Molecules in strong magnetic fields
Trygve Helgaker1 , Mark Hoffmann1,2 , Kai Lange1 , Alessandro Soncini1,3 , Erik Tellgren1
1 Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, Norway
2 Department of Chemistry, University of Dakota, USA
3 School of Chemistry, University of Melbourne, Australia
Laboratoire de Chimie Theorique (LCT),
Université Pierre et Marie Curie (UPMC) – Sorbonne Universités
May 26 2011
1 , Alessandro
1,3 , Erik Tellgren
1 (CTCC, University
Trygve Helgaker1 , Mark Hoffmann1,2 , Kai
Molecules
Lange
in strong
magnetic fields Soncini
UPMC 2011
1 / 28
Outline
1
Molecular magnetism
2
The LONDON code and London orbitals
3
Molecular diamagnetism and paramagnetism
4
The helium atom in strong fields
5
Molecular bonding in strong fields
6
Molecular structure in strong fields
7
Linear response in strong fields
8
Conclusions
Helgaker et al. (CTCC, University of Oslo)
Overview
UPMC 2011
2 / 28
Molecules in weak magnetic fields
I Molecular magnetism is usually studied perturbatively
E (B) = E (0) −
I
I
X
α
mα Bα −
1
2
X
αβ
χαβ Bα Bβ + · · ·
such an approach is highly successful and widely used in quantum chemistry
molecular magnetic properties are accurately described by perturbation theory
I Example: 200 MHz NMR spectra of vinyllithium
experiment
0
RHF
100
MCSCF
0
Helgaker et al. (CTCC, University of Oslo)
0
200
100
200
100
200
B3LYP
100
0
200
Molecular magnetism
UPMC 2011
3 / 28
Molecules in strong magnetic fields
I However, the field dependence of the energy can be complicated
I
the energy of C20 (ring conformation) plotted against B (atomic units)
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
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æææ
æææ
ææ
æ
æ
ææ
æ
æ
ææ
ææ
æææ
æ
ææ
ææææææææææ
ææææææææææ
-0.04
-0.02
0.02
0.04
I such behaviour cannot be described or understood perturbatively
I We have undertaken a nonperturbative study of molecular magnetism
I
I
I
I
I
gives new insight into molecular electronic structure
describes atomic and molecules observed in astrophysics (stellar atmospheres)
mimicks conditions in certain semiconductors
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)
Molecular magnetism
UPMC 2011
4 / 28
The electronic Hamiltonian in an external magnetic field
I The non-relativistic electronic Hamiltonian (atomic units) in a magnetic field B is given by
1
8
H = T + VCoulomb + 12 B · L + B · s +
I
I
I
X
i
(B × ri ) · (B × ri )
T is the kinetic-energy operator
VCoulomb = Vne + Vee + Vnn are the Coulomb operators
L and s are the orbital and spin angular momentum operators
I Coulomb regime: B ≈ 0 a.u.
I earth-like conditions: Coulomb interactions dominate
I magnetic interactions are treated perturbatively
I Landau regime: B 1 a.u.
I astrophysical conditions: magnetic interactions dominate
H = T + 12 BLz + Bsz +
I
I
I
I
1
2
X
i
(xi2 + yi2 )(B 2 /4)
harmonic-oscillator Hamiltonian with force constant B 2 /4: Landau levels
the electrons are confined in the transverse directions by the magnetic field
cyclotron energy proportional to B, cyclotron radius proportional to B −1/2
Coulomb interactions are treated perturbatively
I intermediate regime: B ≈ 1 a.u.
I the Coulomb and magnetic interactions are equally important
I complicated behaviour resulting from interplay of linear and quadratic terms
I the linear term may lower or raise the energy
I the quadratic term always raises the energy
Helgaker et al. (CTCC, University of Oslo)
Molecular magnetism
UPMC 2011
5 / 28
The electronic Hamiltonian in an external magnetic field
I The non-relativistic electronic Hamiltonian (atomic units) in a magnetic field B is given by
H = T + VCoulomb + 12 B · L + B · s +
1
8
X
i
(B × ri ) · (B × ri )
I We distinguish three magnetic-field regimes
I
I
I
Coulomb regime: B ≈ 0 a.u.
intermediate regime: B ≈ 1 a.u.
