Density-Functional-Theory Calculations of Matter in Strong Magnetic Fields: I. Atoms and Molecules

0.28 MB·English

by  Zach Medin

Density-functional-theory calculations of matter in strong magnetic fields. I. Atoms
and molecules
Zach Medin and Dong Lai
Center for Radiophysics and Space Research, Department of Astronomy,
Cornell University, Ithaca, New York 14853, USA
(Received 9 June 2006; published 14 December2006)
We present calculations of the electronic structure of various atoms and molecules in strong
magneticfieldsrangingfromB=1012 Gto2×1015 G,appropriateforradiopulsarsandmagnetars.
Forthesefieldstrengths,themagneticforcesontheelectronsdominateovertheCoulombforces,and
toagoodapproximationtheelectronsareconfinedtothegroundLandaulevel. Ourcalculationsare
basedonthedensityfunctionaltheory,andusealocalmagneticexchange-correlationfunctionwhich
istestedtobereliableinthestrongfieldregime. Numericalresultsoftheground-stateenergiesare
given for H (up to N = 10), He (up to N = 8), C (up to N = 5), and Fe (up to N = 3),
N N N N
7 aswell as forvarious ionized atoms. Fitting formulae fortheB-dependenceof theenergies arealso
0 given. Ingeneral,asN increases,thebindingenergyperatominamolecule,|E |/N,increasesand
N
0 approachesa constant value. Forall thefield strengthsconsidered in this paper,hydrogen,helium,
2 andcarbonmoleculesarefoundtobeboundrelativetoindividualatoms(althoughforBlessthana
n few×1012 G,carbonmoleculesareveryweaklyboundrelativetoindividualatoms). Ironmolecules
a are not bound at B <∼ 1013 G, but become energetically more favorable than individual atoms at
J larger field strengths.

5
PACSnumbers: 31.15.Ew,95.30.Ky,97.10.Ld

2
v
6 I. INTRODUCTION
6
1
Neutron stars (NSs) are endowed with magnetic fields far beyond the reach of terrestrial laboratories [1, 2, 3].
7
0 Most of the 1600 known radio pulsars have surface magnetic fields in the range of 1011 1013 G, as inferred
∼ −
6 from their measured spin periods and period derivatives and the assumption that the spindown is due to magnetic
0 dipole radiation. A smaller population of older, millisecond pulsars have B 108 109 G. For about a dozen
/ accreting neutron stars in binary systems, electron cyclotron features have been∼detec−ted, implying surface fields of
h
p B 1012 1013 G. Animportantdevelopmentinastrophysicsinthe lastdecadecenteredonthe so-calledanomalous
∼ −
- x-ray pulsars and soft gamma repeaters [4]: there has been mounting observational evidence in recent years that
o
supports the idea that these are magnetars, neutron stars whose radiations are powered by superstrong magnetic
r
t fields of order 1014 1015 G or higher [5, 6, 7]. By contrast, the highest static magnetic field currently produced
s in a terrestrial labo−ratory is 5 105 G; transient fields approaching 109 G have recently been generated during
a
×
: high-intensity laser interactions with dense plasmas [8].
v
It is well-known that the properties of matter can be drastically modified by strong magnetic fields found on
i
X neutron star surfaces. The natural atomic unit for the magnetic field strength, B , is set by equating the electron
0
r cyclotronenergyh¯ωBe =h¯(eB/mec)=11.577B12 keV, where B12 =B/(1012 G), to the characteristicatomic energy
a e2/a =2 13.6 eV (where a is the Bohr radius):
0 0
×
m2e3c
B = e =2.3505 109G. (1)
0 ¯h3 ×
Forb=B/B >1,theusualperturbativetreatmentofthemagneticeffectsonmatter(e.g.,Zeemansplittingofatomic
0
energy levels)d∼oes not apply. Instead, in the transversedirection (perpendicular to the field) the Coulombforces act
asaperturbationtothemagneticforces,andtheelectronsinanatomsettleintothegroundLandaulevel. Becauseof
the extreme confinement of the electrons in the transversedirection, the Coulombforce becomes much more effective
in binding the electrons along the magnetic field direction. The atom attains a cylindrical structure. Moreover, it is
possiblefortheseelongatedatomstoformmolecularchainsbycovalentbondingalongthefielddirection. Interactions
between the linear chains can then lead to the formation of three-dimensional condensed matter [9, 10, 11].
