Chemical bonding in strong magnetic fields
Trygve Helgaker1 , Mark Hoffmann1,2 Kai Lange1 , Alessandro Soncini1,3 , and Erik Tellgren1
2
1
CTCC, Department of Chemistry, University of Oslo, Norway
Department of Chemistry, University of North Dakota, Grand Forks, USA
3
School of Chemistry, University of Melbourne, Australia
Computational Molecular Science 2012,
Royal Agricultural College, Cirencester, UK, June 24–27, 2012
1 Alessandro
1
Trygve Helgaker1 , Mark Hoffmann1,2Chemical
Kai Lange
bonding in ,strong
magnetic fields
Soncini1,3 , CMS2012,
and Erik
JuneTellgren
24–27 2012
(CTCC,
1 / 32
Uni
Introduction
perturbative vs. nonperturbative studies
I Molecular magnetism is usually studied perturbatively
I
I
I
such an approach is highly successful and widely used in quantum chemistry
molecular magnetic properties are accurately described by perturbation theory
example: 200 MHz NMR spectra of vinyllithium
experiment
0
RHF
100
200
MCSCF
0
0
100
200
100
200
B3LYP
100
200
0
I We have undertaken a nonperturbative study of molecular magnetism
I
I
I
I
gives insight into molecular electronic structure
describes atoms and molecules observed in astrophysics (stellar atmospheres)
provides a framework for studying the current dependence of the universal density functional
enables evaluation of many properties by finite-difference techniques
Helgaker et al. (CTCC, University of Oslo)
Chemical bonding in strong magnetic fields
CMS2012, June 24–27 2012
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Overview
I background: electronic Hamiltonian and wave functions
I diamagnetism and paramagnetism
I helium atom: atomic distortion and electron correlation in magnetic fields
I H2 and He2 : bonding, structure and orientation in magnetic fields
I molecular structure in strong magnetic fields
I electronic excitations in strong magnetic fields
Helgaker et al. (CTCC, University of Oslo)
Chemical bonding in strong magnetic fields
CMS2012, June 24–27 2012
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Background
the electronic Hamiltonian
I A classical particle of charge q in a magnetic field experiences the Lorentz force:
F = qv × B
I
I
the velocity-dependent force is perpendicular to B and v and induces rotation
kinetic energy of a particle in a perpendicular circular path:
mv 2
1
q2 B 2 r 2
⇒
mv 2 =
r
2
2m
I The one-electron Hamiltonian in the absence of a magnetic field (a.u.):
qvB =
H = 12 p 2 + V ,
p = −i∇
I In a magnetic field, the kinetic momentum π replaces the canonical momentum p:
H = 21 π 2 + B · S + V ,
π = p + A,
B=∇×A
I For a uniform magnetic field in the z direction, expansion of π 2 gives:
H = 12 p 2 + 12 BLz + Bsz + 18 B 2 (x 2 + y 2 ) + V ,
I
I
I
A = 12 B × r
L and s are the orbital and spin angular momentum operators
a paramagnetic linear dependence on the magnetic field
a diamagnetic quadratic dependence on the magnetic field
Helgaker et al. (CTCC, University of Oslo)
Chemical bonding in strong magnetic fields
CMS2012, June 24–27 2012
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Background
three field regimes
I The non-relativistic electronic Hamiltonian (a.u.) in a magnetic field B along the z axis:
H = H0 + 12 BLz + Bsz + 81 B 2 (x 2 + y 2 ) ← linear and quadratic B terms
I
one atomic unit of B corresponds to 2.35 × 105 T = 2.35 × 109 G
I Coulomb regime: B 1 a.u.
I
I
I
earth-like conditions: Coulomb interactions dominate
magnetic interactions are treated perturbatively
earth magnetism 10−10 , NMR 10−4 ; pulsed laboratory field 10−3 a.u.
I Intermediate regime: B ≈ 1 a.u.
I
I
I
white dwarves: up to about 1 a.u.
the Coulomb and magnetic interactions are equally important
complicated behaviour resulting from an interplay of linear and quadratic terms
I Landau regime: B 1 a.u.