Landau regime: B 1 a.u.
I One atomic unit of B corresponds to 2.235 × 105 T = 2.235 × 109 G
I
I
I
I
I
I
earth magnetism: 30–50 µT (10−10 a.u.)
fridge magnet: 1 mT
loudspeaker magnet and medical MRI: 1 T
strongest sustained laboratory fields: 45 T (2 · 10−4 a.u.)
strongest pulsed laboratory fields: 100–1000 T
white dwarfs: 101 –105 T; neutron stars 108 –109 T; magnetars 1010 T
I We shall here consider the intermediate regime
I
I
I
I
Coulomb and magnetic interactions are both described nonperturbatively
requires complex arithmetic
gauge-origin invariance must be ensured (London orbitals)
requires development of a non-standard code (LONDON)
Helgaker et al. (CTCC, University of Oslo)
Molecular magnetism
UPMC 2011
6 / 28
London orbitals
I The external magnetic field is represented by a vector potential
π = −i~∇ + eA(r),
I
I
A(r) = 21 B × (r − O) ,
B(r) = ∇ × A(r)
the position of the gauge origin O is not unique
approximate calculations may suffer from spurious gauge-origin dependence
I Gauge-origin invariance may be enforced by using London atomic orbitals:
I standard atomic orbitals equipped with a complex phase factor
ωlm = exp[iAK (O) · r]χlm (rK )
I London orbitals are correct to first-order in the external magnetic field
I for this reason, basis-set convergence is usually improved, as here illustrated for benzene:
London
origin CM
origin H
basis set
STO-3G
6-31G
cc-pVDZ
aug-cc-pVDZ
STO-3G
6-31G
cc-pVDZ
aug-cc-pVDZ
STO-3G
6-31G
cc-pVDZ
aug-cc-pVDZ
Helgaker et al. (CTCC, University of Oslo)
χxx
−8.1
−8.2
−8.1
−8.0
−35.8
−31.6
−15.4
−9.9
−35.8
−31.6
−15.4
−9.9
χyy
−8.1
−8.2
−8.1
−8.0
−35.8
−31.6
−15.4
−9.9
−176.3
−144.8
−48.0
−20.9
χzz
−23.0
−23.1
−22.3
−22.4
−48.1
−39.4
−26.9
−25.2
−116.7
−88.0
−41.6
−33.9
The LONDON code and London orbitals
Xxxxx
−211
−219
−236
−316
45
29
9
−413
45
29
9
−413
Xyyyy
−211
−219
−236
−316
45
29
9
−413
1477
1588
2935
−3321
Xzzzz
−52
−64
−120
−153
27
−152
−241
−159
−5340
−5866
−3355
−1097
UPMC 2011
7 / 28
Hybrid plane-wave–Gaussians (PWG) orbitals
I London AOs are in fact hybrid plane-wave–Gaussian (PWG) orbitals
ωκ,c (r) = exp(iκ · r) Slm (r) exp(−arA2 )
| {z } |
{z
}
plane wave solid-harmonic Gaussian
I
the wave vector κ is the AO-centered vector potential at the gauge origin
I More generally, PWGs have several uses:
I
I
mixed basis for periodic boundary conditions and scattering studies
gauge-origin independent magnetic properties at zero field
I PWGs necessitate a generalization of GTO integral-evaluation techniques
I
I
at zero field, complex algebra may be avoided
at finite field, complex algebra cannot be avoided
I We have developed and implemented a McMurchie–Davidson PWG scheme
I
I
Tellgren et al., JCP 129, 154114 (2008)
previous work: M. Tachikawa and M. Shiga, Phys. Rev. E 64:056706 (2001)
I Integral evaluation in complex algebra
I
the Boys function with complex arguments
Z 1
2 2n
Fn (z) =
exp(−zt )t dt
← complex argument z
0
I
more terms in the recurrence relations
Helgaker et al. (CTCC, University of Oslo)