Ourmainmotivationforstudyingmatterinsuchstrongmagneticfieldsarisesfromtheimportanceofunderstanding
neutron star surface layers, which play a key role in many neutron star processes and observed phenomena. The-
oretical models of pulsar and magnetar magnetospheres depend on the cohesive properties of the surface matter in
strong magnetic fields [12, 13, 14, 15, 16]. For example, depending on the cohesive energy of the surface matter, an
accelerationzone (“polar gap”) above the polar cap of a pulsar may or may not form. More importantly, the surface
layer directly mediates the thermal radiations from neutron stars. The advent of x-ray telescopes in recent years
2
has made detailed study of neutron star surface emission a reality. Such study can potentially provide invaluable
informationonthe physicalpropertiesandevolutionofNSs: equationofstateatsupernucleardensities,superfluidity,
coolinghistory,magneticfield,surfacecomposition,differentNSpopulations,etc. (see,e.g.,Ref.[17]). Morethantwo
dozen isolated neutron stars (including radio pulsars, radio-quietneutron stars and magnetars) have clearly detected
thermal surface emission [3, 18, 19]. While some neutron stars show featureless spectra, absorption lines or features
havebeendetectedinhalfadozenorsosystems[19]. Indeed,manyoftheobservedneutronstarshavesufficientlylow
surface temperatures and/or high magnetic fields, such that bound atoms or molecules are expected to be present in
theiratmospheres[20,21,22,23]. Itisevenpossiblethattheatmosphereiscondensedintoasolidorliquidformfrom
which radiation directly emerges [11, 23, 24]. Thus, in order to properly interpret various observations of neutron
stars, it is crucial to have a detailed understanding of the properties of atoms, molecules and condensed matter in
strong magnetic fields (B 1011-1016 G).
∼
A. Previous works
H and He atoms at almost all field strengths have been well studied [10, 25, 26], including the nontrivial effect
associated with the center-of-mass motion of a H atom [27]. Neuhauser et al. [28] presented numerical results for
several atoms up to Z =26 (Fe) at B 1012 G based on calculations using a one-dimensional Hartree-Fock method
∼
(see also Ref. [29] for Z up to 10). Some results [based on a two-dimensional (2D) mesh Hartree-Fock method] for
atoms(uptoZ =10)atthefieldstrengthsB/B =0.5 104arealsoavailable[30,31,32]. TheHartree-Fockmethod
0
−
is approximate because electron correlations are neglected. Due to their mutual repulsion, any pair of electrons tend
tobe moredistantfromeachotherthantheHartree-Fockwavefunctionwouldindicate. Inzero-field,thiscorrelation
effectisespeciallypronouncedforthespin-singletstatesofelectronsforwhichthespatialwavefunctionissymmetrical.
Instrongmagneticfields(B B ),theelectronspins(inthegroundstate)areallalignedantiparalleltothemagnetic
0
≫
field, and the multielectron spatial wave function is antisymmetric with respect to the interchange of two electrons.
Thus the error in the Hartree-Fock approach is expected to be less than the 1% accuracy characteristic of zero-field
Hartree-Fockcalculations[28,33]. Othercalculationsofheavyatomsinstrongmagneticfields includeThomas-Fermi
type statistical models [34, 35, 36] and density functional theory [37, 38, 39, 40]. The Thomas-Fermitype models are
usefulinestablishingasymptoticscalingrelations,butarenotadequateforobtainingaccuratebinding andexcitation
energies. The density functional theory can potentially give results as accurate as the Hartree-Fock method after
proper calibration is made [41, 42].
Quantitativeresults for the energiesofhydrogenmoleculesH with N =2,3,4,5in a wide rangeoffield strengths
N
(B B ) were obtained (based on the Hartree-Fock method) by Lai et al. [11, 43] and molecular excitations were
0
≫
studied in Refs. [44, 45] (more complete references can be found in Ref. [11]). Quantum Monte Carlo calculations of
H in strong magnetic fields have been performed [46]. Some numerical results of He for various field strengths are
2 2
also available [11]. Hartree-Fock results of diatomic molecules (from H up to C ) and several larger molecules (up
2 2
to H and He ) at B/B =1000 are given in Ref. [47].
5 4 0
B. Plan of this paper
In this paper and its companion paper [48], we develop a density-functional-theory calculation of the ground-state
energy of matter for a wide range of magnetic field strengths, from 1012 G (typical of radio pulsars) to 2 1015 G
×
(magnetar fields). We consider H, He, C, and Fe, which represent the most likely composition of the outermost layer
of neutron stars (e.g., Ref. [3]). The present paper focuses on atoms (and related ions) and small molecules. Because
of additional complications related to the treatment of band structure, calculations of infinite molecular chains and
condensed matter are presented in Ref. [48].
Ourcalculationsarebasedondensityfunctionaltheory[49,50,51]. Asmentionedabove,the Hartree-Fockmethod
isexpectedtobehighlyaccurate,particularlyinthestrongfieldregimewheretheelectronspinsarealignedwitheach
other. Inthisregimethedensityfunctionalmethodisnotasaccurate,duetothelackofanexactcorrelationfunction
forelectronsinstrongmagneticfields. However,indealingwithsystemswithmanyelectrons,theHartree-Fockmethod
becomes increasingly impractical as the magnetic field increases, since more and more Landau orbitals (even though
electrons remain in the ground Landau level) are occupied and keeping track of the direct and exchange interactions
between electrons in various orbitals becomes computationally rather tedious. Our density-functional calculations
allow us to obtain the energies of atoms and small molecules and the energy of condensed matter using the same
method, thus providing reliable cohesive energy of condensed surface of magnetic neutron stars, a main goal of our
study. Compared to previous density-functional-theory calculations [37, 38, 39, 40], we use an improved exchange-
correlation function for highly magnetized electron gases, we calibrate our density functional code with previous
3
results (when available) based on other methods, and (for calculations of condensed matter) adopt a more accurate
treatment of the band structure. Moreover,our calculations extend to the magnetar-like field regime (B 1015 G).
