Translational Effects on Electronic and Nuclear Ring CurrentsIngo Barth*

Max-Born-Institut, Max-Born-Strasse 2A, 12489 Berlin, Germany

ABSTRACT: In previous works, it was predicted that electronic andnuclear ring currents in degenerate excited states of atomic and molecularsystems persist after the end of driven circularly polarized atto- orfemtosecond laser pulses on relatively long time scales, often on pico- ornanosecond time scales, before spontaneous emission occurs. Although thisconclusion is true in the center of mass frame, it is not true in the laboratoryframe, where the translation has to be considered. In this theoretical work,the analytic formulas for the ring current densities, electric ring currents,mean ring current radii, and induced magnetic fields at the ring center,depending on the translational wavepacket widths, are derived. It showsthat the ring currents and the corresponding induced magnetic fields inthe laboratory frame persist on shorter timecales due to spreading oftranslational wavepackets. The electronic ring currents in 2p± orbitals of thehydrogen-like systems decay on the femtosecond time scale, but the corresponding nuclear ring currents with giant inducedmagnetic fields (for example up to 0.54 MT for 7Li2+) and very small mean ring current radii on the femtometer scale decay onthe very short, zeptosecond time scale, according to the Heisenberg uncertainty principle. The theory is also applied to ringcurrents in many-electron atoms and ions as well as to nuclear ring currents in pseudorotating molecules. For example, in the firsttriply degenerate pseudorotational states |v1l±1⟩ of the tetrahedral molecule OsH4, the ring currents of the heavy central nucleusOs decay on the attosecond time scale.

1. INTRODUCTION

Nonvanishing electronic and nuclear current densities occur notonly in nonstationary states, corresponding to electron cir-culations,1−5 molecular vibrations,6 motors,7 pseudorotations,8,9

rotations, torsions,10 isomerizations, proton transfers,11,12 peri-cyclic rearrangements,13,14 or chemical reactions15−17 but also instationary states.18,19 The existence of stationary current densitiesis guaranteed in electronic degenerate states of atoms, ions,20,21

and linear22,23 or ring-shaped molecules24 or in nucleardegenerate states of pseudorotating molecules.25,26 In thesedegenerate states, the electronic or nuclear wave functions arenot purely real or imaginary but complex, e.g., presentinglinear combinations of two real wave functions, |Ψ±⟩ = |Ψx⟩ ±i|Ψy⟩. If the ground state is nondegenerate, left or right cir-cularly polarized ultrashort (atto- or femtosecond) laserpulses with suitable laser parameters can induce populationtransfer from the ground state to the excited degenerate state|Ψ−⟩ or |Ψ+⟩, carrying negative or positive ring currents,respectively.18,19

Of course, the excited degenerate states have finite lifetimesbut they persist on relatively long (pico- or nanosecond) timescales, before spontaneous emission occurs. Therefore, it waspredicted that the properties of the ring currents, includingelectric ring currents, mean angular momenta, mean ring currentradii, and induced magnetic fields, do not change significantly onthese timescales, even after the end of the circularly polarizedlaser pulses. This prediction is true only in the center of massframe. In the laboratory frame, the translational wave functioncorresponding to the center of mass motion has to be included

in the total wave function. Though the mean angular momentaare conserved, the remaining properties of the ring currents, i.e.,electric ring currents, mean ring current radii, and inducedmagnetic fields in the laboratory frame, depend on translationalwavepacket width. I will show that due to center of mass motionthe electronic and nuclear ring currents decay on shorter(zepto-, atto-, and femtosecond) time scales, depending on thesize of the mean ring current radii.In section 2, the analytic formulas for the ring current densities

(azimuthal components of the current densities), electric ringcurrents, mean angular momenta, mean ring current radii, andinduced magnetic fields are derived. These formulas are definedin section 2.1 for the electronic and nuclear ring currents inorbitals of hydrogen-like atoms. Section 2.2 considers the ringcurrents in the center of mass frame. Then, I use sphericallyGaussian-distributed translational wavepacket in section 2.3 toobtain analytical formulas for 2p± orbitals in the laboratoryframe, depending on the translational wavepacket width. Theseformulas can be then generalized for application to ring currentsin many-electron atoms and ions (section 2.4) as well as tonuclear ring currents in pseudorotating molecules (section 2.5).Section 3 discusses the results and section 4 summarizes thiswork.

Special Issue: Jorn Manz Festschrift

Received: May 31, 2012Revised: August 8, 2012

Article

pubs.acs.org/JPCA

© XXXX American Chemical Society A dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXX

2. THEORY2.1. Basic Equations. In this work, I extend the theory of

electronic ring currents in atomic orbitals20 to include nuclearring currents and translational effects on ring currents.However, the relativistic effects of the electronic and nuclearspins on ring currents are not considered. In degenerate atomicorbitals carrying nonzero angular momenta, not only theelectrons but also the nucleus circulate about the center ofmass. For the hydrogen atom or one-electron ions with atomicnumber , the total electronic and nuclear wave function iswritten as

ψ ψΨ =t tR r R r( , , ) ( , ) ( )nlmtr (1)

where ψtr(R,t) is the time-dependent wave function describingthe center of mass motion and ψnlm(r) are the usual atomicorbitals with quantum numbers n, l, m. In spherical coordiantes,the latter are given by20,27

ψ θ ϕ

θ ϕ

= − − !+ !