I
I
I
I
neutron stars: 103 –104 a.u.
magnetic interactions dominate (Landau levels)
Coulomb interactions are treated perturbatively
relativity becomes important for B ≈ α−2 ≈ 104 a.u.
I We here consider the weak and intermediate regimes (B < 10 a.u.)
I For a review, see D. Lai, Rev. Mod. Phys. 73, 629 (2001)
Helgaker et al. (CTCC, University of Oslo)
Chemical bonding in strong magnetic fields
CMS2012, June 24–27 2012
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Background
London orbitals and size extensivity
I The non-relativistic electronic Hamiltonian in a magnetic field:
H = H0 + 12 BLz + Bsz + 18 B 2 (x − Ox )2 + (y − Oy )2
I The orbital-angular momentum operator is imaginary and gauge-origin dependent
I
L = −i (r − O) × ∇,
AO = 12 B × (r − O)
we must optimize a complex wave function and also ensure gauge-origin invariance
I Gauge-origin invariance is imposed by using London atomic orbitals (LAOs)
ωlm (rK , B) = exp
1
2
iB × (O − K) · r χlm (rK ) ← special integral code needed
I London orbitals are necessary to ensure size extensivity and correct dissociation
I
H2 dissociation
I
FCI/un-aug-cc-pVTZ
I
0 ≤ B⊥ ≤ 2.5
I
diamagnetic system
I
full lines: with LAOs
I
dashed lines: AOs with
mid-bond gauge origin
1.0
0.5
B = 2.5
2
4
6
8
10
-0.5
Helgaker et al. (CTCC, University of Oslo)
B = 1.0
-1.0
Chemical bonding in strong magnetic fields
B = 0.0
CMS2012, June 24–27 2012
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1
Background
A = 20
ẑ ⇥ r
The effect of gauge transformations
= RHF/aug-cc-pVQZ
! A0 = A + r( A(G) · r)
0 = e iA(G)·r
!
I Consider H2 along the z axis in a uniform magnetic field in the y direction
I
the RHF wave function with gauge origin at O = (0, 0, 0) (left) and G = (100, 0, 0) (right)
the (dashed)
density remains
the 0
same
the =
(complex)
wave function changes
Gauge-origin
moved
from
tobutG
100ŷ.
I
0.5
0.4
0.4
0.3
0.3
Gauge transformed wave function, ψ"
0.5
Wave function, ψ
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0.2
0.1
0
−0.1
−0.2
−0.3
Re(ψ)
Im(ψ)
|ψ|2
Re(ψ")
Im(ψ")
−0.4
|ψ"|2
−0.5
−1.5
−1
−0.5
0
0.5
1
Space coordinate, x (along the bond)
Helgaker et al. (CTCC, University of Oslo)
1.5
−0.5
−1.5
−1
−0.5
0
0.5
1
Space coordinate, x (along the bond)
Chemical bonding in strong magnetic fields
1.5
CMS2012, June 24–27 2012
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Background
the London program and overview
I We have developed the London code for calculations in finite magnetic fields
I
I
I
I
complex wave functions and London atomic orbitals
Hartree–Fock theory (RHF, UHF, GHF), FCI theory, and Kohn–Sham theory
energy, gradients, excitation energies
London atomic orbitals require a generalized integral code
Z 1
2 2n
exp(−zt )t dt
← complex argument z
Fn (z) =
0
I
I
C++ code written by Erik Tellgren, Alessandro Soncini, and Kai Lange
C20 is a “large” system
I Overview:
I
I
I
I
I
molecular dia- and paramagnetism
helium atom in strong fields: atomic distortion and electron correlation
H2 and He2 in strong magnetic fields: bonding, structure and orientation
molecular structure in strong magnetic fields
excitations in strong magnetic fields
I Previous work in this area:
I
I
I
I
much work has been done on small atoms
FCI on two-electron molecules H2 and H− (Cederbaum)
Nakatsuji’s free complement method
no general molecular code with London orbitals
