molecules in an external magnetic field - trygve helgaker

Molecular MagnetismMolecules in an External Magnetic Field

Trygve Helgaker

Centre for Theoretical and Computational Chemistry (CTCC),Department of Chemistry, University of Oslo, Norway

Laboratoire de Chimie Theorique,Universite Pierre et Marie Curie, Paris, France

November 27, 2012

Trygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 2012 1 / 34

Molecular Magnetism

I Part 1: The electronic HamiltonianI Hamiltonian mechanics and quantizationI electromagnetic fieldsI scalar and vector potentialsI electron spin

I Part 2: Molecules in an external magnetic fieldI Hamiltonian in an external magnetic fieldI gauge transformations and London orbitalsI magnetizabilitiesI diamagnetism and paramagnetismI induced currentsI molecules and molecular bonding in strong fields

I Part 3: NMR parametersI Zeeman and hyperfine operatorsI nuclear shielding constantsI indirect nuclear spin–spin coupling constants


Hamiltonian in a uniform magnetic field

I The nonrelativistic electronic Hamiltonian (implied summation over electrons):

H = H0 + A (r) · p + B (r) · s + 12A (r)2

I The vector potential of the uniform (static) fields B is given by:

B = ∇× A = const ⇒ AO(r) = 12

B× (r −O) = 12

B× rO

I note: the gauge origin O is arbitrary!

I The orbital paramagnetic interaction:

AO(r) · p = 12

B× (r −O) · p = 12

B · (r −O)× p = 12

B · LO

where we have introduced the angular momentum relative to the gauge origin:

LO = rO × p

I The diamagnetic interaction:

12A2 (B) = 1

8(B× rO) · (B× rO) = 1

8

[B2r2

O − (B · rO)2]

I The electronic Hamiltonian in a uniform magnetic field depends on the gauge origin:

H = H0 +1

2B · LO + B · s +

1

8

[B2r2

O − (B · rO)2]

I a change of the origin is a gauge transformation


Gauge transformation of the Schrodinger equation

I What is the effect of a gauge transformation on the wave function?

I Consider a general gauge transformation for the electron (atomic units):

A′ = A + ∇f , φ′ = φ− ∂f

∂t

I It can be shown this represents a unitary transformation of H − i∂/∂t:(H′ − i

∂

∂t

)= exp (−if )

(H − i

∂

∂t

)exp (if )

I In order that the Schrodinger equation is still satisfied(H′ − i

∂

∂t

)Ψ′ ⇔

(H − i

∂

∂t

)Ψ,

the new wave function must undergo a compensating unitary transformation:

Ψ′ = exp (−if ) Ψ

I All observable properties such as the electron density are then unaffected:

ρ′ = (Ψ′)∗Ψ′ = [Ψ exp(−if )]∗[exp(−if )Ψ] = Ψ∗Ψ = ρ


Gauge-origin transformations

I Different choices of gauge origin in the external vector potential

AO (r) = 12

B× (r −O)

are related by gauge transformations:

AG (r) = AO (r)− AO (G) = AO (r) + ∇f , f (r) = −AO (G) · r

I The exact wave function transforms accordingly and gives gauge-invariant results:

ΨexactG = exp [−if (r)] Ψexact

O = exp [iAO (G) · r] ΨexactO rapid oscillations

I Illustration: H2 on the z axis in a magnetic field B = 0.2 a.u. in the y direction

I wave function with gauge origin at O = (0, 0, 0) (left) and G = (100, 0, 0) (right)

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 = eiA(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 orbitalsTrygve Helgaker (CTCC, University of Oslo) Molecules in an external magnetic field LCT, UPMC, November 27 2012 5 / 34

London atomic orbitals

I The exact wave function transforms in the following manner:

ΨexactG = exp

[i 1

2B× (G−O) · r

]Ψexact

O

I this behaviour cannot easily be modelled by standard atomic orbitals

I Let us build this behaviour directly into the atomic orbitals:

ωlm(rK ,B,G) = exp[i 1

2B× (G− K) · r

]χlm(rK)

I χlm(rK ) is a normal atomic orbital centered at K and quantum numbers lmI ωlm(rK ,B,G) is a field-dependent orbital at K with field B and gauge origin G

I Each AO now responds in a physically sound manner to an applied magnetic field

I indeed, all AOs are now correct to first order in B, for any gauge origin GI the calculations become rigorously gauge-origin independentI uniform (good) quality follows, independent of molecule size

I These are the London orbitals after Fritz London (1937)

I also known as GIAOs (gauge-origin independent AOs or gauge-origin including AOs)

I Questions:

I are London orbitals needed in atoms?I why not attach the phase factor to the total wave function instead?