×

μ

μ μ

−

− −+

μ⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

rna

n ln n l

rna

Lr

naY

( , , )2 ( 1)

2 ( )e

2 2( , )

nlmr na

l

n ll

lm

3/

12 1

(2)

where aμ = 4πε0ℏ2/(μe2) is the Bohr radius for the reduced

mass μ = memn/M, me is the electron mass, mn is the nuclearmass, and M = me + mn is the total mass. Because ring currentsin atomic orbitals are toroidal,18,20 I also use cylindricalcoordinates (ρ, z, ϕ) with relations ρ = r sin θ and z = r cos θ.The position vectors of the center of mass R and of the relativemotion r are connected to the ones of the electron re and of thenucleus rn by

= +m

Mm

MR

r re e n n(3)

= −r r re n (4)

The electronic (i = e) and nuclear (i = n) probability densitiesare defined as

∭ρ = |Ψ | ≠t tr R r r( , ) ( , , ) dnlmi

i2

j i (5)

With the electronic (i = e) and nuclear (i = n) current densities

∭= ℏ Ψ ∇ Ψ*

− Ψ* ∇ Ψ ≠

tm

t t

t t

j r R r R r

R r R r r

( , )i

2( ( , , ) ( , , )

( , , ) ( , , )) d

nlm r

r

ii

i

j i

i

i (6)

one can calculate various physical properties of the ringcurrents, including the mean angular momenta18,26

∭⟨ ⟩ = ×t m tL r j r r( ) ( , ) dnlm nlmi

i ii

i i (7)

and the electric ring currents18,20

∬= ·I t e t dj r S( ) ( , )nlm nlmi

ii

i i (8)

with = −1e and =n . Here, I consider only ringcurrents with its axis of symmetry as z-axis. Hence, Si is the halfplane perpendicular to the x/y-plane at fixed arbitrary azimuthalangle ϕi with the domains ρi ∈ [0,∞) and zi ∈ (−∞, ∞) orwith the domains ri ∈ [0, ∞) and θi ∈ [0, π], thus dSi = dρi dzi

eϕ(ϕi) = ri dri dθi eϕ(ϕi). Note that the unit vector eϕ(ϕi)depends on the azimuthal angle ϕi. The corresponding meanperiods are18

=T te

I t( )

( )nlmnlm

i ii

(9)

The kth moment of mean ring current radii (k ≠ 0) is definedas in refs 18 and 20,

∬ ρ=⎛⎝⎜

⎞⎠⎟R t

eI t

tj r S( )( )

( , ) dk nlmnlm

knlm

k

,i i

i ii

i i

1/

(10)

The usual mean ring current radius corresponds to k = 1, butk = −1 is also considered, because it corresponds to the“inverse” mean ring current radius according to the Biot−Savartlaw ∝ I/R.20 The induced magnetic fields at the ring centerri = r0(t) are

∭μπ

=

=′ × − ′| − ′|

′

t t

e t t

t

B r r

j r r r

r rr

( ( ), )

4

( , ) ( ( ) )

( )d

nlm

nlm

ii 0

0 ii

0

03

(11)

where the joint center r0(t) of the electronic and nuclear ringcurrents is determined by the mean position of the center ofmass, i.e.

∭ ψ= | |t tr R R R( ) ( , ) d0 tr2

(12)

2.2. Center of Mass Frame. In this section, I consider thering currents in the center of mass frame (atomic or molecularframe) or equivalently in the laboratory frame without transla-tion of atoms or ions. In this case, the square-integrable time-independent wave function for the translation can be definedas18,26

ψ δ=R R( ) ( )tr (13)

2.2.1. Probability and Current Densities. With eqs 1−4, thenormalized electronic (i = e) and nuclear (i = n) probabilitydensities (5) are rewritten as (j ≠ i)

∭ρ δμ μ

ψ

μψ

μ

= + | |

=

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

m m

m m

rr r

r r

r

( ) ( ) d (14)

(15)

nlm nlm

nlm

ii

i

j

j

i

2j

i3

3i i

2

The corresponding current densities (6) are evaluated as

∭ δμ μ

ψ ψ ψ ψ

μψ ψ

ψ ψ

μψ ϕ

= ℏ +

× ∇ * − * ∇

=ℏ

∇ *

− * ∇

= ℏ | | ∇

=−

=−

⎛⎝⎜⎜

⎞⎠⎟⎟m m m

m

mm

j r

r r

r r r r r

r r

r r

r

( )

i2

( ( ) ( ) ( ) ( )) d

(16)

i2

( ( ) ( )

( ) ( ))

(17)

( ) (18)

nlm

nlm nlm nlm nlm

nlm nlm

nlm nlmm m

nlmm m

r r

r

rr r

rr r

ii

i

i

j

j

i

j

i2

3

i2

32

i i

i

i

j j i i

i

j j i i

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXB

With ϕ = sgn(y) arccos(x/(x2 + y2)1/2), x = xe − xn, y = ye − yn,and d/dx arccos(x) = −1/(1 − x2)1/2, the gradients of therelative angle ϕ with respect to re and rn are

ϕ∇ = ∇+

= −+

∇+

= −+ ∂

∂ ++ ∂

∂ +

=− +

+

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

y x

x y

x yy

x

x y

x yy x

x

x y yx

x y

y x

x y

e e

e e

sgn( ) arccos (19)

(20)

(21)

(22)

x y

x y

r r

r

2 2

2 2

2 2

2 2

e2 2

e2 2

2 2

e e

e

and by analogy

ϕ∇ =−

+

y x

x y

e ex yr 2 2n (23)

In the center of mass frame, i.e., mnrn = −mere and x = xe − xn =(me/μ)xe = −(mn/μ)xn, y = ye − yn = (me/μ)ye = −(mn/μ)yn,the gradients are simplified to (i ≠ j)

ϕ μ μ ϕρ

∇ | =− +

+= ϕ

=− m

y x

x y m

e e e ( )m m

x yr r r

i

i i

i2

i2

i

i

ii j j i i

(24)

where in the last step ρi2 = xi

2 + yi2 and −yiex + xiey = ρi(−sin ϕi

ex + cos ϕi ey) = ρieϕ(ϕi) were used. With eqs 18 and 24, theelectronic (i = e) and nuclear (i = n) current densities (6)

