helium.So, studying the interaction of light molecular systems with relatively strong magnetic fields could give more information about the physics of white dwarfs and neutron stars.
In this work, we investigate the diamagnetic effect on H + 2 and pHe + -like molecules for magnetic fields from B = 10 −6 au up to B = 1 au.In strong fields, the intermolecular axis generally aligns with the vector B [13], this hypothesis will be adopted in the computations here.The total energy of the studied systems is calculated in the Born-Oppenheimer approximation by solving the three-particle Schrödinger equation in spheroidal coordinates.The resulting differential equation is computed with a two-dimensional finite difference method (more details on the method used are given in [14]).The quadratic Zeeman effect is calculated both by including the diamagnetic term in the total Hamiltonian of the system and in the first-order perturbation theory.Matrix elements for a few low-lying electronic states for hydrogen molecular ions and antiprotonic helium are obtained and tabulated.The latter allows for the computation of the electronic and total energies of the investigated systems for a magnetic field of arbitrary value.
The paper is organized as follows.In Sec. 2, the general theory of systems interacting with an external magnetic field is briefly reviewed.The main results of the work are presented in Sec. 3. In Sebsec.3.1, the total energies of the studied systems are calculated by including the quadratic magnetic term in the interaction Hamiltonian.The results are compared with similar ones from the literature.Calculations of the diamagnetic term in perturbation theory are presented in 3.2 and their errors are estimated.Concluding remarks are given in Sec. 4.

Magnetic interactions in light atomic and molecular systems
In Cartesian coordinates, the non-relativistic Hamiltonian of a three-particle system in a homogeneous magnetic field B in atomic units (au) h = e = m e = 1 is: Here, p i , m i , L i , and Z i (where i = 1, 2, 3) are the particles' momentum, mass, angular momentum and charge.The distances between them are r 1 , r 2 , R = |r 1 − r 2 |, and R ⊥ i is the distance from the i-th particle to the z axis (R ⊥ 3 = ρ), as shown on Figure 1.In general, in the presence of a strong magnetic field, the magnetic field vector aligns with the molecular axis.This justifies investigating the case of a magnetic field parallel to the intermolecular axis (B∥z).Also, we will study the system in the Born-Oppenheimer approximation (the charges Z 1 and Z 2 are fixed).Then the Hamiltonian reduces to: A light molecular system, consisting of two heavy particles with charges Z 1 and Z 2 and an electron, in a homogeneous magnetic field B along the molecular axis.R is the distance between the two heavy particles and ρ -the distance of the electron from the z axis.
Since we are only interested in the internal degrees of freedom, we separate out those belonging to the center of mass.Then the wavefunction describing the system could be factorized as (see for example [15]): where D J mm J (Φ, Θ, ω) is the symmetrized Wigner function.Here, Φ, Θ, ω are the Euler angles of the vector r (connecting the midpoint of R with the third particle) relative to the laboratory frame.With ν and J are denoted the vibrational and total orbital momentum quantum numbers.The function F νJ m (r, R) describes the system in a frame co-rotating with the plane containing the three particles and it can be written in the basis of the electronic wavefunctions ϕ mnµn λ (µ, λ, ω, B; R).The decomposition coefficients χ mnµn λ νJ (R) correspond to the radial wavefunctions of the system. ( The electronic part ϕ mnµn λ (µ, λ, ω, B; R) is written in spheroidal coordinates which allows the Schrödinger equation for the system to be split into multiple differential equations as the electronic Hamiltonian becomes separable in these coordinates (more details can be found in Ref. [15]).This part of the Hamiltonian, describing a charged particle in the Coulomb potential of two stationary particles and an external magnetic field, could be written as a sum of two parts: one independent of the magnetic field H sph 0 and one corresponding to the interaction with the field H sph M (see [8,15]): Here, the following variables and parameters are used: and the volume element is dV = R 3 8 (λ 2 − µ 2 )dλdµdω.In this coordinate system, the vector r is written in prolate spheroidal coordinates {λ, µ, ω} with origin at the middle point of R and foci at its ends.
The electronic energies ε m,n λ ,nµ are found by solving The equation is separable, where the separation constant m = 0, 1, 2, ... is a solution to: and the total electronic wavefunction can be finally written as We have solved the Schrödinger equation for the wavefunction Φ m,n λ ,nµ (λ, µ, B; R) by a twodimensional finite difference method on a non-equidistant grid.The precision of the obtained results reach 12 significant digits for 0.1 au < R < 10 au as can be seen in Table 1.More details of the method used are given in [14].

