Molecules in strong magnetic fields - Universitetet i...

Molecules in strong magnetic fields

Trygve Helgaker1, Mark Hoffmann1,2 Kai Lange1, Alessandro Soncini1,3, and Erik Tellgren1

1CTCC, Department of Chemistry, University of Oslo, Norway2Department of Chemistry, University of North Dakota, Grand Forks, USA

3School of Chemistry, University of Melbourne, Australia

Chimie Theorique, Methodologies, Modelisation (CTMM),Institut Charles Gerhardt (ICG) Montpellier, Universite Montpellier 2,

France, November 19, 2012

Trygve Helgaker1, Mark Hoffmann1,2 Kai Lange1, Alessandro Soncini1,3, and Erik Tellgren1(CTCC, University of Oslo)Molecules in strong magnetic fields CMS2012, November 19 2012 1 / 32

Introductionperturbative vs. nonperturbative studies

I Molecular magnetism is usually studied perturbatively

I such an approach is highly successful and widely used in quantum chemistryI molecular magnetic properties are accurately described by perturbation theoryI example: 200 MHz NMR spectra of vinyllithium

0 100 200

MCSCF

0 100 200 0 100 200

B3LYP

0 100 200

0 100 200

experiment

0 100 200 0 100 200

RHF

0 100 200

I We have undertaken a nonperturbative study of molecular magnetism

I gives insight into molecular electronic structureI describes atoms and molecules observed in astrophysics (stellar atmospheres)I provides a framework for studying the current dependence of the universal density functionalI enables evaluation of many properties by finite-difference techniques

Helgaker et al. (CTCC, University of Oslo) Chemical bonding in strong magnetic fields CMS2012, November 19 2012 2 / 32

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


Backgroundthe electronic Hamiltonian

I A classical particle of charge q in a magnetic field experiences the Lorentz force:

F = qv × B

I the velocity-dependent force is perpendicular to B and v and induces rotationI kinetic energy of a particle in a perpendicular circular path:

qvB =mv2

r⇒ 1

2mv2 =

q2B2r2

2m

I The one-electron Hamiltonian in the absence of a magnetic field (a.u.):

H = 12p2 + V , p = −i∇

I In a magnetic field, the kinetic momentum π replaces the canonical momentum p:

H = 12π2 + B · S + V , π = p + A, B = ∇× A

I For a uniform magnetic field in the z direction, expansion of π2 gives:

H = 12p2 + 1

2BLz + Bsz + 1

8B2(x2 + y2) + V , A = 1

2B× r

I L and s are the orbital and spin angular momentum operatorsI a paramagnetic linear dependence on the magnetic fieldI a diamagnetic quadratic dependence on the magnetic field


Backgroundthree field regimes

I The non-relativistic electronic Hamiltonian (a.u.) in a magnetic field B along the z axis:

H = H0 + 12BLz + Bsz + 1

8B2(x2 + y2) ← 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 earth-like conditions: Coulomb interactions dominateI magnetic interactions are treated perturbativelyI earth magnetism 10−10, NMR 10−4; pulsed laboratory field 10−3 a.u.

I Intermediate regime: B ≈ 1 a.u.

I white dwarves: up to about 1 a.u.I the Coulomb and magnetic interactions are equally importantI complicated behaviour resulting from an interplay of linear and quadratic terms

I Landau regime: B � 1 a.u.

I neutron stars: 103–104 a.u.I magnetic interactions dominate (Landau levels)I Coulomb interactions are treated perturbativelyI 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)


BackgroundLondon orbitals and size extensivity

I The non-relativistic electronic Hamiltonian in a magnetic field:

H = H0 + 12BLz + Bsz + 1

8B2[(x − Ox )2 + (y − Oy )2

]I The orbital-angular momentum operator is imaginary and gauge-origin dependent

L = −i (r −O)×∇, AO = 12

B× (r −O)

I 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[

12iB× (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 withmid-bond gauge origin

2 4 6 8 10

-1.0

-0.5

0.5

1.0

B = 0.0

B = 1.0

B = 2.5


BackgroundThe effect of gauge transformations

I Consider H2 along the z axis in a uniform magnetic field in the y directionI the RHF wave function with gauge origin at O = (0, 0, 0) (left) and G = (100, 0, 0) (right)I the (dashed) density remains the same but the (complex) wave function changes

London orbitals: do we need them?