μ ρψ

μϕ= ℏ

ϕ⎛⎝⎜

⎞⎠⎟

m m mj r

re( ) ( )nlm nlm

ii

i2

i

i i2

i(25)

have cylindrical symmetry and vanish for m = 0; i.e., theirmagnitudes do not depend on the azimuthal angle ϕi. With thedefinition for the reduced mass20

μρψ ϕ= ℏ | | ϕ

mj r r e( ) ( ) ( )nlm nlm

2i (26)

I obtain the important relations for the electron (i = e) andnuclear (i = n) current densities

μ μ=

⎛⎝⎜

⎞⎠⎟

m mj r j

r( )nlm nlm

ii

i2

2i i

(27)

2.2.2. Mean Angular Momenta. The electronic (i = e) andnuclear (i = n) mean angular momenta (7) are

∭∭∭

μψ

μϕ

ρ

μψ

μ

μ ρ

μ

⟨ ⟩ = ℏ ××

= ℏ

= ℏ

= ℏ

ϕ⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

mm

m

mm

m

mm

mm

Lr r e

r

rr e

r r e

e

( )d (28)

d (29)

( ) d (30)

(31)

nlm nlm

nlm z

nlm z

z

i i2

2i i

2i i

ii

i2

2i i

2

i

i

ii i

i

hence

μ⟨ ⟩ = ℏm

mL enlm ze

e (32)

and

μ⟨ ⟩ = ℏm

mL enlm zn

n (33)

In eq 29, the relation ri × eϕ(ϕi) = (ρieρ(ϕi) + ziez) × eϕ(ϕi) =ρiez − zieρ(ϕi) was used and the ρ-component does notcontribute to the integral because of ϕi-oscillations in eρ(ϕi) =cos ϕiex+ sin ϕiey. As expected, the total angular momentum is

⟨ ⟩ = ⟨ ⟩ + ⟨ ⟩ = ℏmL L L enlm nlm nlm ztot e n

(34)

2.2.3. Electric Ring Currents. With eq 27, the electric ringcurrents (8) are rewritten as

∬μ μ=

⎛⎝⎜

⎞⎠⎟I

me

mj

rSdnlm nlm

i i2

2 ii i

i(35)

Using the substitution r = (mi/μ)ri, I obtain dSi = μ2/mi2 dS, thus

∬=I e j r S( ) dnlm nlmi

i (36)

With the definition for the reduced mass20

∬= −I e j r S( ) dnlm nlm (37)

it becomes

= −I Inlm nlmi

i (38)

Hence, the electric ring current for the electron Inlme is exactly

equal to the one for the reduced mass Inlm, i.e.20

πμ= = − ℏ

μI I m e

a nsgn( )

2nlm nlme

2

2

3(39)

For the nucleus, I get

πμ= − = ℏ

μI I m

ea n

sgn( )2nlm

nnlm 2

3

3(40)

thus the ratio of nuclear and electronic electric ring currents is

= −II

nlm

nlm

n

e(41)

i.e., the nuclear electric ring current is exactly times largerthan the electronic one and they are opposite. It also confirmsthat the mean periods for the electron and for the nucleus (9)

∬= | | =| |

−Te

Ij r S( ( ) d )nlm nlm

nlm

i 1

(42)

are identical as expected also in classical mechanics.2.2.4. Mean Ring Current Radii. Again with eqs 27 and 38,

the kth moment of mean ring current radii (10) for m ≠ 0 inthe center of mass frame are

∬μρ

μ= − ·

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟R

eI

m mdj

rSk nlm

nlm

knlm

k

,i i

2

2 ii i

i

1/

(43)

Using the substitution r = (mi/μ)ri, i.e., dSi = μ2/mi2 dS and

ρi = (μ/mi)ρ, it yields

∬μ ρ= −⎛⎝⎜

⎞⎠⎟R

me

Ij r S( ) dk nlm

nlm

knlm

k

,i

i

1/

(44)

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXC

With the definition for the reduced mass (cf. refs 18 and 20)

∬ ρ= −⎛⎝⎜

⎞⎠⎟R

eI

j r S( ) dk nlmnlm

knlm

k

,

1/

(45)

I get

μ=Rm

Rk nlm k nlm,i

i,

(46)

and in particular for the electron

μ=Rm

Rk nlm k nlm,e

e,

(47)

and for the nucleus

μ=Rm

Rk nlm k nlm,n

n,

(48)

The corresponding ratio is

=R

Rmm

k nlm

k nlm

,n

,e

e

n (49)

and it is obvious that the condition of the center of mass issatisfied for all orders k ≠ 0. The analytic expression for Rk=−1,nlmis found in eq 30 of ref 20 and equal to 4aμ/(π ) for 2p±orbitals. The corresponding general formula for 2p± orbitals fororders k > −2 and k ≠ 0 is derived in section 2.3.4 (see eq 131).Thus, for k = 1, it is equal to 3πaμ/(4 ) for 2p± orbitals. In thiswork, the nucleus is considered as a point charge; therefore, themean ring current radius for the nucleus must be much largerthan the nuclear radius Rn ≈ r0A

1/3 where r0 ≈ 1 fm =10−15 m isthe nucleon radius and A is the atomic mass number.2.2.5. Induced Magnetic Fields. The position of the ring

center (12) in the center of mass frame is

∭ δ= =r R R R 0( ) d0 (50)

Thus, the electronic and nuclear ring currents induce themagnetic fields (11) at the ring center (ri = r0 = 0), i.e.