∼
Note that in this paper we neglect the motions of the nuclei, due to electron-nucleus interactions or finite temper-
atures. The center-of-mass motions of the atoms and molecules induce the motional Stark effect, which can change
the internal structure of the bound states (see, e.g., Refs. [11, 27]). Such issues are beyond the scope of this paper.
After summarizing the approximate scaling relations for atoms and molecules in strong magnetic fields in Sec. II,
we describe our method in Sec. III and present numerical results in Sec. IV. Some technical details are given in the
Appendix.
II. BASIC SCALING RELATIONS FOR ATOMS AND MOLECULES IN STRONG MAGNETIC FIELDS
A. Atoms
First consider a hydrogenic atom (with one electron and nuclear charge Z). In a strong magnetic field with
b=B/B Z2, the electron is confined to the ground Landau level (“adiabatic approximation”),and the Coulomb
0
≫
potentialcanbe treatedasa perturbation. The energyspectrumis specifiedby twoquantumnumbers,(m,ν), where
m=0,1,2,... measures the mean transverse separation between the electron and the nucleus ( m is also known as
−
the magnetic quantum number), while ν specifies the number of nodes in the z wavefunction. There are two distinct
typesofstatesintheenergyspectrumE . The“tightlybound”stateshavenonodeintheirzwavefunctions(ν =0).
mν
The transverse size of the atom in the (m,0) state is L ρ = (2m+1)1/2ρ , with ρ = (h¯c/eB)1/2 = b−1/2 (in
⊥ m 0 0
∼
atomic units).1 For ρ 1, the atom is elongated with L L . We can estimate the longitudinal size L by
m z ⊥ z
minimizing the energy, E≪ L−2 ZL−1ln(L /L ) (where the≫first term is the kinetic energy and the second term
∼ z − z z ⊥
is the Coulomb energy), giving
−1
1
L Zln . (2)
z
∼(cid:18) Zρ (cid:19)
m
The energy is given by
1 b 2
E Z2 ln (3)
m0 ∼− (cid:20) Z2 (cid:18)2m+1(cid:19)(cid:21)
for b (2m+1)Z2. Another type of state of the atom has nodes in the z wave functions (ν > 0). These states are
≫
“weakly bound”, and have energies given by E Z2n−2 Ry, where n is the integer part of (ν+1)/2. The sizes
mν
of the wave functions are ρ perpendicular to the≃fi−eld and L ν2/Z along the field (see Ref. [11] and references
m z
∼
therein for more details).
A multielectron atom (with the number of electrons N and the charge of the nucleus Z) can be constructed by
e
placing electrons at the lowest available energy levels of a hydrogenic atom. The lowest levels to be filled are the
tightly bound states with ν = 0. When a /Z √2N 1ρ , i.e., b 2Z2N , all electrons settle into the tightly
0 e 0 e
≫ − ≫
bound levels with m = 0,1,2, ,N 1. The energy of the atom is approximately given by the sum of all the
e
··· −
eigenvalues of Eq. (3). Accordingly, we obtain an asymptotic expression for N 1 [52]:
e
≫
b 2
E Z2N ln . (4)
∼− e(cid:18) 2Z2N (cid:19)
e
For intermediate-strong fields (but still strong enough to ignore Landau excitations), Z2N−2/3 b 2Z2N ,
e e
≪ ≪
manyν >0statesoftheinnerLandauorbitals(stateswithrelativelysmallm)arepopulatedbytheelectrons. Inthis
regimeaThomas-Fermitypemodelfortheatomisappropriate,i.e.,theelectronscanbetreatedasaone-dimensional
Fermi gas in a more or less spherical atomic cell (see, e.g., Refs. [53, 54]). The electrons occupy the ground Landau
level, with the z momentum up to the Fermi momentum p n/b, where n is the number density of electrons inside
F
∼
the atom(recallthatthe degeneracyofaLandauleveliseB/hc b). The kineticenergyofelectronsper unitvolume
is ε bp3 n3/b2, and the total kinetic energy is E R3n3/∼b2 N3/(b2R6), where R is the radius of the atom.
k ∼ F ∼ k ∼ ∼ e
1 Unlessotherwisespecified, weuse atomicunits, inwhichlength isina0 (Bohr radius), massin me, energyine2/a0 =2Ry, and field
strengthinunitsofB0.
4
The potential energy is E ZN /R (for N < Z). Therefore the total energy of the atom can be written as
p e e
E N3/(b2R6) ZN /R. M∼in−imizing E with res∼pect to R yields
∼ e − e
R (N2/Z)1/5b−2/5, E (Z2N )3/5b2/5. (5)
∼ e ∼− e
For these relations to be valid, the electrons must stay in the ground Landau level; this requires Z/R ¯hω = b,
Be
≪
which corresponds to b Z2N−2/3.
e
≫
B. Molecules
In a strong magnetic field, the mechanism of forming molecules is quite different from the zero-field case [9, 43].