The LONDON code and London orbitals
UPMC 2011
8 / 28
PWG product rule and overlap integrals
I The Gaussian product rule still holds
∗
2
2
Ωκλ
ab (r) = exp(iκ · r) exp(−arA ) exp(iλ · r) exp(−brB )
|
{z
}|
{z
}
PWG at A
PWG at B
2
ab
= exp(− a+b
RAB
) exp[−i(κ − λ) · r] exp[−(a + b)rP2 ]
{z
}
|
{z
}|
PWG at P = (aA + bB)/(a + b)
prefactor
I Integration over all space yields
Z
π 3/2 exp − ab R 2
2
AB
a+b
(κ−λ)
Ωκλ
ab (r) dr = exp − 4(a+b) + i(κ − λ) · P
(a + b)3/2
|
{z
}|
{z
}
plane-wave contribution
standard Gaussian overlap
I As for standard Gaussians, Coulomb integrals reduce to the Boys function
J=
Z Z exp(−iκ · r ) exp(−pr 2 ) exp(iλ · r ) exp(−qr 2 )
1
2
1P
2Q
2
= exp − κ4p −
r12
λ2
4q
− iκ · P − iλ · Q
dr1 dr2
h
2 i
2π 5/2
pq
F0 p+q
P0 − Q0
√
pq p + q
|
{z
}
P0 = P − iκ/2p, Q0 = Q − iλ/2q
Helgaker et al. (CTCC, University of Oslo)
The LONDON code and London orbitals
UPMC 2011
9 / 28
The LONDON program
I An ab initio program for finite-field calculations with London orbitals
I Some features of the present LONDON code:
I
I
I
I
I
I
Hartree–Fock theory (RHF, UHF, GHF)
FCI theory and (soon) MCSCF theory
Kohn–Sham theory
many first-order properties (x − Cx )m (∂/∂x)n
molecular gradients
excitation energies using response theory (TDHF)
I The code has been written by Erik Tellgren, Kai Lange, and Alessandro Soncini
I
I
I
mostly C++, some Fortran 77
modular but not highly optimized yet
C20 is a “large” system
Helgaker et al. (CTCC, University of Oslo)
The LONDON code and London orbitals
UPMC 2011
10 / 28
Molecular diamagnetism and paramagnetism
I When a magnetic field is applied to a molecule, one of two things can happen:
I
154114-8
the energy is lowered: molecular paramagnetism
the energy is raised: molecular diamagnetism
I
Tellgren, Soncini, and Helgaker
a)
b)
−3
14
x 10
0.1
12
I Open-shell molecules are paramagnetic
0.08
I
10
the molecule reorients the permanent magnetic moment and lowers the energy
8
0.06
I Closed-shell molecules may be diamagnetic or paramagnetic
I
6
0.04
the induced magnetic moment may oppose or enhance the external field
most molecules are diamagnetic (Lenz’ law)
in some systems, orbital unquenching results in paramagnetism
4
0.02
I
I
0
−0.1
−0.05
2
0
0.05
0
−0.1
0.1
I RHF
154114-8
Tellgren, Soncini, and Helgaker
Chem. Phys. 129,
154114 !2008"
calculations
of the field-dependence for J.three
closed-shell
systems:
a)
b)
c)
−3
14
x 10
d)
0
−0.002
0.1
12
0.08
0.02
−0.004
0.01
−0.006
10
−0.008
0
8
0.06
−0.01
6
−0.01
−0.012
0.04
−0.014
4
−0.02
−0.016
0.02
0
−0.1
2
−0.05
0
0.05
0.1
0
−0.1
−0.018
−0.03
−0.02
−0.05
0
0.05
0.1
−0.1
−0.05
0
0.05
0.1
0
d)
ac) benzene (aug-cc-pVDZ): typical
diamagnetic quadratic dependence on an out-of-plane field
b cyclobutadiene (aug-cc-pVDZ): diamagnetic non-quadratic dependence on an out-of-plane field
linear magnetizability
c BH (aug-cc-pVTZ): paramagnetic dependence on a perpendicular
field are in fact positive and large enough to
0
−0.002
0.02
−0.004
−0.006
−0.008
Helgaker et al.