∭∭

∭

μπ

μμ

πμ

μμ

π

= = −′ × ′

| ′|′

= − ×′ × ′

| ′|′

= −×

| |

e

m e m

m e

B r 0j r r

rr

j r r

rr

j r r

rr

( )4

( )d (51)

4

( / )d (52)

4

( )d (53)

nlmnlm

nlm

nlm

ii

0 ii

3

i2

20 i i

3

i 0 i3

where eq 27 and the substitution r = (mi/μ)r′ were used. Withthe definition for the reduced mass20

∭μπ

= =×

| |e

B r 0j r r

rr( )

4

( )dnlm

nlm03

(54)

I obtain for l > 0

μ= = − =

mB r 0 B r 0( ) ( )nlm nlm

ii

i i

(55)

as expected in the current loop model,20,28 using |B(0)| ∝ I/Rand eqs 38 and 46. In particular, the opposite magnetic fieldsat the ring center induced by electronic and nuclear ringcurrents are

μμπμ

= = =

= − ℏ+ +μ

mB

em

am

n l l l

B r 0 r 0

e

( ) ( )

(2 1)(2 2)

nlm nlm

z

ee

e

0 e2 3

3

3(56)

and

μμπμ

= = − =

= ℏ+ +μ

m

em

am

n l l l

B r 0 B r 0

e

( ) ( )

(2 1)(2 2)

nlm nlm

nz

nn

n

02 3

4

3(57)

(cf. ref 20). The ratio of the corresponding z-components is

==

= −B

Bm

m

r 0

r 0

( )

( )z nlm n

z nlm

,n

,e

e

n

e (58)

Compared to the magnetic field induced by the electronic ringcurrent, the nuclear ring current with nuclear charge and verysmall ring current radius induces, in fact, very strong magneticfield at the ring center with ratio of m m/ n e. Therefore, thetotal induced magnetic field at the ring center (r = r0 = 0)

μ

= = = + =

=−

=m m

B r 0 B r 0 B r 0

B r 0

( ) ( ) ( )

( )

nlm nlm nlm

nlm

tot ee

nn

e n

(59)

is dominated by the magnetic field induced by the nuclear ringcurrent.

2.3. Translational Effects. In this section, I consider thesystem (atom or ion) in the laboratory frame where theGaussian-distributed center of mass moves in the z-direction.The translational wave function in cylindrical coordinates(P, Φ, Z) is then written as the product of the 1D Gaussianwavepacket along the z-axis and the 2D Gaussian wavepacketperpendicular to the z-axis, i.e.

ψ ψ ψ=t P t Z tR( , ) ( , ) ( , )P Ztr tr tr (60)

With the mean momenta pP = 0 and pZ and the wavepacketwidths σP and σZ at the initial time t = 0, the Gaussianwavepackets are defined as29

ψσ

πσ= + ℏ σ

−− + ℏ⎜ ⎟⎛

⎝⎞⎠P t

tM

( , )2

i2

eP PP

P t Mtr

21

/4( i /(2 ))P2 2

(61)

and

ψσ

πσ= + ℏ

× σ σ σ

−

− ℏ − − ℏ + ℏ

⎜ ⎟⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠Z t

tM

( , )2

i2

e

Z ZZ

p Z p t M

tr

1/22

1/2

( / ) [( 2i / ) /4( i /(2 ))]Z Z Z Z Z2 2 2 2 2 2

(62)

The corresponding densities are

ψπσ

| | = σ−P tt

( , )1

2 ( )eP

P

P ttr

22

/2 ( )P2 2

(63)

and

ψπ σ

| | = σ− −Z tt

( , )1

2 ( )eZ

Z

Z p t M ttr

2 ( / ) /2 ( )Z Z2 2

(64)

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXD

where σP(t) and σZ(t) are the time-dependent wavepacketwidths defined as

σ σσ

= + ℏ⎛⎝⎜

⎞⎠⎟t

tM

( ) 12P P

P2

2

(65)

and

σ σσ

= + ℏ⎛⎝⎜

⎞⎠⎟t

tM

( ) 12Z Z

Z2

2

(66)

Therefore, the electronic and nuclear ring currents depend onthe Gaussian widths σP(t) and σZ(t) and on the translationalmomentum pZ.2.3.1. Ring Current Densities. It is indeed possible to obtain

an analytic formula for the electronic and nuclear probabilitydensities (5) but the main propose of this work is to evaluateintegrals for electronic (i = e) and nuclear (i = n) currentdensities (6) (i ≠ j)

∭

∭

ψ ψ ψ

ψ ψ

ψ ψ ψ

ψ ψ

= ℏ | | ∇ *

− * ∇

+ ℏ | | ∇ *

− * ∇

tm

t

mt t

t t

j r R r r

r r r

r R R

R R r

( , )i

2( , ) ( ( ) ( )

( ) ( )) d

i2

( ) ( ( , ) ( , )

( , ) ( , )) d

nlm nlm nlm

nlm nlm

nlm

r

r

r

r

ii

itr

2

j

i

2tr tr

tr tr j

i

i

i

i (67)

The first term is due to the ring currents of the orbitals, and thesecond term is due to the translational motion and does notcontribute to the ring currents. It can be shown analytically thatthe ϕ-component of the second term vanishes, that the firstterm has zero ρ- and z-components, and that all componentsdo not depend on the azimuthal angle ϕi. Therefore, with thesubstitution

= −Mm

mm

rR r

jj

i i

j (68)

(cf. eq 3), the ϕ-components of the current densities (denotedas ring current densities) are evaluated as

∭

∭

μψ ψ ψ

ψ ψ ϕ

μψ ψ ϕ

ϕ

=ℏ

| | ∇ *

− * ∇ ·

= ℏ | | | | ∇ ·

ϕ

ϕ

ϕ

= −

= −

j tm

t

mm t

r

R r r

r r e R

R r

e R

( , )i

2( , ) ( ( ) ( )

( ) ( )) ( ) d

(69)

( , ) ( ) ( )