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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Closed-shell paramagnetic molecules
154114-8
Tellgren, Soncini, and Helgaker
a)
molecular diamagnetism and paramagnetism
b)
−3
14
x 10
0.1
I The Hamiltonian has paramagnetic and diamagnetic parts:
H = H0 +
1
BLz
2
+ Bsz +
1 2 2
B (x
8
12
2
0.08
+ y ) ← linear and quadratic B terms
0.06
I Most closed-shell molecules are diamagnetic
I their energy increases in an applied magnetic field
0.04
I induced currents oppose the field according to Lenz’s law
8
6
4
0.02
2
I Some closed-shell systems are paramagnetic
0
I their energy decreases in a magnetic field
−0.1
I relaxation of the wave function lowers the energy
154114-8
10
−0.05
0
0.05
0.1
0
−0.1
−0.05
J. Chem. Phys. 129, 154114 !2008"
Tellgren, Soncini, and Helgaker
I RHF calculations of the field dependence of the energy for two closed-shell systems:
c)
d)
a)
b)
−3
014
x 10
−0.002
0.1
0.02
12
−0.004
0.01
−0.00610
0.08
−0.008
0.06
−0.01
0
8
−0.01
−0.012 6
0.04
−0.014
−0.016
0.02
−0.02
4
−0.018 2
−0.03
−0.02
0
−0.1
I
I
−0.05
0
0.05
0.1
0
−0.1
−0.1
−0.05
−0.05
00
0.05
0.05
0.1
0.1
0
0.05
0.1
0.15
FIG. 1. Energy as a function of the magnetic field for different systems. Triangles represent results from finit
fitting polynomials. !a" Benzene !with the aug-cc-pVDZ basis" illustrates the typical case of diamagnetic quadra
field. !b" Cyclobutadiene !aug-cc-pVDZ" deviates from the typical case by exhibiting a nonquadratic depe
monohydride
!aug-cc-pVTZ" is an interesting case of positive magnetizability for a perpendicular field, exhib
d)
hydride !aug-cc-pVTZ" in a larger range of perpendicular fields, exhibiting a clearly nonperturbative behavior.
left: benzene: diamagnetic dependence on an out-of-plane field, χ < 0
c) paramagnetic dependence on a perpendicular field, χ > 0
right: BH:
0
Helgaker et al. (CTCC, University
of Oslo)
−0.002
Molecules in strong magnetic
fields
0.02
CMS2012, June 24–27 2012
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! 0.003
0.015
Closed-shell paramagnetic molecules
0.01
! 0.004
0.005
diamagnetic transition at stabilizing field strength Bc
0.02
0.04
0.06
0.08
0.1
0.12
B !au"
! 0.005
I However, all systems become diamagnetic in sufficiently strong fields:
d)
c)
a)
W!W0 !au"
B !au"
BH
0.05
0.1
0.15
0.2
0.25
0.3
STO!3G, Bc " 0.24
DZ,
Bc " 0.22
aug!DZ, Bc " 0.23
! 0.01
! 0.02
W!W0 !au"
W!W0 !au"
C8H8: total energy
C12H12: total energy
0.002
0.004
0.0015
STO!3G, Bc " 0.035
6!31G,
Bc " 0.034
cc!pVDZ, Bc " 0.032
STO!3G, Bc " 0.018
6!31G,
Bc " 0.018
cc!pVDZ, Bc " 0.016
0.001
0.002
0.0005
! 0.03
0.01
0.02
0.03
0.04
0.05
0.005
0.06
0.01
0.015
0.02
0.025
0.03
B !au"
! 0.0005
! 0.002
! 0.04
b)
! 0.001
W!W0 !au"
CH#
0.1
0.2
0.3
0.4
0.5
0.6
B !au"
I The transition occurs at a characteristic stabilizing critical field strength Bc
! 0.02
I
! 0.04
I
I
! 0.06
I
! 0.08
STO3!G, Bc " 0.43 to principal axis for BH
Bc ≈ 0.22 perpendicular
c " 0.44
Bc ≈ 0.032 DZ,
alongBthe
principal axis for antiaromatic octatetraene C8 H8
!DZ, Bthe
c " 0.45
Bc ≈ 0.016 aug
along
principal axis for antiaromatic [12]-annulene C12 H12
Bc is inversely proportional to the area of the molecule normal to the field
! 0.1
I
we estimate that Bc should be observable for C72 H72
I
! 0.12 We may in principle separate such molecules by applying a field gradient
c)
W!W0 !au"
MnO4!