(CTCC, University of Oslo)
0.0
FIG. 1. Energy as a function of the magnetic field for different systems. Triangles represent re
fitting polynomials. !a" Benzene !with the aug-cc-pVDZ basis" illustrates the typical case of diam
field. !b" Cyclobutadiene !aug-cc-pVDZ" deviates from the typical case by exhibiting a no
monohydride !aug-cc-pVTZ" is an interesting case of positive magnetizability for a perpendic
hydride !aug-cc-pVTZ" in a larger range of perpendicular fields, exhibiting a clearly nonpertur
0.01
Molecular
diamagnetism and paramagnetism
0
make even the average magnetizability positive !paramagnetic". It is therefore interesting to verify via our finite-field
London-orbital approach whether this very small system is
UPMC
2011 magindeed characterized by a particularly
large nonlinear
For the
hypermagne
fitting proce
data points
11gree
/ 28
of the
! 0.003
0.015
0.01
Stabilizing field strength for paramagnetic molecules
! 0.004
0.005
0.02 0.04 0.06 0.08
I However, plots over a larger range reveal
that
0.1
0.12
B !au"
! 0.005
all closed-shell systems become diamagnetic in sufficiently strong fields:
a)
d)
c)
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 B ≈ 0.22 perpendicular to principal axis for BH
c
STO3!G, Bc " 0.43
I B ≈ 0.032 along the principal axis for antiaromatic octatetraene C H
! 0.04
c
8 8
DZ,
Bc " 0.44
I B ≈ 0.016 along the principal axis for antiaromatic [12]-annulene C H
c
12 12
aug!DZ, Bc " 0.45
! 0.06
I The stabilizing field strength decreases with increasing molecular size
I strongest laboratory fields attainable: 10−3 a.u.
I B is inversely proportional to the area of the molecule normal to the field
! 0.1
c
I we estimate that B should be observable for C H
c
72 72
! 0.12
I We may in principle separate such molecules by applying a field gradient
! 0.08
c)
W!W0 !au"
0.1
MnO4!
0.2
0.3
0.4
0.5
0.6
0.7
B !au"
STO!3G, Bc " 0.45
! 0.1
Helgaker et al. (CTCC, UniversityWachters,
of Oslo) Bc " 0.50 Molecular diamagnetism and paramagnetism
UPMC 2011
12 / 28
Analytical model for the diamagnetic transition
I The diamagnetic transition arises from an interplay between linear and quadratic terms
I
it can be understood from a simple two-level model
I Molecular orbitals relevant for BH:
1sB , 2σBH , 2px , 2py , 2pz
(molecule along z axis)
I Ground and excited states:
2
|0i = |1sB2 2σBH
2pz2 |,
2
|xi = |1sB2 2σBH
2pz 2px |,
2
|y i = |1sB2 2σBH
2pz 2py |
I Let us apply a perpendicular magnetic field in the y direction:
HB = 21 BLy + 18 B 2 (x 2 + z 2 )
I This leads to the following Hamiltonian matrix:
H(B) =
h0|H|0i
hx|H|0i
h0|H|xi
hx|H|xi
=
E0 − 21 χ0 B 2
−iµB
iµB
E1 − 21 χ1 B 2
where we have introduced
χ0 = − 14 h0|x 2 + z 2 |0i,
Helgaker et al. (CTCC, University of Oslo)
χ1 = − 41 hx|x 2 + z 2 |xi,
Molecular diamagnetism and paramagnetism
µ = − 21 ih0|Ly |xi
UPMC 2011
13 / 28
Energy levels of the two-level model
I Two-level Hamiltonian with χ0 = −28, χ0 = −16 and ∆ = (E1 − E0 )/2 = 0.01:
H(B) =