Example: H2 molecule, on the x-axis, in the field B = 110 z.

A = 120 z ⇥ r �! A0 = A + r(�A(G) · r)

= RHF/aug-cc-pVQZ �! 0 = e�iA(G)·r (10)

Gauge-origin moved from 0 to G = 100y.

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Space coordinate, x (along the bond)

Wav

e fun

ction

, ψ

Re(ψ)Im(ψ)|ψ|2

−1.5 −1 −0.5 0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Space coordinate, x (along the bond)

Gaug

e tra

nsfor

med w

ave f

uncti

on, ψ"

Re(ψ")Im(ψ")|ψ"|2

Erik Tellgren Ab initio finite magnetic field calculations using London orbitalsHelgaker et al. (CTCC, University of Oslo) Chemical bonding in strong magnetic fields CMS2012, November 19 2012 7 / 32

Backgroundthe London program and overview

I We have developed the London code for calculations in finite magnetic fields

I complex wave functions and London atomic orbitalsI Hartree–Fock theory (RHF, UHF, GHF), FCI theory, and Kohn–Sham theoryI energy, gradients, excitation energiesI London atomic orbitals require a generalized integral code

Fn(z) =

∫ 1

0

exp(−zt2)t2ndt ← complex argument z

I C++ code written by Erik Tellgren, Alessandro Soncini, and Kai LangeI C20 is a “large” system

I Overview:

I molecular dia- and paramagnetismI helium atom in strong fields: atomic distortion and electron correlationI H2 and He2 in strong magnetic fields: bonding, structure and orientationI molecular structure in strong magnetic fieldsI excitations in strong magnetic fields

I Previous work in this area:

I much work has been done on small atomsI FCI on two-electron molecules H2 and H− (Cederbaum)I Nakatsuji’s free complement methodI no general molecular code with London orbitals

Helgaker et al. (CTCC, University of Oslo) Molecules in strong magnetic fields CMS2012, November 19 2012 8 / 32

Closed-shell paramagnetic moleculesmolecular diamagnetism and paramagnetism

I The Hamiltonian has paramagnetic and diamagnetic parts:

H = H0 + 12BLz + Bsz + 1

8B2(x2 + y2) ← linear and quadratic B terms

I Most closed-shell molecules are diamagneticI their energy increases in an applied magnetic fieldI induced currents oppose the field according to Lenz’s law

I Some closed-shell systems are paramagneticI their energy decreases in a magnetic fieldI relaxation of the wave function lowers the energy

I RHF calculations of the field dependence of the energy for two closed-shell systems:

linear magnetizability are in fact positive and large enough tomake even the average magnetizability positive !paramag-netic". It is therefore interesting to verify via our finite-fieldLondon-orbital approach whether this very small system isindeed characterized by a particularly large nonlinear mag-netic response. The geometry used for the calculations is thatoptimized at the multiconfigurational SCF level in Ref. 51,corresponding to a bond length of rBH=1.2352 Å.

For the parallel components of the magnetizability andhypermagnetizability, we are able to obtain robust estimatesusing the fitting described above, leading to the values !# =!2.51 a.u. and X# =35.25 a.u., respectively, from aug-cc-pVTZ calculations. The same values are obtained both withLondon orbitals and any common-origin calculation that em-ploys a gauge origin on the line passing through the B and Hatoms since in this case, due to the cylindrical symmetry, theLondon orbitals make no difference.

For the perpendicular components, the estimates of thehypermagnetizability we obtain using the above mentionedfitting procedure are not robust, varying with the number ofdata points included in the least-squares fitting and the de-gree of the polynomial. Using 41 uniformly spaced field val-ues in the range !0.1–0.1 a.u. and a fitting polynomial oforder 16, we arrive at reasonably converged values of !!