0.3 University
0.4 0.5 of0.6
0.7
Helgaker et0.1
al. 0.2
(CTCC,
Oslo)
B !au"
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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Closed-shell paramagnetic molecules
paramagnetism and double minimum explained
I Ground and (singlet) excited states of BH along the z axis
2
|zzi = |1sB2 2σBH
2pz2 |,
2
|zxi = |1sB2 2σBH
2pz 2px |,
2
|zy i = |1sB2 2σBH
2pz 2py |
I All expectation values increase quadratically in a perpendicular field in the y direction:
n H0 + 21 BLy + 81 B 2 (x 2 + z 2 ) n = En +
1
8
2
n x + z 2 n B 2 = En − 12 χn B 2
I The |zzi ground state is coupled to the low-lying |zxi excited state by this field:
zz H0 + 12 BLy + 18 B 2 (x 2 + z 2 ) xz =
0.09
0.09
0.06
0.06
0.03
0.03
-0.1
0.1
-0.1
-0.03
0.09
0.09
0.06
0.06
0.03
0.03
0.1
-0.03
hzz |Ly | xzi B 6= 0
0.1
-0.03
-0.1
1
2
-0.1
0.1
-0.03
I A paramagnetic ground-state with a double minimum is generated by strong coupling
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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Closed-shell paramagnetic molecules
C20 : more structure
æ
æ
-756.680
æ
æ
æ
æ
-756.685
æ
æ
æ
æ
æ
æ
-756.690
æ
æ
æ
æ
æ
æ
-756.695
æ
æ
æ
æ
æ
æ
-756.700
æ
æ
æ
æ
-756.705
æ
æ
æ
æ
æ
æ
æ
æ
-756.710
æ
æ
ææææææææææ
æ
æ
ææææ
ææ
æ
æ
ææ
ææ
æ
æ
æ
æ
æ
æææ
æ
ææ
ææææææææææ
ææææææææææ
-0.04
Helgaker et al. (CTCC, University of Oslo)
-0.02
Molecules in strong magnetic fields
0.02
0.04
CMS2012, June 24–27 2012
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Closed-shell paramagnetic molecules
induced electron rotation
I The magnetic field induces a rotation of the electrons about the field direction:
I
Example
2: Non-perturbative phenomena
Example 2: Non-perturbative phenomena
the amount of rotation is the expectation value of the kinetic angular-momentum operator
h0|Λ|0i = 2E 0 (B),
Λ = r × π,
π =p+A
BH properties (aug-cc-pVDZ) as function of perpendic
I Paramagnetic closed-shell molecules (here BH):
BH properties (aug-cc-pVDZ) as function of perpendic
Energy
Angular momentum
Energy
1
!25.11
!25.11
Nuclear shielding
Angular momentum
Nuclear shielding in
Boron
1
L (B )
!25.14
!25.14
I
I
0.20.2
00
!0.5
!0.5
00
0.4
0.4
0.2
0.2
0.4
00
there is no rotation at the field-free energy maximum: B = 0
Orbital
energies
HOMO!LUMO
gapfor B > 0
energies
HOMO!LUMO
gap
the onset of paramagnetic Orbital
rotation
(against the field) reduces
the energy
the strongest paramagnetic rotation occurs at the 0.5
energy
inflexion point
0.5
0.4
0.4
the rotation comes0.1
to0.1
a halt at the stabilizing field strength: B = Bc
0.48
the onset of diamagnetic rotation (with the field)0.48
increases the energy for B > B0.35
c
0.35
Helgaker et al. (CTCC, University of Oslo) !0.2
x
(B )
0.46
0.46
0.44
energy 0.44
according
to Lenz’s law
gap
!0.1
LUMO
LUMO
increases
the
HOMO
HOMO
0.42
Molecules in strong magnetic
fields
0.42
!