−∆ − 12 χ0 B 2
−iµB
iµB
∆ − 12 χ1 B 2
=
−0.01 + 14B 2
−iµB
iµB
0.01 + 8B 2
I Eigenvalues:
1
1
W0/1 (B) = − (χ0 + χ1 )B 2 ∓
2
2
q
[2∆ + (χ0 − χ1 )B 2 ]2 + 4µ2 B 2
I Plots for different values of µ:
I uncoupled (0), diamagnetic (0.2), nonmagnetic (0.374), and paramagnetic (0.6)
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
0.1
-0.1
-0.03
I
0.1
-0.03
0.1
-0.03
magnetizability (negative second derivative at B = 0):
00
2
χ(0) = −W0 (0) = χ0 + µ /2∆ = −7, −5, 0, 11
Helgaker et al. (CTCC, University of Oslo)
Molecular diamagnetism and paramagnetism
UPMC 2011
14 / 28
Example 2: A two-state model
Two-level model fitted to experimental data
Setting ∆E = 0.097 (zero field excitation energy) and fittin
yields:
I The two-level model contains four parameters
I
I
may be fitted to experimental data
provides
excellent fits,
to polynomial
fits
parameters
tosuperior
the ground
energies
BH singlet energies (aug!cc!pVDZ)
0.25
x
Energy E(B )
0.2
0.15
0.1
g.s.
exc!1
exc!2
fitted
fitted
0.05
0
!0.05
0
0.05
Helgaker et al. (CTCC, University of Oslo)
0.1
0.15
0.2
0.25
Field Bx
0.3
Molecular diamagnetism and paramagnetism
0.35
0.4
0.45
UPMC 2011
15 / 28
Induced magnetic moment and kinetic angular momentum
I The induced magnetic moment M and the kinetic angular momentum Λkin are related as
M = −E 0 (B) = − 12 h0|Λ|0i ,
I
I
Λ = r × π,
π =p+A
the kinetic angular momentum represents the amount of rotation in the system
it differs from the canonical momentum L = r × p in an electromagnetic field
I Diamagnetic closed-shell molecules:
I h0|Λ|0i aligns with the field, increasing the energy −m · B =
1
2 h0|Λ|0i
·B
I Paramagnetic closed-shell molecules:
I h0|Λ|0i first aligns against the field, decreasing the energy
I h0|Λ|0i goes through a minimum at the inflection point E 00 (B) = 0
I h0|Λ|0i then increases again until it vanishes at Bc
I h0|Λ|0i finally aligns with the field, making the system diamagnetic
Example 2: Non-perturbative phenomena
BH properties (aug-cc-pVDZ) as function of perpendicular field:
Energy
Angular momentum
Nuclear shielding integral
1
!25.11
Boron
Hydrogen
0.4
Lx / |r ! Cnuc|
0.5
!25.13
Lx(Bx)
E(Bx)
!25.12
!25.14
0
0.2
0
!25.15
!0.2
!25.16
!0.5
0
I
0.2
0.4
0
Orbital energies
0.2
0.4
0
HOMO!LUMO gap
0.4
Singlet excitation energies
there is no net induced rotation at the0.5energy minimum
0.4
0.1
0.48
0
Helgaker et al. (CTCC, University of Oslo)
0.2
Molecular diamagnetism
and paramagnetism
0.46
0.35
UPMC 2011
16 / 28
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
ææ
ææ
æææ
æ
æ
æ
ææææææææææ
æææææææææ
-0.04
Helgaker et al. (CTCC, University of Oslo)
-0.02
0.02
Molecular diamagnetism and paramagnetism
0.04
UPMC 2011
17 / 28
The helium atom in strong fields
I The helium energy behaves in simple manner in magnetic fields (left)
I
an initial quadratic diamagnetic behaviour is followed by linear increase in B
this is expected from the Landau levels (harmonic potential increases as B 2 )
I