=7.1 a.u. and X!=!8"103 a.u. for the magnetizability andhypermagnetizability, respectively, at the aug-cc-pVTZ level.In Fig. 1!c", we report a plot of the aug-cc-pVTZ energy asfunction of field !triangles". For comparison, we report inFig. 1!a" the corresponding benzene plot. As the linear re-sponse for BH is paramagnetic, the curvature of the magneticfield energy dependence is clearly reversed. More impor-tantly, whereas it is evident from Fig. 1!a" that the curve forbenzene is to a very good approximation parabolic so thatthe nonlinearities arise from small corrections that are not

a)

!0.1 !0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

b)

!0.1 !0.05 0 0.05 0.10

2

4

6

8

10

12

14x 10

!3

c)

!0.1 !0.05 0 0.05 0.1

!0.02

!0.018

!0.016

!0.014

!0.012

!0.01

!0.008

!0.006

!0.004

!0.002

0

d)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

!0.03

!0.02

!0.01

0

0.01

0.02

FIG. 1. Energy as a function of the magnetic field for different systems. Triangles represent results from finite-field calculations and solid lines are quarticfitting polynomials. !a" Benzene !with the aug-cc-pVDZ basis" illustrates the typical case of diamagnetic quadratic dependence in response to an out-of-planefield. !b" Cyclobutadiene !aug-cc-pVDZ" deviates from the typical case by exhibiting a nonquadratic dependence on an out-of-plane field. !c" Boronmonohydride !aug-cc-pVTZ" is an interesting case of positive magnetizability for a perpendicular field, exhibiting nonquadratic behavior. !d" Boronmono-hydride !aug-cc-pVTZ" in a larger range of perpendicular fields, exhibiting a clearly nonperturbative behavior.

154114-8 Tellgren, Soncini, and Helgaker J. Chem. Phys. 129, 154114 !2008"

Downloaded 28 Oct 2008 to 129.240.80.34. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

linear magnetizability are in fact positive and large enough tomake even the average magnetizability positive !paramag-netic". It is therefore interesting to verify via our finite-fieldLondon-orbital approach whether this very small system isindeed characterized by a particularly large nonlinear mag-netic response. The geometry used for the calculations is thatoptimized at the multiconfigurational SCF level in Ref. 51,corresponding to a bond length of rBH=1.2352 Å.

For the parallel components of the magnetizability andhypermagnetizability, we are able to obtain robust estimatesusing the fitting described above, leading to the values !# =!2.51 a.u. and X# =35.25 a.u., respectively, from aug-cc-pVTZ calculations. The same values are obtained both withLondon orbitals and any common-origin calculation that em-ploys a gauge origin on the line passing through the B and Hatoms since in this case, due to the cylindrical symmetry, theLondon orbitals make no difference.

For the perpendicular components, the estimates of thehypermagnetizability we obtain using the above mentionedfitting procedure are not robust, varying with the number ofdata points included in the least-squares fitting and the de-gree of the polynomial. Using 41 uniformly spaced field val-ues in the range !0.1–0.1 a.u. and a fitting polynomial oforder 16, we arrive at reasonably converged values of !!

=7.1 a.u. and X!=!8"103 a.u. for the magnetizability andhypermagnetizability, respectively, at the aug-cc-pVTZ level.In Fig. 1!c", we report a plot of the aug-cc-pVTZ energy asfunction of field !triangles". For comparison, we report inFig. 1!a" the corresponding benzene plot. As the linear re-sponse for BH is paramagnetic, the curvature of the magneticfield energy dependence is clearly reversed. More impor-tantly, whereas it is evident from Fig. 1!a" that the curve forbenzene is to a very good approximation parabolic so thatthe nonlinearities arise from small corrections that are not

a)

!0.1 !0.05 0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

b)

!0.1 !0.05 0 0.05 0.10

2

4

6

8

10

12

14x 10

!3

c)

!0.1 !0.05 0 0.05 0.1

!0.02

!0.018

!0.016

!0.014

!0.012

!0.01

!0.008

!0.006

!0.004

!0.002

0

d)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

!0.03

!0.02

!0.01

0

0.01

0.02

FIG. 1. Energy as a function of the magnetic field for different systems. Triangles represent results from finite-field calculations and solid lines are quarticfitting polynomials. !a" Benzene !with the aug-cc-pVDZ basis" illustrates the typical case of diamagnetic quadratic dependence in response to an out-of-planefield. !b" Cyclobutadiene !aug-cc-pVDZ" deviates from the typical case by exhibiting a nonquadratic dependence on an out-of-plane field. !c" Boronmonohydride !aug-cc-pVTZ" is an interesting case of positive magnetizability for a perpendicular field, exhibiting nonquadratic behavior. !d" Boronmono-hydride !aug-cc-pVTZ" in a larger range of perpendicular fields, exhibiting a clearly nonperturbative behavior.