( ) d

(70)

nlm

nlm nlm

nlm nlmm M m

nlm

m M m

r

rr R r

r

r R r

,i

i

i2

3 tr2

i

i2

3 tr2 2

i

i

ij j i i

i

j j i i

In the laboratory frame, i.e., mjrj = MR−miri, I obtain x = xe −xn = (me/μ)(xe − X) = (mn/μ)(X − xn), y = ye − yn = (me/μ)·(ye − Y) = (mn/μ)(Y − yn). With eqs 22 and 23, the gradients are(i ≠ j)

ϕ μ∇ | =− − + −

− + −= − m

y Y x X

x X y Y

e e( ) ( )

( ) ( )m M mx y

r r R ri

i i

i2

i2i j j i i

(71)

Using the substitutions ri′ = ri − R, xi′ = ρi′ cos ϕi′, yi′ = ρi′ sin ϕi′,and −sin ϕi′ex + cos ϕi′ey = eϕ(ϕi′), the ring current densities arerewritten as

∭μψ

ψμ

ϕ ϕρ

= ℏ | − ′ |

×′ ′ ·

′′

ϕ

ϕ ϕ⎛⎝⎜

⎞⎠⎟

j tm

m t

m

r r r

r e er

( , ) ( , )

( ) ( )d

nlm

nlm

,i

ii

2 tr i i2

i i2

i i

ii

(72)

or using eq 25

∭ ψ ϕ= | − ′ | ′ · ′ϕ ϕj t tr r r j r e r( , ) ( , ) ( ) ( ) dnlm nlm,i

i tr i i2 i

i i i (73)

i.e., the ring current densities in the laboratory frame are obtainedby averaging the ring current densities in the center of mass frameover the translational motion. Then, using the substitution r′ =(mi/μ)ri′ and the relation

ϕ ϕ ϕ ϕϕ ϕ

ϕ ϕ ϕ ϕ

ϕ ϕ

′ · = − ′ + ′× − +

= ′ + ′

= ′ −

ϕ ϕe e e ee e

( ) ( ) ( sin cos )( sin cos )

(74)

sin sin cos cos (75)

cos( ) (76)

x y

x y

i i i i

i i

i i i i

i i

I obtain from eq 72

∭ ψ μ

ψϕ ϕ

ρ

= ℏ − ′

× | ′ |′ −′

′

ϕ

⎛⎝⎜

⎞⎠⎟j t

mm m

tr rr

r r

( , ) ,

( )cos( )

d

nlm

nlm

,i

ii

tr ii

2

2 i

(77)

With eqs 60, 63, and 64, R = ri − (μ/mi)r′,

μ μ

ρ μρ μρ ρϕ ϕ

μ

= − ′ + −′

= + ′ −′

′ −

= − ′

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

P xx

my

ym

m m

Z zz

m

(78)

2cos( ) (79)

(80)

ii

2

ii

2

i2

i

2i

ii

ii

dr′ = ρ′ dρ′ dz′ dϕ′, and the fact that |ψnlm(r′)|2 does notdepend on the azimuthal angle ϕ′, the ring current densities areexpressed as

∫∫

∫

π σ σ

ρ

ψ ρ ϕ

ϕ

ϕ ϕ

= ℏ

× ′

× ′

× | ′ ′ ′ |

× ′

× ′ −

ϕρ σ

μ ρ σ

σ

πμρ ρ σ ϕ ϕ

−

∞− ′

− − ′ −

′ ′−

j tm

m t t

z

z

r( , )(2 ) ( ) ( )

e

d e

d e

( , , )

d e

cos( )

nlmP Z

t

m t

z m m p t M t

nlm

m t

,i

i 3/2i

2/2 ( )

0

/2 ( )

( / / ) /2 ( )

2

0

2[ / ( ) ]cos( )

i

P

P

z Z Z

P

i2 2

2 2i2 2

i i2 2

i i2

i

(81)

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXE

Using the substitution ϕ = ϕ′ − ϕi and the integral representa-tion of the modified Bessel function of the first kind30

∫πϕ ϕ=

πϕI z( )

12

d e cosz1

0

2cos

(82)

it yields

∫

∫

π σ σ

ρμρ ρσ

ψ ρ ϕ

= ℏ

× ′′

× ′

× | ′ ′ ′ |

ϕρ σ

μ ρ σ

μ σ

−

∞− ′

−∞

∞− − ′ −

⎛⎝⎜

⎞⎠⎟

j tm

m t t

Im t

z

z

r( , )2 ( ) ( )

e

d e( )

d e

( , , )

nlmP Z

t

m t

P

z m p t M t

nlm

,i

ii

2/2 ( )

0

/2 ( )1

i

i2

( / / ) /2 ( )

2

P2

P

z Z Z

i2

2 2i2 2

i i2 2

(83)

Therefore, the ring current densities do not depend on theazimuthal angle ϕi; i.e., they have cylindrical symmetry andvanish for m = 0. Because |ψnlm(ρ′,z′,ϕ′)|2 has the exponentialfunction exp(−2 (ρ′2 + z′2)1/2/(naμ)), the z′- and ρ′-integralshave to be evaluated numerically.However, there is a special case in which the integrals can be

evaluated analytically. In this case, the mean momentum of thecenter of mass is zero pP = pZ = 0 and the translational wavefunction is spherical, i.e., σ = σP = σZ; thus σ(t) = σP(t) = σZ(t).The expression (83) in spherical coordinates, using ρi = ri sin θi,zi = ri cos θi, ρ′ = r′ sin θ′, z′ = r′ cos θ′, and dρ′ dz′ = r′ dr′dθ′, is simplified to

∫

∫π σ

θμ θ θ

σ

ψ θ ϕ

= ℏ × ′ ′

× ′′ ′

× | ′ ′ ′ |

ϕ

σ μ σ

πμ θ θ σ

−∞

− ′

′ ′⎛⎝⎜

⎞⎠⎟

j t

mm t

r r

Ir r

m t

r

r( , )