I The orbital energies behave in a more complicated manner (middle)
I
the initial behaviour is determined by the angular momentum
beyond B ≈ 1, all energies increase with increasing field
HOMO–LUMO gap increases, suggesting a decreasing importance of electron correlation
I
I
I The atom becomes squeezed in the magnetic field (right)
I
this is particularly true for the transverse directions
the cyclotron radius is proportional to B −1/2
special basis sets are need for strong fields (Landau orbitals)
I
I
1
6
0
2
ï2
0
ï2
1
2
3
4
5
Field, B [au]
6
7
8
Helgaker et al. (CTCC, University of Oslo)
9
0.7
4
ï1
0
0.8
Quadrupole moment [bohr2]
8
ï3
Helium atom, RHF/augïccïpVTZ
Helium atom, RHF/augïccïpVTZ
10
2
Orbital energy, ¡ [Hartree]
Energy, E [Hartree]
Helium atom, RHF/augïccïpVTZ
3
Qxx
Qyy
Qzz
0.6
0.5
0.4
0.3
0.2
0
1
2
3
4
5
Field, B [au]
6
7
8
9
The helium atom in strong fields
10
0.1
0
1
2
3
4
5
Field, B [au]
6
7
UPMC 2011
8
9
18 / 28
The helium natural occupation numbers in strong fields
I The natural occupation numbers are eigenvalues of the one-electron density matrix
I The FCI occupation numbers approach 2 and 0 in strong fields
I
I
decreasing importance of dynamical correlation in magnetic fields
both electrons rotate in the same direction about the field direction
He/aug-cc-pVTZ
Cummulative occupation number
2.000
1.995
1.990
1.985
1.980 0
Helgaker et al. (CTCC, University of Oslo)
1
2
B (a.u.)
The helium atom in strong fields
3
4
UPMC 2011
19 / 28
The hydrogen molecule in strong fields
I Magnetic fields strongly affect chemical bonding and potential energy surfaces
I Potential energy curves of H2 in a parallel field B = 0, 5, 10 a.u.
I
FCI/aug–cc-pVDZ calculations with energy set to zero at full dissociation
H2 /aug-cc-pVDZ, lowest singlet, field parallel to bond
0.5
H2 /aug-cc-pVDZ, lowest triplet, field parallel to bond
0.0
0.0
0.5
0.5
E (Ha)
E (Ha)
1.0
1.5
2.0
1.5
B = 0.0 a.u.
B = 5.0 a.u.
B = 10.0 a.u.
2.5
1
2
3
4
R (bohr)
5
6
7
1.0
8
2.0
B = 0.0 a.u.
B = 5.0 a.u.
B = 10.0 a.u.
1
2
3
4
R (bohr)
5
6
7
8
I H2 becomes strongly bound in strong fields
I singlet dissociation energy of about 7000 kJ/mol at field strength 10 a.u.
I triplet dissociation energy of about 5000 kJ/mol at field strength 10 a.u.
I The low-spin triplet component (ββ) becomes the ground state at about 0.25 a.u.
I
the Zeeman interaction contributes BMs to the energy
Helgaker et al. (CTCC, University of Oslo)
Molecular bonding in strong fields
UPMC 2011
20 / 28
The hydrogen molecule in strong fields
I The energy of H2 in the singlet state increases in a parallel field (top left plot)
I the field induces a rotation of the electrons h0|Λ |0iabout the molecular axis (top right plot)
z
I occupation numbers indicate reduced importance of dynamical correlation (bottom plots)
I the electrons rotate in tandem in the same direction, reducing the chance of near encounters
H2 , singlet, molecular energy
1.5
H2 , singlet, induced angular momentum
1.6
1.4
1.0
1.2
1.0
L (a.u.)