154114-8 Tellgren, Soncini, and Helgaker J. Chem. Phys. 129, 154114 !2008"

Downloaded 28 Oct 2008 to 129.240.80.34. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

I left: benzene: diamagnetic dependence on an out-of-plane field, χ < 0I right: BH: paramagnetic dependence on a perpendicular field, χ > 0


Closed-shell paramagnetic moleculesdiamagnetic transition at stabilizing field strength Bc

I However, all systems become diamagnetic in sufficiently strong fields:

a)

b)

c)

0.05 0.1 0.15 0.2 0.25 0.3B !au"

!0.04

!0.03

!0.02

!0.01

W!W0 !au" BH

aug!DZ, Bc " 0.23DZ, Bc " 0.22STO!3G, Bc " 0.24

0.1 0.2 0.3 0.4 0.5 0.6B !au"

!0.12

!0.1

!0.08

!0.06

!0.04

!0.02

W!W0 !au" CH#

aug!DZ, Bc " 0.45DZ, Bc " 0.44STO3!G, Bc " 0.43

0.1 0.2 0.3 0.4 0.5 0.6 0.7B !au"

!0.4

!0.3

!0.2

!0.1

W!W0 !au" MnO4!

Wachters, Bc " 0.50STO!3G, Bc " 0.45

0.02 0.04 0.06 0.08 0.1 0.12B !au"

0.005

0.01

0.015

0.02

0.025

0.03

W!W0 !au" C4H4: total energy

cc!pVDZ6!31GSTO!3G

0.005 0.01 0.015 0.02 0.025 0.03B !au"

!0.001

!0.0005

0.0005

0.001

0.0015

0.002


cc!pVDZ, Bc " 0.0166!31G, Bc " 0.018STO!3G, Bc " 0.018

0.01 0.02 0.03 0.04 0.05 0.06

!0.002

0.002

0.004



0.02 0.04 0.06 0.08 0.1 0.12

!0.005

!0.004

!0.003

!0.002

!0.001

W!W0 !au" C4H4: #!energy


a) b)

d)c)

I The transition occurs at a characteristic stabilizing critical field strength Bc

I Bc ≈ 0.22 perpendicular to principal axis for BHI Bc ≈ 0.032 along the principal axis for antiaromatic octatetraene C8H8I Bc ≈ 0.016 along the principal axis for antiaromatic [12]-annulene C12H12

I Bc is inversely proportional to the area of the molecule normal to the fieldI we estimate that Bc should be observable for C72H72

I We may in principle separate such molecules by applying a field gradient


Closed-shell paramagnetic moleculesparamagnetism and double minimum explained

I Ground and (singlet) excited states of BH along the z axis

|zz〉 = |1s2B2σ2

BH2p2z |, |zx〉 = |1s2

B2σ2BH2pz2px |, |zy〉 = |1s2

B2σ2BH2pz2py |

I All expectation values increase quadratically in a perpendicular field in the y direction:⟨n∣∣H0 + 1

2BLy + 1

8B2(x2 + z2)

∣∣ n⟩ = En + 18

⟨n∣∣x2 + z2

∣∣ n⟩B2 = En − 12χnB

2

I The |zz〉 ground state is coupled to the low-lying |zx〉 excited state by this field:⟨zz∣∣H0 + 1

2BLy + 1

8B2(x2 + z2)

∣∣ xz⟩ = 12〈zz |Ly | xz〉B 6= 0

-0.1 0.1

-0.03

0.03

0.06

0.09

-0.1 0.1

-0.03

0.03

0.06

0.09

-0.1 0.1

-0.03

0.03

0.06

0.09

-0.1 0.1

-0.03

0.03

0.06

0.09

I A paramagnetic ground-state with a double minimum is generated by strong coupling


Closed-shell paramagnetic moleculesC20: more structure

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ææ

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ææ

æææææææææææ

æææ

ææ

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æ

æ

æ

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æ

æ

æ

æ

æ

æ

æ

æ

æ

-0.04 -0.02 0.02 0.04

-756.710

-756.705

-756.700

-756.695

-756.690

-756.685

-756.680


Closed-shell paramagnetic moleculesinduced electron rotation

I The magnetic field induces a rotation of the electrons about the field direction:I the amount of rotation is the expectation value of the kinetic angular-momentum operator

〈0|Λ|0〉 = 2E ′(B), Λ = r × π, π = p + A

I Paramagnetic closed-shell molecules (here BH):

Example 2: Non-perturbative phenomena

BH properties (aug-cc-pVDZ) as function of perpendicular field:

0 0.2 0.4

!25.16

!25.15

!25.14

!25.13

!25.12

!25.11

E(B

x)

Energy

0 0.2 0.4

!0.5

0

0.5

1

Lx(B

x)

Angular momentum

0 0.2 0.4

!0.2

0

0.2

0.4

Lx /

|r !