!(Bx)
!(Bx)
diamagnetic
rotation!0.1
always
gap
! (Bx )
0
0
I Diamagnetic closed-shell
molecules:
I
0.2
0.2
!0.2
!0.2
"(B)x)
"(B
x
I
x
00
!25.16
!25.16
I
nuc
0.5
!25.15
!25.15
I
LLx / /|r|r!!CC
| |
nuc
0.5
!25.13
x )x
Lx(B
x
E(Bx)
E(Bx)
!25.12
!25.13
0 0
Boron
Hydrog
Hydroge
0.4
0.4
!25.12
0.2
0.2
Singlet
Singletexcitation
excitationen
0.3
0.3
0.25
0.25
CMS2012, June 24–27
0.2 2012
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The helium atom
total energy of the 1 S(1s2 ), 3 S(1s2s), 3 P(1s2p) and 1 P(1s2p) states
I Electronic states evolve in a complicated manner in a magnetic field
I
I
the behaviour depends on orbital and spin angular momenta
eventually, all energies increase diamagnetically
1.6
1.8
<XYZ>
2.0
2.2
2.4
2.6
2.8
3.00.0
Helgaker et al. (CTCC, University of Oslo)
0.2
0.4
B [a.u]
0.6
Molecules in strong magnetic fields
0.8
1.0
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The helium atom
orbital energies
I The orbital energies behave in an equally complicated manner
I the initial behaviour is determined by the angular momentum
I beyond B ≈ 1, all energies increase with increasing field
I HOMO–LUMO gap increases, suggesting a decreasing importance of electron correlation
Helium atom, RHF/augïccïpVTZ
10
8
Orbital energy, ¡ [Hartree]
6
4
2
0
ï2
0
1
Helgaker et al. (CTCC, University of Oslo)
2
3
4
5
Field, B [au]
6
Molecules in strong magnetic fields
7
8
9
10
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The helium atom
natural occupation numbers and electron correlation
I The FCI occupation numbers approach 2 and 0 strong fields
I diminishing importance of dynamical correlation in magnetic fields
I the two electrons rotate in the same direction about the field direction
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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The helium atom
atomic size and atomic distortion
I Atoms become squeezed and distorted in magnetic fields
I Helium 1s2 1 S (left) is prolate in all fields
I Helium 1s2p 3 P (right) is oblate in weak fields and prolate in strong fields
I
√
transversal size proportional to 1/ B, longitudinal size proportional to 1/ log B
I Atomic distortion affects chemical bonding
I
which orientation is favored?
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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The H2 molecule
2
3 +
∗
potential-energy curves of the 1 Σ+
g (1σg ) and Σu (1σg 1σu ) states (MS = 0)
I FCI/un-aug-cc-pVTZ curves in parallel (full) and perpendicular (dashed) orientations
I The singlet (blue) and triplet (red) energies increase diamagnetically in all orientations
1.0
1.0
B = 0.
B = 0.75
0.5
0.5
1
2
3
4
1
-0.5
-0.5
-1.0
-1.0
1.0
2
3
4
1.0
B = 1.5
B = 2.25
0.5
0.5
1
2
3
4
1
-0.5
-0.5
-1.0
-1.0
2
3
4
I The singlet–triplet separation is greatest in the parallel orientation (larger overlap)
I the singlet state favors a parallel orientation (full line)
I the triplet state favors a perpendicular orientation (dashed line) and becomes bound
I parallel orientation studied by Schmelcher et al., PRA 61, 043411 (2000); 64, 023410 (2001)
I Hartree–Fock studies by Žaucer and and Ažman (1977) and by Kubo (2007)
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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The H2 molecule
lowest singlet and triplet potential-energy surfaces E (R, Θ)
I Polar plots of the singlet (left) and triplet (right) energy E (R, Θ) at B = 1 a.u.