2 ( )e d e

d esin sin

( )

( , , )

nlm

r t r m t

r r m t

nlm

,i

i

i3

/2 ( )

0

/2 ( )

0

cos cos / ( )1

i i

i2

2

i2 2 2 2

i2 2

i i i2

(84)

Although the θ′- and r′-integrals are analytically executable forall quantum numbers n, l, m, only the strongest ring currents;i.e., the ones in 2p± orbitals are considered here (cf. ref 20).The wave functions (2) for 2p± orbitals are

ψ θ ϕπ

θ=μ

ϕ±

− ±μ⎛⎝⎜⎜

⎞⎠⎟⎟r

ar( , , )

18

e sin er a21 1

5/2/2 i

(85)

and the corresponding ring current densities (84) are

∫∫

π σ

θ θ

μ θ θσ

= ± ℏ

× ′ ′

× ′ ′

×′ ′

ϕμ

σ

μ σ

πμ θ θ σ

±−

∞− ′ − ′

′ ′

μ

⎛⎝⎜

⎞⎠⎟

j tm a t

r r

Ir r

m t

r( , )32(2 ) ( )

e

d e

d sin e

sin sin( )

r t

r m t r a

r r m t

,21 1i

i

5

3/2i

5 3/2 ( )

0

3 [ /2 ( ) ] ( / )

0

2 cos cos / ( )

1i i

i2

i2 2

2 2i2 2

i i i2

(86)

With abbreviations

σ=btr

( )2 i (87)

σμ

=μ

cm t

a( )

2i

(88)

the electronic (i = e) and nuclear (i = n) ring current densitiesof the 2p± orbitals (86) is written in the analytic form as

μ θπ

π

= ±ℏ

× − + −

+ + − +

−

ϕ ±

−

− −

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎡⎣⎢⎢⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠

⎤⎦⎥⎥

j tc

b m re

bc

bc c

b

bc

bc c

b

cb

r( , )sin

32

1 12

erfc1

2e

1 12

erfc1

2e

2e

c

c b

c b

c b

,21 1i

i

5i

2i2

i4

2/

2/

(1/4 )

2

2 2

(89)

(Appendix A). Because of the appearance of the function sin θiin eq 89, the electronic and nuclear ring currents in 2p± orbitalsin the laboratory frame are toroidal.In the limit of zero translational wavepacket width (σ(t) = 0),

i.e., b = 0, c = 0, c/b = miri/(μaμ), the second term as well asthe third term of eq 89 vanish, because erfc(∞) = 0 andexp(−∞) = 0. With erfc(−∞) = 2, the current densities of the2p± orbitals in the laboratory frame in the limit σ(t) = 0 arethen equal to the ones in the center of mass frame

πμθ= ±

ℏϕ

μ

μ±

− μjm

arr( )

64e sinm r a

,21 1i

i

5i3

4 5 i/

ii i

(90)

(cf. eqs 25 and 85).2.3.2. Mean Angular Momenta. In this section, I show that

the translation does not affect the electronic and nuclear meanangular momenta (7) because these quantities are conserved inquantum mechanics. With

ρ ϕϕ

ϕ

ϕ ρ

ρ ϕ

× = +× ++

= − ++ −

ρ

ρ ρ

ϕ ϕ

ϕ ρ ϕ

ρ ϕ

t zj t j t

j t

z j t j t

z j t j t

r j r e er e r e

r e

r e r e

r r e

( , ) ( ( ) )( ( , ) ( ) ( , )

( , ) ( ))

(91)

( , ) ( ) ( , )

[ ( , ) ( , )] ( )

(92)

nlm z

nlm z nlm z

nlm

nlm nlm z

nlm z nlm

ii

i i i i

,i

i i ,i

i

,i

i i

i ,i

i i i ,i

i

i ,i

i i ,i

i i

and the fact that all components of the current densities do notdepend on the azimuthal angle ϕi, the mean angular momenta(7) are rewritten as

∭ ρ⟨ ⟩ = ϕt m j t eL r r( ) ( , ) dnlm nlm zi

i i ,i

i i (93)

Its ρ- and ϕ-components are exactly zero due to ϕi-oscillationsin eρ(ϕi) and eϕ(ϕi). Using the general formula for the ringcurrent densities (77), I get

∭∭ ψ

ρ

ρ ψ μ ϕ ϕ

⟨ ⟩ = ℏ| ′ |

′′

× − ′ ′ −⎛⎝⎜

⎞⎠⎟

t m

mt

Lr

r

rr

r e

( )( )

d

, cos( ) d

nlmnlm

z

i2

i tr ii

2

i i

(94)

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXF

With the substitution R = ri − (μ/mi)r′, i.e., dri = dR = P dP dZdΦ, the term cos(ϕ′ − ϕi) is rewritten as

ϕ ϕϕ ϕ ϕ ϕ

ρ ρ

ρ ρμ μ

ρ ρμρ

ρϕ ϕ μρ

ρϕ μρ

′ −= ′ + ′

=′

′ + ′

=′

′ + ′ + ′ +′

=′

′ + ′ + ′

= ′ Φ + ′ Φ + ′

= ′ − Φ + ′

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

x x y y

x Xx

my Y

ym

x X y Ym

Pm

Pm

cos( )cos cos sin sin

(95)

1( ) (96)

1(97)

1(98)

1(cos cos sin sin ) (99)

1cos( ) (100)

i

i i

ii i

i i i

i

2

i

i i

i i

Because the translational wave function does not depend on Φ,the first term of eq 100 vanishes after Φ-integration, thus

∭ ∭μ ψ ψ

⟨ ⟩

= ℏ | ′ | ′ | |

t

mm t e

L

r r R R

( )

( ) d ( , ) d

nlm

nlm z

i

i

2tr

2

(101)