Cnuc|

Nuclear shielding integral

BoronHydrogen

0 0.2 0.4

!0.3

!0.2

!0.1

0

0.1

!(B

x)

Orbital energies

LUMOHOMO

0 0.2 0.4

0.38

0.4

0.42

0.44

0.46

0.48

0.5

! gap(B

x)

HOMO!LUMO gap

0 0.2 0.40.1

0.15

0.2

0.25

0.3

0.35

0.4

"(B

x)

Singlet excitation energies

Example 2: Non-perturbative phenomena

BH properties (aug-cc-pVDZ) as function of perpendicular field:

0 0.2 0.4

!25.16

!25.15

!25.14

!25.13

!25.12

!25.11

E(B

x)Energy

0 0.2 0.4

!0.5

0

0.5

1

Lx(B

x)

Angular momentum

0 0.2 0.4

!0.2

0

0.2

0.4

Lx /

|r !

Cnuc|

Nuclear shielding integral

BoronHydrogen

0 0.2 0.4

!0.3

!0.2

!0.1

0

0.1

!(B

x)

Orbital energies

LUMOHOMO

0 0.2 0.4

0.38

0.4

0.42

0.44

0.46

0.48

0.5

! gap(B

x)

HOMO!LUMO gap

0 0.2 0.40.1

0.15

0.2

0.25

0.3

0.35

0.4

"(B

x)

Singlet excitation energies

I there is no rotation at the field-free energy maximum: B = 0I the onset of paramagnetic rotation (against the field) reduces the energy for B > 0I the strongest paramagnetic rotation occurs at the energy inflexion pointI the rotation comes to a halt at the stabilizing field strength: B = BcI the onset of diamagnetic rotation (with the field) increases the energy for B > Bc

I Diamagnetic closed-shell molecules:I diamagnetic rotation always increases the energy according to Lenz’s law


The helium atomtotal energy of the 1S(1s2), 3S(1s2s), 3P(1s2p) and 1P(1s2p) states

I Electronic states evolve in a complicated manner in a magnetic fieldI the behaviour depends on orbital and spin angular momentaI eventually, all energies increase diamagnetically

0.0 0.2 0.4 0.6 0.8 1.0B [a.u]

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

<X

YZ

>


The helium atomorbital energies

I The orbital energies behave in an equally complicated mannerI the initial behaviour is determined by the angular momentumI beyond B ≈ 1, all energies increase with increasing fieldI HOMO–LUMO gap increases, suggesting a decreasing importance of electron correlation

0 1 2 3 4 5 6 7 8 9 102

0

2

4

6

8

10

Field, B [au]

Orb

ital e

nerg

y,

[Har

tree]

Helium atom, RHF/aug cc pVTZ


The helium atomnatural occupation numbers and electron correlation

I The FCI occupation numbers approach 2 and 0 strong fieldsI diminishing importance of dynamical correlation in magnetic fieldsI the two electrons rotate in the same direction about the field direction


The helium atomatomic size and atomic distortion

I Atoms become squeezed and distorted in magnetic fieldsI Helium 1s2 1S (left) is prolate in all fieldsI Helium 1s2p 3P (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?


The H2 moleculepotential-energy curves of the 1Σ+

g (1σ2g) and 3Σ+

u (1σg1σ∗u ) 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 2 3 4

-1.0

-0.5

0.5

1.0

B = 0.

1 2 3 4

-1.0

-0.5

0.5

1.0

B = 0.75

1 2 3 4

-1.0

-0.5

0.5

1.0

B = 1.5

1 2 3 4

-1.0

-0.5

0.5

1.0

B = 2.25

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 boundI parallel orientation studied by Schmelcher et al., PRA 61, 043411 (2000); 64, 023410 (2001)I Hartree–Fock studies by Zaucer and and Azman (1977) and by Kubo (2007)


The H2 moleculelowest 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.