90°
90°
135°
45°
1
180°
225°
2
3
4
135°
45°
5
0°
315°
1
180°
2
3
5
0°
225°
270°
4
315°
270°
I Bond distance Re (pm), orientation Θe (◦ ), diss. energy De , and rot. barrier ∆E0 (kJ/mol)
B
0.0
1.0
Re
74
66
Helgaker et al. (CTCC, University of Oslo)
θe
–
0
singlet
De
459
594
∆E0
0
83
Re
∞
136
Molecules in strong magnetic fields
triplet
θe
De
–
0
90
12
∆E0
0
12
CMS2012, June 24–27 2012
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The H2 molecule
a new bonding mechanism: perpendicular paramagnetic bonding
I Consider a minimal basis of London AOs, correct to first order in the field:
1σg/u = Ng/u (1sA ± 1sB ) ,
2
1
1sA = Ns e−i 2 B×RA ·r e−arA
I In the helium limit, the bonding and antibonding MOs transform into 1s and 2p AOs:
lim 1σg = 1s,
(
2p0 ,
lim 1σu =
B
R→0
2py ≈ 2p−1 ,
2px − i 4a
R→0
all orientations
parallel orientation
perpendicular orientation
I The magnetic field modifies the MO energy level diagram
I
I
perpendicular orientation: antibonding MO is stabilized, while bonding MO is destabilized
parallel orientation: both MOs are unaffected relative to the AOs
1Σ*u H´L
0.05
1Σg HÞL
1Σ*u H´L
0.5
1Σg H´L
1.0
1Σ*u HÞL
2.0
1sA
-0.05
-0.10
1.5
1sB
1Σu* HÞL
1Σg HÞL
-0.15
1Σg H´L
I Molecules of zero bond order may therefore be stabilized by the magnetic field
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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The H2 molecule
a new bonding mechanism: perpendicular paramagnetic bonding
I The triplet H2 is bound by perpendicular stabilization of antibonding MO 1σu∗
I note: Hartree–Fock theory gives a qualitatively correct description
EêkJmol-1
EêkJmol-1
-10 500
H2
-5500
HF
-10 700
-5525
FCI
-5550
-10 900
Rêpm
100
I
I
300
100
however, there are large contributions from dynamical correlation
method
UHF/un-aug-cc-pVTZ
FCI/un-aug-cc-pVTZ
I
200
B
2.25
2.25
Re
93.9 pm
92.5 pm
De
28.8 kJ/mol
38.4 kJ/mol
UHF theory underestimates the dissociation energy but overestimates the bond length
basis-set superposition error of about 8 kJ/mol
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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The H2 molecule
Zeeman splitting of the lowest triplet state
I The spin Zeeman interaction contributes BMs to the energy, splitting the triplet
I
lowest singlet (blue) and triplet (red) energy of H2 :
H1,+1L
B = 2.25
2
1
H1,0L
1
2
H0,0L
B = 2.25
3
4
H1,ML
B = 0.00
-1
H0,0L
-2
H1,-1L
B = 2.25
I The ββ triplet component becomes the ground state at B ≈ 0.4 a.u.
I
eventually, all triplet components will be pushed up in energy diamagnetically. . .