Consider hydrogen as an example. The spin of the electron in a H atom is aligned antiparallel to the magnetic field
(flipping the spin would cost h¯ω ), therefore two H atoms in their ground states (m = 0) do not bind together
Be
accordingtothe exclusionprinciple. Instead,oneHatomhastobe excitedtothem=1state. ThetwoHatoms,one
in the ground state (m = 0), another in the m = 1 state then form the ground state of the H molecule by covalent
2
bonding. Since the activation energy for exciting an electron in the H atom from the Landau orbital m to (m+1)
is small, the resulting H molecule is stable. Similarly, more atoms can be added to form H , H , .... The size of
2 3 4
the H molecule is comparable to that of the H atom. The interatomic separation a and the dissociation energy D
2
of the H molecule scale approximately as a (lnb)−1 and D (lnb)2, although D is numerically smaller than the
2
∼ ∼
ionization energy of the H atom.
Consider the molecule Z , formed out of N neutral atoms Z (each with Z electrons and nuclear charge Z). For
N
sufficiently large b (see below), the electrons occupy the Landau orbitals with m = 0, 1, 2,...,NZ 1, and the
−
transversesizeofthe moleculeis L (NZ/b)1/2. Leta be the atomic spacingandL Na the size ofthe molecule
⊥ z
in the z direction. The energy per “a∼tom” in the molecule, E =E /N, can be written∼as E Z(Na)−2 (Z2/a)l,
N
∼ −
where l ln(a/L ). Variation of E with respect to a gives
⊥
∼
b
a (ZN2l)−1, E Z3N2l2, with l ln . (6)
∼ ∼− ∼ (cid:18)N5Z3(cid:19)
This above scaling behavior is valid for 1 N N . The “critical saturation number” N is reached when a L ,
s s ⊥
≪ ≪ ∼
or when [43]
1/5
b
N . (7)
s ∼(cid:18)Z3(cid:19)
Beyond N , it becomes energetically more favorable for the electrons to settle into the inner Landau orbitals (with
s
smallerm)withnodesintheirlongitudinalwavefunctions(i.e.,ν =0). ForN >N ,theenergyperatomasymptotes
s
to a value E Z9/5b2/5, and size of the atom scales as L 6a Z1/5b−2∼/5, independent of N — the molecule
⊥
∼ − ∼ ∼
essentially becomes one-dimensional condensed matter.
The scaling relations derived above are obviously crude — they are expected to be valid only in the asymptotic
limit, ln(b/Z3) 1. For realistic neutron stars, this limit is not quite reached. Thus these scaling results should
≫
only serve as a guide to the energies of various molecules. For a given field strength, it is not clear from the above
analysiswhethertheZ moleculeisboundrelativetoindividualatoms. Toanswerthisquestionrequiresquantitative
N
calculations.
III. DENSITY-FUNCTIONAL CALCULATIONS: METHODS AND EQUATIONS
Our calculations will be based on the “adiabatic approximation,” in which all electrons are assumed to lie in the
ground Landau level. For atoms or molecules with nucleus charge number Z, this is an excellent approximation for
b Z2. Even under more relaxed condition, b Z4/3 (assuming the number of electrons in each atom is N Z)
e
≫ ≫ ∼
this approximation is expected to yield a reasonable total energy of the system and accurate results for the energy
difference between different atoms and molecules; a quantitative evaluation of this approximation in this regime is
beyond the scope of this paper (but see Refs. [30, 31, 32]).
In the adiabatic approximation, the one-electron wave function (“orbital”) can be separated into a transverse
(perpendicular to the external magnetic field) component and a longitudinal (along the magnetic field) component:
Ψ (r)=W (r )f (z). (8)
mν m ⊥ mν
5
Here W is the ground-state Landau wave function [55] given by
m
1 ρ m ρ2
W (r )= exp − exp( imφ), (9)
m ⊥ ρ √2πm!(cid:18)√2ρ (cid:19) (cid:18)4ρ2(cid:19) −
0 0 0
where ρ =(h¯c/eB)1/2 is the cyclotronradius (or magnetic length), and f is the longitudinal wave function which
0 mν
must be solved numerically. We normalize f over all space:
mν
∞
dz f (z)2 =1, (10)
mν
Z | |
−∞
so that dr Ψ (r)2 =1. The density distribution of electrons in the atom or molecule is
mν
| |
R
n(r)= Ψ (r)2 = f (z)2 W 2(ρ), (11)
mν mν m
| | | | | |
Xmν Xmν
where the sum is over all the electrons in the atom or molecule, with each electron occupying an (mν) orbital. The
notation W 2(ρ)= W (r )2 is used here because W is a function of ρ and φ but W 2 is a function only of ρ.
m m ⊥ m m
| | | | | |
In an external magnetic field, the Hamiltonian of a free electron is
1 e 2 ¯heB
Hˆ = p+ A + σ , (12)
z
2m c 2m c
e (cid:16) (cid:17) e
where A= 1B r is the vectorpotential ofthe externalmagnetic fieldand σ is the z componentPaulispin matrix.