0°

45°

90°

135°

180°

225°

270°

315°

1

2

3

4

5

0°

45°

90°

135°

180°

225°

270°

315°

1

2

3

4

5

I Bond distance Re (pm), orientation Θe (◦), diss. energy De, and rot. barrier ∆E0 (kJ/mol)

singlet tripletB Re θe De ∆E0 Re θe De ∆E0

0.0 74 – 459 0 ∞ – 0 01.0 66 0 594 83 136 90 12 12


The H2 moleculea 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) , 1sA = Ns e−i 1

2B×RA·r e−ar2

A

I In the helium limit, the bonding and antibonding MOs transform into 1s and 2p AOs:

limR→0

1σg = 1s, all orientations

limR→0

1σu =

{2p0, parallel orientation

2px − i B4a

2py ≈ 2p−1, perpendicular orientation

I The magnetic field modifies the MO energy level diagramI perpendicular orientation: antibonding MO is stabilized, while bonding MO is destabilizedI parallel orientation: both MOs are unaffected relative to the AOs

0.5 1.0 1.5 2.0

-0.15

-0.10

-0.05

0.05

1Σu*H´L

1Σu*HÞL

1ΣgH´L

1ΣgHÞL

1sA 1sB

1Σu*H´L

1Σu*HÞL

1ΣgH´L

1ΣgHÞL

I Molecules of zero bond order may therefore be stabilized by the magnetic field


The H2 moleculea new bonding mechanism: perpendicular paramagnetic bonding

I The triplet H2 is bound by perpendicular stabilization of antibonding MO 1σ∗uI note: Hartree–Fock theory gives a qualitatively correct description

100 200 300Rêpm

-5550

-5525

-5500

EêkJmol-1

FCI

HF

H2

100 200 300Rêpm

-10900

-10700

-10500

EêkJmol-1

FCI

HF

He2

I however, there are large contributions from dynamical correlation

method B Re De

UHF/un-aug-cc-pVTZ 2.25 93.9 pm 28.8 kJ/molFCI/un-aug-cc-pVTZ 2.25 92.5 pm 38.4 kJ/mol

I UHF theory underestimates the dissociation energy but overestimates the bond lengthI basis-set superposition error of about 8 kJ/mol


The H2 moleculeZeeman splitting of the lowest triplet state

I The spin Zeeman interaction contributes BMs to the energy, splitting the tripletI lowest singlet (blue) and triplet (red) energy of H2:

1 2 3 4

-2

-1

1

2

B = 0.00

B = 2.25H1,-1L

B = 2.25H1,0L

H0,0L

H1,ML

H0,0L

B = 2.25H1,+1L

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. . .


The H2 moleculeevolution of lowest three triplet states

I We often observe a complicated evolution of electronic statesI a (weakly) bound 3Σ+

u (1σg1σ∗u ) ground state in intermediate fieldsI a covalently bound 3Πu(1σg2πu) ground state in strong fields

0 1 2 3 4 5 6 7 8R (bohr)

2.0

1.5

1.0

0.5E (

Ha)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50


The H2 moleculeelectron rotation and correlation

I The field induces a rotation of the electrons 〈0|Λz |0〉 about the molecular axis

I increased rotation increases kinetic energy, raising the energyI concerted rotation reduces the chance of near encountersI natural occupation numbers indicate reduced importance of dynamical correlation


The helium dimerthe 1Σ+

g (1σ2g1σ∗u

2) singlet state

I The field-free He2 is bound by dispersion in the ground stateI our FCI/un-aug-cc-pVTZ calculations give De = 0.08 kJ/mol at Re = 303 pm

I In a magnetic field, He2 shrinks and becomes more strongly boundI perpendicular paramagnetic bonding (dashed lines) as for H2I for B = 2.5, De = 31 kJ/mol at Re = 94 pm and Θe = 90◦

1 2 3 4 5 6 7R

-5.5

-5.0

-4.5

-4.0

E

B = 0.0

B = 0.5

B = 1.0

B = 1.5

B = 2.0

B = 2.5


The helium dimerthe 3Σ+

u (1σ2g1σ∗u 2σg) triplet state

I the covalently bound triplet state becomes further stabilized in a magnetic fieldI De = 178 kJ/mol at Re = 104 pm at B = 0I De = 655 kJ/mol at Re = 80 pm at B = 2.25 (parallel orientation)I De = 379 kJ/mol at Re = 72 pm at B = 2.25 (perpendicular orientation)