Quadratic magnetic effect calculations
The simplest molecular system that is expected to exist in the atmosphere of white dwarfs and in the accretion disks of massive astrophysical objects is the hydrogen molecular ion H + 2 .As the other most common element in the universe is helium, systems containing it are assumed to form in the conditions mentioned above.This motivates us to study the effects of the magnetic field on three-particle atoms and molecules composed of the lightest and most abundant chemical elements.We will investigate two systems with charges Z 1 = Z 2 = 1, Z 3 = −1 (H + 2 , HD + ,...) and Z 1 = 2, Z 2 = Z 3 = −1 (for example the helium atom and, to a greater extent, in terms of the approximations made here -the exotic atom pHe + ).Only the electronic states with m = 0 are computed here.In this case, the first term in H sph M does not contribute to the electronic energies, so only the diamagnetic term H sph M d in Eq. ( 5) is taken into account.We have computed the electronic wavefunctions and the total energies of both systems mentioned above for a few low-laying electronic states without external fields and with the inclusion of a magnetic field up to B = 1 au (2.35051756758(71) × 10 5 T).In Table 1 are given the results for a few values of B and the equilibrium distance R eq in each case.
The results are compared with existing data from the literature.As can be seen, our method gives results that match up to 12 th digit (the same digits are marked in boldface) with other similar calculations in the studied range of magnetic field strengths.On Figure 2 are shown the energies E T as a function of the internuclear distance R for three states (depicted on the pictures with different color and thickness of the lines) of both H + 2 (Left) and pHe + (Right) systems.The solid lines correspond to energies at zero magnetic field and the dotted lines -to E T when the magnetic field is B = 0.5 au.At that field strength, the effect of the diamagnetic term is relatively weak for the ground state.This is clearly noticeable in the case of pHe + -like atoms and molecules where the lines, with the H sph M d term and without it, almost coincide.For the excited states, the quadratic Zeeman effect is considerably stronger.Table 1.The total energy E T of the ground state of hydrogen molecular ion obtained using the method described in Sec. 2, by Doma et al. [16] and by Vincke and Baye [13].The calculations were performed for a wide range of magnetic field strengths at equilibrium distance R eq .All quantities are in atomic units.

B
R R in atomic units as a function of the internuclear distance in the Born-Oppenheimer approximation for several low-lying electronic states of H + 2 (Left) and pHe + (Right).The dark blue, red and dark yellow curves correspond to 1sσ g , 1sσ g , and 2pσ u (Left) and 1σ, 2σ, 3σ (Right) states.The solid and dotted lines are the terms for B = 0 au and B = 0.5 au respectively.