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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The H2 molecule
evolution of lowest three triplet states
I We often observe a complicated evolution of electronic states
I a (weakly) bound 3 Σ+ (1σ 1σ ∗ ) ground state in intermediate fields
g
u
u
I a covalently bound 3 Π (1σ 2π ) ground state in strong fields
u
g
u
0.5
2.50
2.25
2.00
1.0
1.75
E (Ha)
1.50
1.25
1.5
1.00
0.75
2.0
0.50
0.25
0
Helgaker et al. (CTCC, University of Oslo)
1
2
3
4
5
R (bohr)
6
Molecules in strong magnetic fields
7
8
0.00
CMS2012, June 24–27 2012
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The H2 molecule
electron rotation and correlation
I The field induces a rotation of the electrons h0|Λz |0i about the molecular axis
I
I
I
increased rotation increases kinetic energy, raising the energy
concerted rotation reduces the chance of near encounters
natural occupation numbers indicate reduced importance of dynamical correlation
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
24 / 32
The helium dimer
2
∗2
the 1 Σ+
g (1σg 1σu ) singlet state
I The field-free He2 is bound by dispersion in the ground state
I our FCI/un-aug-cc-pVTZ calculations give D = 0.08 kJ/mol at R = 303 pm
e
e
I In a magnetic field, He2 shrinks and becomes more strongly bound
I
I
perpendicular paramagnetic bonding (dashed lines) as for H2
for B = 2.5, De = 31 kJ/mol at Re = 94 pm and Θe = 90◦
-4.0
-4.5
B = 2.0
-5.0
B = 1.5
B = 1.0
-5.5
B = 0.5
B = 0.0
1
Helgaker et al. (CTCC, University of Oslo)
2
3
4
Molecules in strong magnetic fields
5
6
7
CMS2012, June 24–27 2012
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The helium dimer
2
∗
the 3 Σ+
u (1σg 1σu 2σg ) triplet state
I the covalently bound triplet state becomes further stabilized in a magnetic field
I
I
I
De = 178 kJ/mol at Re = 104 pm at B = 0
De = 655 kJ/mol at Re = 80 pm at B = 2.25 (parallel orientation)
De = 379 kJ/mol at Re = 72 pm at B = 2.25 (perpendicular orientation)
-4.8
-5.0
B = 0.0
-5.2
-5.4
B = 0.5
-5.6
B = 1.0
B = 2.0
-5.8
0
1
2
3
4
5
I The molecule begins a transition to diamagnetism at B ≈ 2
I
eventually, all molecules become diamagnetic
T =
1
2
Helgaker et al. (CTCC, University of Oslo)
(σ · π)2 =
1
2
(σ · (p + A))2 =
1
2
Molecules in strong magnetic fields
2
σ · (p + 12 B × r)
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The helium dimer
the lowest quintet state
I In sufficiently strong fields, the ground state is a bound quintet state
I
at B = 2.5, it has a perpendicular minimum of De = 100 kJ/mol at Re = 118 pm
-3.5
-4.0
B = 0.0
-4.5
-5.0
B = 0.5
-5.5
B = 1.0
-6.0
B = 1.5
-6.5
B = 2.5
0
1
2
3
4
5
6
I In strong fields, anisotropic Gaussians are needed for a compact description for B 1
I
without such basis sets, calculations become speculative
Helgaker et al. (CTCC, University of Oslo)
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
27 / 32
Molecular structure
Hartree–Fock calculations on helium clusters
I We have studied helium clusters in strong magnetic fields (here B = 2)
I RHF/u-aug-cc-pVTZ level of theory
I all structures are planar and consist of equilateral triangles
I suggestive of hexagonal 2D crystal lattice (3 He crystallizes into an hcp structure at about 10 MPa)
I He3 and He6 bound by 3.7 and 6.8 mE per atom
h
I ‘vibrational frequencies’ in the range 200–2000 cm−1 (for the 4 He isotope)
M
OOOOLLLLDDDDEEEENNNN
M
M
M
M
M
MOOOLLLDDDEEENNN
defaults used
2.053
2.049
first point
2.048
2.044
2.042
2.061
1.951
2.293
2.060
2.084
Helgaker et al. (CTCC, University of Oslo)
2.086
Molecules in strong magnetic fields
CMS2012, June 24–27 2012
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Molecular structure
FCI calculations on the H+
3 ion
I We are investigating H+ in a magnetic field