E (Ha)
0.5
0.0
0.8
0.6
0.5
0.4
1.0
1.50.0
0.5
1.0
1.5
2.0
B (a.u.)
2.5
3.0
3.5
0.00.0
4.0
H2 /cc-pVTZ, optimized geometry, field parallel to bond
2.00
1.98
Cummulative occupation number
Cummulative occupation number
2.00
0.2
R =R ∗
R =1.4 a0
1.96
1.94
1.92
1.90 0
1
Helgaker et al. (CTCC, University of Oslo)
2
B (a.u.)
3
4
R =R ∗
R =1.4 a0
0.5
1.0
1.5
2.0
B (a.u.)
2.5
3.0
3.5
4.0
H2 /cc-pVTZ, R = 1.4 bohr, field parallel to bond
1.98
1.96
1.94
1.92
1.90 0
Molecular bonding in strong fields
1
2
B (a.u)
3
4
UPMC 2011
21 / 28
The helium dimer in strong fields
I We have also performed FCI/aug-cc-pVDZ calculations on the helium dimer
I
energy set to zero for full dissociation in all cases
I We have considered the lowest singlet, triplet and quintet states
I
in the absence of a field, only the triplet state is covalently bound
in the presence of a field, all states become strongly bound
I
He2 /aug-cc-pVDZ, lowest singlet, field parallel to bond
He2 /aug-cc-pVDZ, lowest triplet, field parallel to bond
He2 /aug-cc-pVDZ, lowest quintet, field parallel to bond
0.3
0.0
0.2
0.0
0.1
0.2
1
2
3
4
R (bohr)
5
6
7
0.4
B = 0.0 a.u.
B = 5.0 a.u.
B = 10.0 a.u.
B = 15.0 a.u.
0.8
8
0.1
0.2
0.6
B = 0.0 a.u.
B = 5.0 a.u.
B = 10.0 a.u.
B = 15.0 a.u.
1.0
E (Ha)
E (Ha)
E (Ha)
0.0
0.5
1
2
3
4
R (bohr)
5
6
7
B = 0.0 a.u.
B = 5.0 a.u.
B = 10.0 a.u.
B = 15.0 a.u.
0.3
0.4
8
0.5
1
2
3
4
R (bohr)
5
6
7
8
I In magnetic fields, spin multiplets are split
I
within each multiplet, the low-spin component has lowest energy
I Increasing magnetic fields favor low-spin components of high-spin states
I
I
in weak fields, the ground state is a singlet
in strong fields, the ground state is the low-spin quintet
Helgaker et al. (CTCC, University of Oslo)
Molecular bonding in strong fields
UPMC 2011
22 / 28
F2 in a perpendicular magnetic field
I For larger molecules, we have so far only performed HF calculations
I The magnetic field changes the shape of the potential-energy curve
-198.4
B=0.00
B=0.45
B=0.05
-198.5
B=0.10
B=0.30
-198.6
-198.7
2.5
I
I
3.0
3.5
4.0
4.5
5.0
5.5
diamagnetic behaviour in the molecular limit
paramagnetic then diamagnetic behaviour in the atomic limit
I The bond length increases while dissociation energy decreases with increasing field
Helgaker et al. (CTCC, University of Oslo)
Molecular structure in strong fields
UPMC 2011
23 / 28
The structure of NH3 in strong fields
I Magnetic field applied along the symmetry axis
NH3, HF/6−31G**
NH3, HF/6−31G**
100.15
108.3
100.1
108.2
100.05
108.1
H−N−H angle [degrees]
d(N−H) [pm]
100
99.95
99.9
99.85
99.8
108
107.9
107.8
107.7
99.75
107.6
99.7
99.65
0
0.02
0.04
0.06
0.08
B [au]
0.1
0.12
0.14
0.16
107.5
0
0.02
0.04
0.06
0.08
B [au]
0.1
0.12
0.14
0.16
I As expected the molecule shrinks
I bonds contract by 0.7 pm in a field of 0.15 a.u.