2 × z
For electrons in Landau levels, with their spins aligned parallel/antiparallel to the magnetic field, the Hamiltonian
becomes
pˆ2 1 1
Hˆ = z + n + ¯hω ¯hω , (13)
L Be Be
2m (cid:18) 2(cid:19) ± 2
e
where n = 0,1,2, is the Landau level index; for electrons in the ground Landau level, with their spins aligned
L
···
antiparallel to the magnetic field (so n =0 and σ 1),
L z
→−
pˆ2
Hˆ = z . (14)
2m
e
The total Hamiltonian for the atom or molecule then becomes
pˆ2
Hˆ = z,i +V , (15)
2m
Xi e
where the sum is over all electrons and V is the total potential energy of the atom or molecule. From this we can
derive the total energy of the system.
Note that we use nonrelativistic quantum mechanics in our calculations, even when h¯ω > m c2 or B > B =
Be e Q
B /α2 =4.414 1013 G. This is valid for two reasons: (i) The free-electron energy in relativis∼tic theory is ∼
0
×
B 1/2
E = c2p2+m2c4 1+2n . (16)
(cid:20) z e (cid:18) LB (cid:19)(cid:21)
Q
For electrons in the ground Landau level (n = 0), Eq. (16) reduces to E m c2+p2/(2m ) for p c m c2; the
L ≃ e z e z ≪ e
electron remains nonrelativistic in the z direction as long as the electron energy is much less than m c2; (ii) Eq. (9)
e
indicates thatthe shape ofLandautransversewavefunctionis independent ofparticlemass,andthus Eq.(9)is valid
intherelativistictheory. Ourcalculationsassumethatthe longitudinalmotionofthe electronisnonrelativistic. This
is valid at all field strengths and for all elements considered with the exception of iron at B > 1015 G. Even at
B = 2 1015 G (the highest field considered in this paper), however, we find that the most-bo∼und electron in any
Fe atom×or molecule has a longitudinal kinetic energy of only 0.2m c2 and only the three most-bound electrons
e
have longitudinal kinetic energies >0.1m c2. Thus relativistic c∼orrections are small in the field strengths considered
e
in this paper. Moreover, we expec∼t our results for the relative energies between Fe atoms and molecules to be much
more accurate than the absolute energies of either the atoms or the molecules.
6
Consider the molecule Z , consisting of N atoms, each with an ion of charge Z and Z electrons. In the lowest-
N
energy state of the system, the ions are aligned along the magnetic field. The spacing between ions, a, is chosen to
be constant across the molecule. In the density functional theory, the total energy of the system can be represented
as a functional of the total electron density n(r):
E[n]=E [n]+E [n]+E [n]+E [n]+E [n]. (17)
K eZ dir exc ZZ
Here E [n] is the kinetic energyofa systemofnoninteractingelectrons,andE ,E , andE arethe electron-ion
K eZ dir ZZ
Coulomb energy, the direct electron-electron interaction energy, and the ion-ion interaction energy, respectively,
N n(r)
E [n]= Ze2 dr , (18)
eZ − Z r z
Xj=1 | − j|
e2 n(r)n(r′)
E [n]= drdr′ , (19)
dir 2 ZZ r r′
| − |
N−1 Z2e2
E [n]= (N j) . (20)
ZZ
− ja
Xj=1
The location of the ions in the above equations is represented by the set z , with
j
{ }
a
z =(2j N 1) ˆz. (21)
j
− − 2
The term E represents exchange-correlationenergy. In the local approximation,
exc
E [n]= drn(r)ε (n), (22)
exc exc
Z
where ε (n) = ε (n)+ε (n) is the exchange and correlation energy per electron in a uniform electron gas of
exc ex corr
density n. For electrons in the ground Landau level, the (Hartree-Fock) exchange energy can be written as follows
[56]:
ε (n)= πe2ρ2nF(t), (23)
ex − 0
where the dimensionless function F(t) is
F(t)=4 ∞dx tan−1 1 xln 1+ 1 e−4tx2, (24)
Z (cid:20) (cid:18)x(cid:19)− 2 (cid:18) x2(cid:19)(cid:21)
0
and
2
n
t= =2π4ρ6n2, (25)
(cid:18)n (cid:19) 0
B
[n = (√2π2ρ3)−1 is the density above which the higher Landau levels start to be filled in a uniform electron gas].