With the fact that the wave functions ψtr(R,t) and ψnlm(r) arenormalized, it yields

μ⟨ ⟩ = ℏtm

mL e( ) nlm zi

i (102)

(cf. eq 31). The total angular momentum is

⟨ ⟩ = ⟨ ⟩ + ⟨ ⟩ = ℏmL L L enlm nlm nlm ztot e n

(103)

(cf. eq 34). Therefore, the mean angular momenta in the centerof mass frame and in the laboratory frame are equal, conserved,and independent of the translational wavepacket widths andmean translational momenta.2.3.3. Electric Ring Currents. With the formula for the ring

current densities (83), the electric ring currents (8) areexpressed as

∫

∫

∫

∫

π σ σρ

ψ ρ ϕ

ρμρ ρσ

=ℏ

′

× ′ | ′ ′ ′ |

×′

×

μ ρ σ

ρ σ

μ σ

∞− ′

−∞

∞

∞−

−∞

∞− − ′ −

⎛⎝⎜

⎞⎠⎟

I tem

m t t

z z

Im t

z

( )2 ( ) ( )

d e

d ( , , )

d e( )

d e

nlmP Z

m t

nlm

t

P

z z m p t M t

i i

i2 0

/2 ( )

2

0 i/2 ( )

1i

i2

i( / / ) /2 ( )

P

P

Z Z

2 2i2 2

i2 2

i i2 2

(104)

With the substitution z = (zi − μz′/mi − Pzt/M)/(21/2σP(t)),the zi-integral is easily carried out, i.e.

∫∫σ

π σ

=

=

μ σ

−∞

∞− − ′ −

−∞

∞−

z

t z

t

d e

2 ( ) d e

2 ( )

z z m p t M t

Zz

Z

i( / / ) /2 ( )Z Zi i

2 2

2

(105)

thus

∫

∫

∫

σρ

ψ ρ ϕ

ρμρ ρσ

=ℏ

′

× ′ | ′ ′ ′ |

×′

μ ρ σ

ρ σ

∞− ′

−∞

∞

∞−

⎛⎝⎜

⎞⎠⎟

I tem

m t

z z

d Im t

( )( )

d e

d ( , , )

e( )

nlmP

m t

nlm

t

P

i i

i2 0

/2 ( )

2

0 i/2 ( )

1i

i2

P

P

2 2i2 2

i2 2

(106)

The electric ring currents as the integral of the ring currentdensities over the half-plane along the axis of symmetry (z-axis)are independent of the distribution of the translationalwavepacket along the z-axis, in particular they are independentof the translational wavepacket width σZ(t) and the meanmomentum pZ. With the Taylor series of the modified Besselfunction30

∑=!Γ +=

∞ +⎜ ⎟⎛⎝

⎞⎠I u

j ju

( )1

( 2) 2j

j

10

2 1

(107)

the substitution u = ρi/(21/2σP(t)), the definition of the Gamma

function30

∫ ∫Γ = =∞

− −∞

− −j u u u u( ) d e 2 d ej u j u

0

1

0

2 1 2

(108)

the recursion relation Γ(j+1) = j Γ(j), and the Taylor series ofthe exponential function30

∑=!=

∞ uj

eu

j

j

0 (109)

the ρi-integral in eq 105 is evaluated as

∫

∫

∫

∑

∑

∑

∑

ρμρ ρσ

ρμρ ρσ

σ μρσ

σ μρσ

σμρ

μ ρσ

σμρ

′

=!Γ +

×′

=!Γ +

′

×

=Γ +!Γ +

′

=′ + !

′

=′

−

ρ σ

ρ σ

μ ρ σ

∞−

=

∞

∞ +−

=

∞ +

∞+ −

=

∞ +

=

∞ +

′

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Im t

j j

m t

tj j m t

u u

t jj j m t

m tj m t

m t

d e( )

1( 2)

d2 ( )

e

(110)

2 ( )1

( 2) 2 ( )

d e

(111)

( )2

( 1)( 2) 2 ( )

(112)

( ) 1( 1) 2 ( )

(113)

( )(e 1) (114)

t

P

j

P

jt

Pj P

j

j u

P

j P

j

P

j P

P m t

0 i/2 ( )

1i

i2

0

0 ii

i2

2 1/2 ( )

0 i

2 1

0

2 1

0 i

2 1

i2

0

2 2

i2 2

j 1

i2

/2 ( )

P

P

P

i2 2

i2 2

2

2 2i2 2

hence

∫

∫μ

ρρ

ψ ρ ϕ

=ℏ

′′

−

× ′ | ′ ′ ′ |

μ ρ σ∞

− ′

−∞

∞

I tem

z z

( ) d1

(1 e )

d ( , , )

nlmm t

nlm

i i

0

/2 ( )

2

P2 2

i2 2

(115)

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXG

Using eqs 26 and 36, the formula for the electric ring currentsare simplified to

∫∫

ρ

ρ

= − ′

× ′ ′ ′

μ ρ σ

ϕ

∞− ′

−∞

∞

I t I e

z j z

( ) d e

d ( , )

nlm nlmm t

nlm

i ii

0

/2 ( )

,

P2 2

i2 2

(116)

Thus, the electric ring currents in the laboratory frame areequal to the ones in the center of mass frame minus the onesweighted by the translational exponential function exp(−μ2ρ′2/(2mi

2σP(t)2)) perpendicular to the z-axis. In the limit σP(t) = 0;

i.e., there is no translation perpendicular to the z-axis, thesecond term of eq 116 vanishes, and the electric ring currents inthe laboratory frame and in the center of mass frame are equal.If σP tends to infinity, then the exponential function tends tounity and the second term is equal to Inlm

i ; therefore, the electricring currents tend to zero because in the limit σP(t) → ∞ theindividual ring currents with finite ring current radii are spreadeverywhere and the effect of the ring currents with respect tothe z-axis vanishes.Now I focus on the strong ring currents in 2p± orbitals.