Computation of the H +
2 and pHe + diamagnetic terms in perturbation theory As an alternative to the calculations in Sec.3.1, the contribution of the diamagnetic term ) in the magnetic interaction Hamiltonian to the total energy E T could be computed perturbatively.This is possible if B 2 /8 is treated as a small parameter, which is fulfilled for the magnetic field strengths considered here.To obtain the corrections in the firstorder perturbation theory, the electronic Schrödinger equation is solved for the Hamiltonian H sph 0 without the magnetic terms.Then the resulting wavefunctions for a particular state Φ m,n λ ,nµ 0 (λ, µ, ω, B = 0; R) are used in the computation of the matrix elements corresponding to the quadratic magnetic effect: Φ m,n λ ,nµ 0 By multiplying the quantity Φ m,n λ ,nµ 0 HM Φ m,n λ ,nµ 0 on the coefficient B 2 /8, one can easily calculate the diamagnetic term contribution to the electronic energy for an arbitrary magnetic field strength.As a result, the total energy becomes a simple expression that has a trivial dependence on B: Here, ε m,n λ ,nµ 0 is the electronic energy of the (m, n λ , n µ ) state at zero magnetic field.In Table 2 are given the results for the diamagnetic term contribution to the electronic energies of three low-lying states for both systems studied here.The calculations for states with quantum numbers (m, n λ , n µ ) equal to (0, 0, 0), (0, 1, 0), and (0, 0, 1) corresponding to 1sσ g , 2sσ g , 2pσ u of H + 2 , and 1σ, 2σ, 3σ (pHe + ) are presented.The matrix elements Φ m,n λ ,nµ 0 HM Φ m,n λ ,nµ 0 from the interaction Hamiltonian given by Eq. ( 5) (proportional to the squared distance operator ρ 2 in Cartesian coordinates) are computed for internuclear distance 0 ≤ R ≤ 10 au.
To evaluate the accuracy of the results of the perturbative calculations of the quadratic Zeeman effect, we compare them with the exact solution of Eq. (7).First, to compute the contribution of the diamagnetic term to the total electronic energy for a particular state (m, n λ , n µ ), from the total electronic energy ε m,n λ ,nµ we subtract the energy ε m,n λ ,nµ 0 obtained with the same Schrödinger equation for zero magnetic field: The corresponding contribution computed in perturbation theory is given by Eq. (11).Then the relative error between them is calculated as: On Figure 3, the quantity |∆ε/ε| is plotted for a wide range of magnetic field strengths 10 −6 ≤ B ≤ 1 au for H + 2 (Left) and pHe + (Right).The calculations are conducted for the points marked on the graphs with circles, triangles, and diamonds.For better visibility, they are connected with lines.As can be seen on Figure 3, both computations of the diamagnetic term in the studied range of field strengths give similar results as the relative error between them is significantly smaller than the one for the majority of the interval.For magnetic fields around B = 10 −3 au, they coincide up to the 7 th digit.For much weaker or stronger fields |∆ε/ε| is high mainly due to the following reasons.First, for B > 1 au the perturbation calculations become unreliable as the small parameter B 2 /8 becomes close to one.In addition, the computations explained in the previous subsection, become unstable for stronger magnetic fields.Table 2. Matrix elements of the diamagnetic term for three low-lying electronic states of both studied systems and internuclear distance 0.1 < R < 10 au.The states 1sσ g , 2sσ g , 2pσ u of H + 2 and 1σ, 2σ, 3σ of pHe + correspond to quantum numbers (m, n λ , n µ ) equal to (0, 0, 0), (0, 1, 0), and (0, 0, 1).Table 3.Total energy E T of H + 2 -and pHe + -like systems for few low-lying electronic states for different values of the magnetic field B. In the third column in boldface are marked the digits that are estimated to be accurate in our calculations of the total energy at zero magnetic field.In the last three columns, the underlined digits are those that change due to the diamagnetic interaction.three-particle systems in the Born-Oppenheimer approximation and in the case of a magnetic field along the molecular axis.The computation of electronic energy is done by solving the Schrödinger equation in spheroidal coordinates.A two-dimensional finite difference method on a non-equidistant grid is used.The diamagnetic effect is calculated both directly by including it in the interaction Hamiltonian, and by perturbation theory for a wide range of magnetic field strengths.In the latter case, the diamagnetic matrix elements are obtained for a few low-lying electronic states of H + 2 and pHe + for a large number of internuclear distances.We have shown that those results could be used to easily and accurately calculate the electronic terms and the total energy of the investigated systems for magnetic fields between a few and a few hundred thousand Tesla.

Figure
Figure2.E T = ε m,n λ ,nµ + Z 1 Z 2 /R in atomic units as a function of the internuclear distance in the Born-Oppenheimer approximation for several low-lying electronic states of H + 2 (Left) and pHe + (Right).The dark blue, red and dark yellow curves correspond to 1sσ g , 1sσ g , and 2pσ u (Left) and 1σ, 2σ, 3σ (Right) states.The solid and dotted lines are the terms for B = 0 au and B = 0.5 au respectively.