B 0
For small t, F(t) can be expanded as follows [57]:
2t 13 8t2 67
F(t) 3 γ ln4t+ γ ln4t + γ ln4t + (t3lnt), (26)
≃ − − 3 (cid:18) 6 − − (cid:19) 15 (cid:18)30 − − (cid:19) O
where γ = 0.5772 is Euler’s constant. We have found that the condition t 1 is well satisfied everywhere for
almostallmolecule·s··inourcalculations. The notableexceptionsarethe carbonm≪oleculesatB =1012 Gandthe iron
molecules at B = 1013 G, which have t < 1 near the center of the molecule. These molecules are expected to have
higher t values than the other molecules∼in our calculations, as they have large Z and low B.2
2 Fortheuniformgasmodel,t∝Z6/5Ne−2/5B−3/5.
7
The correlationenergy of uniform electron gas in strong magnetic fields has not be calculated in general,except in
the regime t 1 and Fermi wavenumber k = 2π2ρ2n 1 [or n (2π3ρ2a )−1]. Skudlarski and Vignale [58] use
≪ F 0 ≫ ≫ 0 0
the random-phase approximationto find a numerical fit for the correlation energy in this regime (see also Ref. [59]):
e2
ε = [0.595(t/b)1/8(1 1.009t1/8)]. (27)
corr
−ρ −
0
In the absence of an “exact” correlation energy density we employ this strong-field-limit expression. Fortunately,
because we are concerned mostly with finding energy changes between different states of atoms and molecules, the
correlation energy term does not have to be exact. The presence or the form of the correlation term has a modest
effectontheatomicandmolecularenergiescalculatedbuthasverylittleeffectontheenergydifference betweenthem
(see Appendix B for more details on various forms of the correlationenergy and comparisons).
Variation of the total energy with respect to the total electron density, δE[n]/δn = 0, leads to the Kohn-Sham
equations:
¯h2
2+V (r) Ψ (r)=ε Ψ (r), (28)
eff mν mν mν
(cid:20)−2m ∇ (cid:21)
e
where
N Ze2 n(r′)
V (r)= +e2 dr′ +µ (n), (29)
eff − r z Z r r′ exc
Xj=1 | − j| | − |
with
∂(nε )
exc
µ (n)= . (30)
exc
∂n
Averaging the Kohn-Sham equations over the transverse wave function yields a set of one-dimensional equations:
¯h2 d2 N W 2(ρ) W 2(ρ)n(r′)
−2m dz2 − Ze2Z dr⊥ | rm|z +e2ZZ dr⊥dr′ | mr| r′
e Xj=1 | − j| | − |

+ dr W 2(ρ)µ (n) f (z)=ε f (z). (31)
⊥ m exc mν mν mν
Z | | (cid:19)
These equations are solved self-consistently to find the eigenvalue ε and the longitudinal wave function f (z) for
mν mν
each orbital occupied by the ZN electrons. Once these are known, the total energy of the system can be calculated
using
e2 n(r)n(r′) N−1 Z2e2
E[n]= ε drdr′ + drn(r)[ε (n) µ (n)]+ (N j) . (32)
mν − 2 ZZ r r′ Z exc − exc − ja
Xmν | − | Xj=1
Details ofourmethod usedincomputing the variousintegralsandsolvingthe aboveequationsaregiveninAppendix
A.
Note that for a given system, the occupations of electrons in different (mν) orbitals are not known a priori, and
must be determined as part of the procedure of finding the minimum energy state of the system. In our calculation,
we firstguess n ,n ,n ,..., the number ofelectronsin the ν =0, 1, 2,...orbitals,respectively (e.g., the electrons in
0 1 2
the ν =0orbitalshavem=0,1,2,...,n 1). Note thatn +n +n + =NZ. We findthe energyofthe system
0 0 1 2
− ···
for this particularsetofelectronoccupations. We then varythe electronoccupationsandrepeatthe calculationuntil
the true minimum energy state is found. Obviously, in the case of molecules, we must vary the ion spacing a to
determine the equilibrium separation and the the ground-state energy of the molecule. Graphical examples of how
the ground state is chosen are given in the next section.
IV. RESULTS
In this section we present our results for the parallel configuration of H (up to N = 10), He (up to N = 8),
N N
C (up to N =5), and Fe (up to N =3) at various magnetic field strengths between B =1012 G and 2 1015 G.
N N
×
8
For each molecule (or atom), data is given in tabular form on the molecule’s ground-state energy, the equilibrium
separation of the ions in the molecule, and its orbital structure (electron occupation numbers n ,n ,n ,...). In
0 1 2
some casesthe first-excited-stateenergiesaregivenas well,whenthe ground-stateandfirst-excited-stateenergiesare
similar in value. We also provide the ground-state energies for selected ionization states of C and Fe atoms; among
other uses, these quantities are needed for determining the ion emission from a condensed neutron star surface [48].
All of the energies presentedin this section are calculated to better than 0.1% numericalaccuracy(see Appendix A).
For each of the molecules and ions presented in this section we provide numerical scaling relations for the ground-
state energy as a function of magnetic field, in the form of a scaling exponent β with E Bβ . We have provided
N ∝ 12
this information to give readers easy access to energy values for fields in between those listed in the tables. The
ground-stateenergyis generallynotwellfitby aconstantβ overthe entiremagneticfieldrangecoveredbythis work,
sowehaveprovidedβ valuesoverseveraldifferentmagneticfieldranges. Notethatthetheoreticalvalueβ =2/5(see
Sec. II) is approached only in certain asymptotic limits.