0 1 2 3 4 5R

-5.8

-5.6

-5.4

-5.2

-5.0

-4.8

E

B = 2.0

B = 1.0

B = 0.5

B = 0.0

I The molecule begins a transition to diamagnetism at B ≈ 2I eventually, all molecules become diamagnetic

T = 12

(σ · π)2 = 12

(σ · (p + A))2 = 12

(σ · (p + 1

2B× r)

)2


The helium dimerthe 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

1 2 3 4 5 6

-7

-6

-5

-4

B = 2.5

B = 2.0

B = 1.5

B = 1.0

B = 0.5

B = 0.0

I In strong fields, anisotropic Gaussians are needed for a compact description for B � 1

I without such basis sets, calculations become speculative


Molecular structureHartree–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 theoryI all structures are planar and consist of equilateral trianglesI suggestive of hexagonal 2D crystal lattice (3He crystallizes into an hcp structure at about 10 MPa)I He3 and He6 bound by 3.7 and 6.8 mEh per atomI ‘vibrational frequencies’ in the range 200–2000 cm−1 (for the 4He isotope)

2.086 2.086 2.084

2.060 2.060

2.084

1.951 1.951

2.293 2.293

2.048

2.042

2.048

2.042 2.044

2.061 2.061

2.053

2.044

2.049

2.053

2.049

defaults used first point

M O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E NM O L D E N


Molecular structureFCI calculations on the H+

3 ion

I We are investigating H+3 in a magnetic field

I Warke and Dutta, PRA 16, 1747 (1977)I equilateral triangle for B < 1; linear chain for B > 1I ground state singlet for B < 0.5; triplet for B > 0.5I the triplet state does not become bound until fields B > 1

I Potential energy curves for lowest singlet (left) and triplet (right) states

I electronic energy as function of bond distances for equilateral triangles

0.5 1.0 1.5 2.0 2.5R (bohr)

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

E (

Ha)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

0.5 1.0 1.5 2.0 2.5R (bohr)

3

2

1

0

1

2

3

E (

Ha)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50


Molecular structureHartree–Fock calculations on ammonia and benzene

I Ammonia for 0 ≤ B ≤ 0.06 a.u. at the RHF/cc-pVTZ level of theory

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21.878

1.88

1.882

1.884

1.886

1.888

1.89

B [au]

dN

H [

bo

hr]

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2107.7

107.8

107.9

108

108.1

108.2

108.3

108.4

108.5

108.6

108.7

B [au]

∠ H

−N

−H

[d

eg

ree

s]

0 0.01 0.02 0.03 0.04 0.05 0.06

8.35

8.4

8.45

8.5

8.55

8.6

B [au]

Ein

v [

mE

h]

B ⊥ mol.axis

B || mol.axis

I bond length (left), bond angle (middle) and inversion barrier (right)I ammonia shrinks and becomes more planar (from shrinking lone pair?)I 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 directionI agrees with perturbational estimates by Caputo and Lazzeretti, IJQC 111, 772 (2011)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16138.3

138.35

138.4

138.45

138.5

138.55

138.6

138.65

138.7

138.75

B [au]

d(C

−C

) [p

m]

C6H

6, HF/6−31G*

C−C other (c)C−C || B (d)


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.)

I Oscillator strength for transition to green CH3 stateI length (red) and velocity (blue) gauges

0.00 0.15 0.30 0.45B⊥ [a.u.]

−39.65

−39.60

−39.55

−39.50

−39.45

−39.40

−39.35

−39.30

−39.25

En

[Ha]

0.00 0.15 0.30 0.45B⊥ [a.u.]

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

fE1


ConclusionsSummary 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 atoms and molecules shrinkI molecules are stabilized by magnetic fieldsI 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 all paramagnetic molecules attain a global minimum at a characteristic field BcI Bc decreases with system size and should be observable for C72H72

I An important goal is to study the universal density functional in magnetic fields

F [ρ, j] = supu,A

(E [u,A]−

∫ρ(r)u(r) dr −

∫j(r) · A(r)dr

)I Support: The Norwegian Research Council and the European Research Council

Helgaker et al. (CTCC, University of Oslo) Conclusions CMS2012, November 19 2012 32 / 32

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