London orbitals

I Let us consider the FCI dissociation of H2 in a magnetic field

I full lines: London atomic orbitalsI dashed lines: AOs with gauge origin between atomsI dotted lines: AOs with gauge origin on one of the atoms

I Without London orbitals, the FCI method is not size extensive in magnetic fields


Expansion of the molecular energy in a magnetic field

I Assume zero nuclear magnetic moments and expand the molecular electronic energy in theexternal magnetic induction B:

E (B) = E0 + BTE(10) +1

2BTE(20)B + · · ·

I The molecular magnetic moment at B is now given by

Mmol (B)def= −dE (B)

dB= −E(10) − E(20)B + · · · = Mperm + ξB + · · · ,

where we have introduced the permanent magnetic moment and the magnetizability:

Mperm = −E(10) = − dE

dB

∣∣∣∣B=0

← permanent magnetic moment

I describes the first-order change in the energy but vanishes for closed-shell systems

ξ = −E(20) = − d2E

dB2

∣∣∣∣B=0

← molecular magnetizability

I describes the second-order energy and the first-order induced magnetic moment

I First-order energies for imaginary and triplet perturbations vanish for closed-shell sysetems:⟨c.c.

∣∣∣ Ωimaginary

∣∣∣ c.c.⟩≡⟨

c.c.∣∣∣ Ωtriplet

∣∣∣ c.c.⟩≡ 0


The magnetizability

I The electronic Hamiltonian in a uniform magnetic field:

H = H0 +1

2B · LO + B · s +

1

8

[B2r2

O − (B · rO)2]

I The molecular magnetizability of a closed-shell system:

ξ = −d2E

dB2= −

⟨0

∣∣∣∣∂2H

∂B2

∣∣∣∣ 0

⟩+ 2

∑n

⟨0∣∣∣ ∂H∂B

∣∣∣ n⟩⟨n ∣∣∣ ∂H∂B

∣∣∣ 0⟩

En − E0

=1

4

⟨0∣∣∣rOrTO −

(rTO rO

)I3

∣∣∣ 0⟩

︸︷︷︸diamagnetic term

+1

2

∑n

〈0 |LO| n〉⟨n∣∣LT

O

∣∣ 0⟩

En − E0︸︷︷︸paramagnetic term

I The isotropic part of the diamagnetic term is given by:

ξdia =1

3Tr ξdia = −1

6

⟨0∣∣x2

O + y2O + z2

O

∣∣ 0⟩

= −1

6

⟨0∣∣r2

O

∣∣ 0⟩← system surface

I Only the orbital Zeeman interaction contributions to the paramagnetic term:

S |0〉 ≡ 0 ← singlet state

I for 1S systems (closed-shell atoms), the paramagnetic term vanishes altogether:

12

LO

∣∣1S⟩ ≡ 0 ← gauge origin at nucleus

I In most (but not all) systems the diamagnetic term dominates


Hartree–Fock magnetizabilities and hypermagnetizabilities

I London orbitals are correct to first-order in the external magnetic field

I for this reason, basis-set convergence is usually improved,I RHF magnetizabilities and hypermagnetizabilities of benzene:

basis set χxx χyy χzz Xxxxx Xyyyy Xzzzz

London STO-3G −8.1 −8.1 −23.0 −211 −211 −526-31G −8.2 −8.2 −23.1 −219 −219 −64cc-pVDZ −8.1 −8.1 −22.3 −236 −236 −120aug-cc-pVDZ −8.0 −8.0 −22.4 −316 −316 −153

origin CM STO-3G −35.8 −35.8 −48.1 45 45 276-31G −31.6 −31.6 −39.4 29 29 −152cc-pVDZ −15.4 −15.4 −26.9 9 9 −241aug-cc-pVDZ −9.9 −9.9 −25.2 −413 −413 −159

origin H STO-3G −35.8 −176.3 −116.7 45 1477 −53406-31G −31.6 −144.8 −88.0 29 1588 −5866cc-pVDZ −15.4 −48.0 −41.6 9 2935 −3355aug-cc-pVDZ −9.9 −20.9 −33.9 −413 −3321 −1097

I The RHF model overestimates the magnitude of magnetizabilities by 5%–10%:

10−30 JT−2 HF exp. diff.H2O −232 −218 −6.4%NH3 −289 −271 −6.6%CH4 −315 −289 −9.0%CO2 −374 −349 −7.2%PH3 −441 −435 −1.4%


Normal distributions of errors for magnetizabilities

I Normal distributions of magnetizability errors for 27 molecules in the aug-cc-pCVQZ basis relative toCCSD(T)/aug-cc-pCV[TQ]Z values (Lutnæs et al., JCP 131, 144104 (2009))


Mean absolute errors for magnetizabilities

I Mean relative errors (MREs, %) in magnetizabilities of 27 molecules realtive to the CCSD(T)/aug-cc-pCV[TQ]Z values.The DFT results are grouped by functional type. The heights of the bars correspond to the largest MRE in each category.(Lutnæs et al., JCP 131, 144104 (2009))


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

æ

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æ

ææ

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ææ

æææææ

ææææææææææææææ

ææ

æææææææææææ

æææ

ææ

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æ

æ

æ

æ

æ

æ

æ

æ

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æ

æ

æ

æ

æ

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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


Molecules in strong magnetic fieldsthree field regimes

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

H = H0 + 12 BLz + Bsz + 1

8 B2(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)


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 fields

I 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 states

I a (weakly) bound 3Σ+u (1σg1σ∗u ) ground state in intermediate fields

I 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 axisI 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 stateI 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 1I 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 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)


molecules in an external magnetic field - trygve helgaker

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