We discuss here briefly a few trends in the data: All of the molecules listed in the following tables are bound. The
Fe and Fe molecules at B = 5 are not bound, so we have not listed them here, but we have listed the Fe atom
2 3 12
at this field strength for comparison with other works. All of the bound molecules listed below have ground-state
energies per atom that decrease monotonically with increasing N, with the exception of H at B =1, which has a
N 12
slight upward glitch in energy at H (see Table I). Additionally, these energies approach asymptotic values for large
4
N — the molecule essentially becomes one-dimensional condensed matter [48]. The equilibrium ion separations also
approach asymptotic values for large N, but there is no strong trend in the direction of approach: sometimes the
equilibrium ion separations increase with increasing N, sometimes they decrease, and sometimes they oscillate back
and forth.
In general, we find that for a given molecule (e.g., Fe ), the number of electrons in ν > 0 states decreases as the
3
magnetic field increases. This is because the characteristic transverse size ρ B−1/2 decreases, so the electrons
0
∝
prefer to stay in the ν =0 states. For a given field strength, as the number of electrons in the system NN increases
e
(e.g., from Fe to Fe ), more electrons start to occupy the ν >0 states since the average electron-nucleus separation
2 3
ρ (2m+1)1/2B−1/2 becomes too large for large m. For large enough N the value of n , the number of electrons
m 0
∝
in ν =0 states, levels off, approachingits infinite chain value (see Ref. [48]). Similar trends happen with n , n , etc.,
1 2
though much more slowly.
There are two ways that we have checked the validity of our results by comparison with other works. First, we
have repeated several of our atomic and molecular calculations using the correlation energy expression empirically
determined by Jones [37]:
e2
ε = (0.0096lnρ3n+0.122). (33)
corr −ρ 0
0
The results we then obtain for the atomic ground-state energies agree with those of Jones [37, 38]. For example, for
Fe at B = 5 we find an atomic energy of 108.05 eV and Jones gives an energy of 108.18 eV. The molecular
12
− −
ground-state energies per atom are of course not the same as those for the infinite chain from Jones’s work, but
they are comparable, particularly for the large molecules. For example, we find for He at B = 5 that the energy
8 12
per atom is 1242 eV and Jones finds for He that the energy per cell is 1260 eV. (See Appendix B for a brief
∞
− −
discussion of why in our calculations we chose to use the Skudlarski-Vignale correlation energy expression over that
of Jones.)
Second, we have compared our hydrogen, helium, and carbon molecule results to those of Refs. [43, 47]. Because
these works use the Hartree-Fock method, we cannot compare absolute ground-stateenergies with theirs, but we can
compare energy differences. We find fair agreement, though the Hartree-Fock results are consistently smaller. Some
of these comparisons are presented in the following subsections.
A. Hydrogen
Our numerical results for H are given in Table I and Table II. Note that at B = 1, H is less bound than
12 4
H , and thus E = E /N is not a necessarily a monotonically decreasing function of N at this field strength. For
3 N
the H molecule, two configurations, (n ,n ) = (4,0) and (3,1), have very similar equilibrium energies (see Fig. 1),
4 0 1
althoughtheequilibriumionseparationsaredifferent. Therealgroundstatemaythereforebe a“mixture”ofthe two
configurations; such a state would presumably give a lower ground-state energy for H , and make the energy trend
4
monotonic.
Hartree-FockresultsforHmoleculesaregivenin[43]. ForH ,H ,andH ,theenergies(peratom)are,respectively:
2 3 4
184.3, 188.7, 185.0 eV at B = 1; 383.9, 418.8, 432.9 eV at B = 10; and 729.3, 847.4, 915.0 eV at
12 12
− − − − − − − − −
9
TABLEI:Ground-stateenergies, ion separations, andelectron configurations ofhydrogen molecules, overarange ofmagnetic
field strengths. In some cases the first-excited-state energies are also listed. Energies are given in units of eV, separations in
units of a0 (the Bohr radius). For molecules (HN) the energy per atom is given, E = EN/N. All of the H and H2 molecules
listed here have electrons only in the ν =0 states. For the H3 and larger molecules here, however, the molecular structure is
morecomplicated, andisdesignated bythenotation (n0,n1,...),wheren0 isthenumberofelectronsin theν =0orbitals, n1
is the numberof electrons in theν =1 orbitals, etc.
H H2 H3 H4 H5
B12 E E a E a (n0,n1) E a (n0,n1) E a (n0,n1)
1 -161.4 -201.1 0.25 -209.4 0.22 (3,0) -208.4 0.21 (4,0) -213.8 0.23 (4,1)
-191.1 0.34 (2,1) -207.9 0.26 (3,1) -203.1 0.200 (5,0)
10 -309.5 -425.8 0.125 -469.0 0.106 (3,0) -488.1 0.096 (4,0) -493.5 0.090 (5,0)
-478.9 0.112 (4,1)
100 -540.3 -829.5 0.071 -961.2 0.057 (3,0) -1044.5 0.049 (4,0) -1095.5 0.044 (5,0)
1000 -869.6 -1540.5 0.044 -1818.0 0.033 (3,0) -2049 0.028 (4,0) -2222 0.024 (5,0)
H6 H8 H10
B12 E a (n0,n1) E a (n0,n1,n2) E a (n0,n1,n2)
1 -214.1 0.23 (4,2) -215.8 0.23 (5,2,1) -216.2 0.22 (6,3,1)
-213.4 0.21 (5,1) -215.3 0.25 (4,3,1) -216.0 0.23 (5,3,2)
10 -496.5 0.101 (5,1) -507.1 0.095 (8,2,0) -509.3 0.091 (7,3,0)
-490.8 0.86 (6,0) -504.1 0.089 (7,1,0) -506.8 0.087 (8,2,0)
100 -1125.0 0.041 (6,0) -1143.0 0.038 (8,0,0) -1169.5 0.038 (9,1,0)
-1139.5 0.043 (7,1,0) -1164.0 0.042 (8,2,0)
1000 -2351 0.22 (6,0) -2518 0.0190 (8,0,0) -2600 0.0170 (10,0,0)
-2542 0.0200 (9,1,0)
TABLE II: Fit of the ground-state energies of hydrogen molecules to the scaling relation E ∝Bβ . The scaling exponent β is
12
fit for each molecule HN over threemagnetic field ranges: B12 =1−10, 10−100, and 100−1000.