3
I
I
I
I
Warke and Dutta, PRA 16, 1747 (1977)
equilateral triangle for B < 1; linear chain for B > 1
ground state singlet for B < 0.5; triplet for B > 0.5
the triplet state does not become bound until fields B > 1
I Potential energy curves for lowest singlet (left) and triplet (right) states
electronic energy as function of bond distances for equilateral triangles
2.0
2.50
1.5
2.25
2.25
1.75
0.5
1.50
0.0
1.25
0.5
1.00
0.75
1.0
0.50
1.5
0.25
2.00.5
0.00
1.0
1.5
R (bohr)
Helgaker et al. (CTCC, University of Oslo)
2.0
2.5
2.50
2
2.00
1.0
E (Ha)
3
2.00
1.75
1
1.50
E (Ha)
I
0
1.25
1.00
1
0.75
0.50
2
0.25
30.5
Molecules in strong magnetic fields
1.0
1.5
R (bohr)
2.0
2.5
0.00
CMS2012, June 24–27 2012
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Molecular structure
Hartree–Fock calculations on ammonia and benzene
I Ammonia for 0 ≤ B ≤ 0.06 a.u. at the RHF/cc-pVTZ level of theory
1.89
8.6
108.7
B ⊥ mol.axis
B || mol.axis
108.6
1.888
108.4
1.884
1.882
8.5
108.3
Einv [mEh]
∠ H−N−H [degrees]
1.886
dNH [bohr]
8.55
108.5
108.2
108.1
8.45
108
8.4
107.9
1.88
107.8
8.35
1.878
I
I
I
0
0.02
0.04
0.06
0.08
0.1
B [au]
0.12
0.14
0.16
0.18
0.2
107.7
0
0.02
0.04
0.06
0.08
0.1
B [au]
0.12
0.14
0.16
0.18
0.2
0
0.01
0.02
0.03
B [au]
0.04
0.05
0.06
bond length (left), bond angle (middle) and inversion barrier (right)
ammonia shrinks and becomes more planar (from shrinking lone pair?)
in the parallel orientation, the inversion barrier is reduced by −0.001 cm−1 at 100 T
I Benzene in a field of 0.16 along two CC bonds (RHF/6-31G**)
I it becomes 6.1 pm narrower and 3.5 pm longer in the field direction
I agrees with perturbational estimates by Caputo and Lazzeretti, IJQC 111, 772 (2011)
C6H6, HF/6−31G*
138.75
138.7
138.65
d(C−C) [pm]
138.6
C−C other (c)
C−C || B (d)
138.55
138.5
138.45
138.4
138.35
138.3
Helgaker et al. (CTCC, University of Oslo)
0
0.02
0.04
0.06
0.08
B [au]
Molecules in strong magnetic fields
0.1
0.12
0.14
0.16
CMS2012, June 24–27 2012
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Linear response theory in finite magnetic fields
I We have implemented linear response theory (RPA) in finite magnetic fields
I Lowest CH3 radical states in magnetic fields (middle)
I transitions to the green state are electric-dipole allowed (but becomes transparent at 0.3 a.u.)
−39.25
0.07
−39.30
0.06
−39.35
0.05
−39.40
0.04
f E1
En [Ha]
I Oscillator strength for transition to green CH3 state
I length (red) and velocity (blue) gauges
−39.45
−39.50
0.02
−39.55
0.01
−39.60
−39.65
0.00
0.03
0.15
0.30
B⊥ [a.u.]
Helgaker et al. (CTCC, University of Oslo)
0.45
0.00
0.00
Molecules in strong magnetic fields
0.15
0.30
B⊥ [a.u.]
0.45
CMS2012, June 24–27 2012
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Conclusions
Summary and outlook
I We have developed the LONDON program for molecules in magnetic field
I We have studied He, H2 and He2 in magnetic fields
I
I
I
atoms and molecules shrink
molecules are stabilized by magnetic fields
preferred orientation in the field varies from system to system
I We have studied molecular structure in magnetic fields
I
bond distances are typically shortened in magnetic fields
I We have studied closed-shell paramagnetic molecules in strong fields
I
I
all paramagnetic molecules attain a global minimum at a characteristic field Bc
Bc decreases with system size and should be observable for C72 H72
I An important goal is to study the universal density functional in magnetic fields
Z
Z
F [ρ, j] = sup E [u, A] − ρ(r)u(r) dr − j(r) · A(r) dr
u,A
I Support: The Norwegian Research Council and the European Research Council
Helgaker et al. (CTCC, University of Oslo)
Conclusions
CMS2012, June 24–27 2012
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