I Less intuitively, the molecule becomes more planar
I could this be caused by the shrinking lone pair?
Helgaker et al. (CTCC, University of Oslo)
Molecular structure in strong fields
UPMC 2011
24 / 28
The structure of C6 H6 in strong fields
I At B = 0.16, the molecule is 6.1 pm narrower and 3.5 pm longer in the field direction
I in agreement with perturbational estimates by Caputo and Lazzeretti IJQC (online 31 aug 2010)
C6H6, HF/6−31G*
107.6
C−H || B
C−H other
107.5
107.4
d(C−H) [pm]
107.3
107.2
107.1
107
106.9
106.8
0
0.02
0.04
0.06
C6H6, HF/6−31G*
0.08
B [au]
0.1
0.12
0.14
0.16
C6H6, HF/6−31G*
138.75
120.5
C−C−C (a)
C−C−H (b)
138.7
120
138.65
d(C−C) [pm]
bond angle [degrees]
119.5
138.6
C−C other (c)
C−C || B (d)
138.55
138.5
138.45
119
118.5
118
138.4
117.5
138.35
138.3
0
0.02
0.04
0.06
Helgaker et al. (CTCC, University of Oslo)
0.08
B [au]
0.1
0.12
0.14
0.16
117
0
0.02
0.04
Molecular structure in strong fields
0.06
0.08
B [au]
0.1
0.12
0.14
0.16
UPMC 2011
25 / 28
Linear response theory in finite magnetic fields
I We have implemented linear response theory (TDHF) in finite magnetic fields
I lowest CH3 radical states in magnetic fields (middle)
I
transitions to the green state are electric-dipole allowed
I oscillator strength for transition to green CH3 state
length (red) and velocity (blue) gauges
−39.25
0.07
−39.30
0.06
−39.35
0.05
−39.40
0.04
f E1
En [Ha]
I
−39.45
−39.50
0.02
−39.55
0.01
−39.60
−39.65
0.00
Helgaker et al. (CTCC, University of Oslo)
0.03
0.15
0.30
B⊥ [a.u.]
0.45
0.00
0.00
Linear response in strong fields
0.15
0.30
B⊥ [a.u.]
0.45
UPMC 2011
26 / 28
TDHF spectrum of BH
I Singlet and triplet energies vs. field for BH [RHF/aug-cc-pVDZ]
I the field is perpendicular to the bond axis
I instabilities at B
⊥ ≈ 0.1, 0.16 a.u.
−24.7
−24.7
−24.8
−24.8
−24.9
−24.9
−25.0
−25.0
−25.1
−25.1
0.00
0.15
Helgaker et al. (CTCC, University of Oslo)
0.30
0.45
0.00
Linear response in strong fields
0.15
0.30
0.45
UPMC 2011
27 / 28
Conclusions
I We have developed the LONDON program
I complex orbitals and wave functions
I restricted, unrestricted and generalized Hartree–Fock and Kohn–Sham theories
I FCI theory
I London atomic orbitals for gauge-origin independence
I expectation values of one-electron operators
I molecular gradients
I linear response theory (TDHF)
I The LONDON program may be used for
I finite-difference alternative to analytical derivatives
I studies of molecules in strong magnetic fields
I studies of exchange–correlation functional in magnetic fields
I We have studied the behaviour of paramagnetic molecules in strong fields
I all paramagnetic molecules attain a global minimum at a characteristic field B
c
I Bc decreases with system size and should be observable for C72 H72
I This behaviour can be understood from a simple two-level model
I explains the existence of a global minimum at B
c
I at B , the induced angular momentum vanishes
c
I We have studied excitation energies in magnetic fields
I excitation energies typically increase in magnetic fields
I We have studied molecular structure in magnetic fields
I atoms are squeezed in magnetic fields—in particular, perpendicular to the field
I bond distances are typically reduced in magnetic fields
Helgaker et al. (CTCC, University of Oslo)
Conclusions
UPMC 2011
28 / 28