β
B12 H H2 H3 H4 H5 H6 H8 H10
1-10 0.283 0.326 0.350 0.370 0.363 0.365 0.371 0.372
10-100 0.242 0.290 0.312 0.330 0.346 0.355 0.353 0.361
100-1000 0.207 0.269 0.277 0.293 0.307 0.320 0.343 0.347
B =100. Thus, our density-functional-theorycalculationtends to overestimatethe energy E by about 10%. Note
12
| |
that the Hartree-Fock results also reveal a non-monotonic behavior of E at N = 4 for B = 1, in agreement with
12
our density-functional result. Demeur et al. [47] calculated the energies of H –H at B =2.35; their results exhibit
2 5 12
similar trends.
B. Helium
Our numerical results for He are given in Table III and Table IV.
Theenergies(peratom)ofHeandHe basedonHartree-Fockcalculations[11]are,respectively: 575.5, 601.2eV
2
− −
at B = 1; 1178, 1364 eV at B = 10; 2193, 2799 eV at B = 100; and 3742, 5021 eV at B = 1000.
12 12 12 12
− − − − − −
At B = 2.35, Demeur et al. [47] find that the energies (per atom) of He, He , He , and He are, respectively:
12 2 3 4
753.4, 812.6, 796.1, 805.1 eV. Using our scaling relations, we find for that same field that the energies of He,
− − − −
He , He , and He (we do not have an He result) are: 791, 871, 889, 901 eV. Thus, our density-functional
2 3 5 4
− − − −
theory calculation tends to overestimate the energy E by about 10%.
| |
10
-160
H
2
H
3
H
-180
) 4
V
H
e 10
(

N
/
N
E -200
-220
0.1 0.15 0.2 0.25 0.3 0.35
a (a )
0
FIG.1: MolecularenergyperatomversusionseparationforvarioushydrogenmoleculesatB12 =1. TheenergyoftheHatom
isshown asahorizontallineat−161.4eV. Thetwolowest-energy configurationsofH4 havenearlythesameminimumenergy,
so the curvesfor both configurations are shown here.
C. Carbon
Our numerical results for C are given in Table V, Table VI, and Table VII.
The only previous result of C molecules is that by Demeur et al. [47], who calculated C only at B = 2.35. At
2 12
this field strength, our calculation shows that C is bound relative to C atom (E = 5994, 6017 eV for C, C ),
2 2
− −
whereas Demeur et al. find no binding (E = 5770, 5749 eV for C, C ). Thus our result differs qualitatively from
2
− −
[47]. We also disagree on the ground-state occupation at this field strength: we find (n ,n ) = (9,3) while Demeur
0 1
et al. find (n ,n )=(7,5). We suggest that if Demeur et al. used the occupation (n ,n )=(9,3) they would obtain
0 1 0 1
a lower-energy for C , though whether C would then be bound remains uncertain. Since the numerical accuracy of
2 2
our computation is 0.1% of the total energy (thus, about 6 eV for B =2.35), our results for B < a few should be
12 12
treated with caution. ∼
Figure 2 gives some examples of the longitudinal electron wave functions. One wave function of each node type
in the molecule (ν = 0 to 4) is represented. Note that on the atomic scale each wave function is nodeless in nature;
that is, there are no nodes at the ions, only in between ions. The exception to this is at the centralion, where due to
symmetry considerations the antisymmetric wavefunctions must have nodes. [The nodes for (m,ν)=(0,2)are near,
but not at, the ions j = 2 and j = 4. This is incidental.] This is not surprising when one considers that all of the
electronsinatomiccarbonatthisfieldstrengtharenodeless. Theentiremolecularwavefunctioncanbethoughtofas
a string of atomic wave functions, one around each ion, each modified by some phase factor to give the overallnodal
nature of the wave function. Indeed, for atoms at field strengths that are low enough to allow ν > 0 states, we find
that their corresponding molecules have electron wave functions with nodes at the ions. Atomic Fe at B = 10, for
12
example, has an electron wave function with one node at the ion, and Fe at B =10 has an electron wave function
2 12
with a node at each ion.