Using eq 85, the corresponding ring current densities for thereduced mass (26) is

θπμ

θ= ± ℏϕ

μ±

− μj ra

r( , )64

e sinr a,21 1

5

5/

(117)

The electronic (i = e) and nuclear (i = n) electric ring currents(116) in 2p± orbitals are then rewritten in spherical coordinates(dρ′ dz′ = r′ dr′ dθ′) as

∫

∫πμ

θ θ

= ∓ℏ

′ ′

× ′ ′

μπ

μ θ σ

± ±

∞− ′

− ′ ′

μI t Ie

ar r( )

64d e

d sin e

r a

r m t

21 1i

21 1i i

5

5 0

2 /

0

sin /2 ( )P2 2 2

i2 2

(118)

and have the simple analytical form as

= − + −± ±I t I c c c( ) (1 [1 e Ei( )])c21 1i

21 1i 2 2 22

(119)

(Appendix B). As expected, the function in the parentheses ofeq 119 is in the range from 1 for c = 0 to 0 for c → ∞; i.e., theelectric ring currents decrease with increasing translationalwavepacket width σP(t) = σ(t) ∝ c or equivalently withincreasing time t due to the spreading of the translationalwavepacket perpendicular to the z-axis. Because c isproportional to the electronic or nuclear mass mi (eq 88),the nuclear electric ring currents decrease faster than theelectronic ones. The corresponding mean periods (9), usingeqs 38 and 42,

=− + −

±T tT

c c c( )

1 [1 e Ei( )]nlm

c21 1i

i

2 2 22

(120)

increase with wavepacket width σP(t) = σ(t) and withtime t.2.3.4. Mean Ring Current Radii. For m ≠ 0, the kth moment

of mean ring current radii (10) (k ≠ 0) is obtained in a similarway as in eqs 104−106, i.e.

∫

∫

∫

σρ

ψ ρ ϕ

ρ ρμρ ρσ

=ℏ

′

× ′ | ′ ′ ′ |

×′

μ ρ σ

ρ σ

∞− ′

−∞

∞

∞−

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎥

R tem

I t m t

z z

Im t

( )( ) ( )

d e

d ( , , )

d e( )

k nlmnlm P

m t

nlm

k t

P

k

,i i

ii

2 0

/2 ( )

2

0 i i/2 ( )

1i

i2

1/

P

P

2 2i2 2

i2 2

(121)

The mean ring current radii are also independent of thedistribution of the translational wavepacket along the z-axis;i.e., the wavepacket width σZ(t) and the mean momentum pZ donot affect the mean ring current radii. Using eqs 107 and 108, theρi-integral is evaluated in a similar way as in eqs 110−114, i.e.

∫

∫

∫

∑

∑

∑

∑

ρ ρμρ ρσ

μρσ

ρμρ ρσ

σ

μρσ

σ

μρσ

μρ σ

μ ρσ

μρ σ

μ ρσ

′

=!Γ +

′

×′

=!Γ +

× ′

=

×Γ + +

!Γ +′

= ′ Γ +

×!

Γ + + ΓΓ + Γ +

× ′

= ′ Γ +

× + ′

ρ σ

ρ σ

∞−

=

∞ −

∞ + +−

+

=

∞

+ ∞+ + −

+

=

∞ +

=

∞

⎜ ⎟

⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

Im t

j j m t

m t

tj j

m tu u

t

j kj j m t

mt

k

jj kk j m t

mt

k

Fk

m t

d e( )

1( 2) 2 ( )

d2 ( )

e

(122)

( 2 ( ))1

( 2)

2 ( )d e

(123)

12

( 2 ( ))

( /2 1)( 2) 2 ( )

(124)

2( 2 ( ))

21

1 ( /2 1) (2)( /2 1) ( 2) 2 ( )

(125)

2( 2 ( ))

21

21;2;

2 ( )

(126)

k t

P

j P

k

P

j kt

Pk

j

P

jj k u

Pk

j P

j

Pk

j P

Pk

P

0 i i/2 ( )

1i

i2

0 i2

0 ii

i2

2 1/2 ( )

1

0

i

2 1

0

2 1

1

0 i

2 1

i

0

2 2

i2 2

j

i

11

2 2

i2 2

P

P

i2 2

i2 2

2

for k > −2, where the confluent hypergeometric function of thefirst kind is defined as

∑=!

Γ + ΓΓ Γ +=

∞

F a b uj

j a ba j b

u( ; ; )1 ( ) ( )

( ) ( )j

j11

0 (127)

Thus, with eq 26, I obtain from eq 121

∫

∫

σμ

σ

ρ ρ

μ ρσ

ρ

=Γ +

× ′ ′

× + ′

× ′ ′ ′

μ ρ σ

ϕ

∞− ′

−∞

∞

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

R t tk

I t m t

Fk

m t

z j z

( ) 2 ( )e ( /2 1)

2 ( ) ( )

d e

21;2;

2 ( )

d ( , )

k nlm Pnlm P

m t

P

nlm

k

,i i

2

ii2 2

0

2 /2 ( )

11

2 2

i2 2

,

1/

P2 2

i2 2

(128)

for k > −2 and k ≠ 0.

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Focusing on the ring currents in 2p± orbitals with its ringcurrent densities (117), the formula for the mean ring currentradii (128) is then rewritten in spherical coordinates as

∫∫

σμ

π σ

θ θ

μ θσ

= ±ℏ Γ +

× ′ ′

× ′ ′

× + ′ ′

μ

πμ θ σ

±±

∞− ′

− ′ ′

μ

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎥

R t te k

I t m a t

r r

Fk r

m t

( ) 2 ( )( /2 1)

128 ( ) ( )

d e

d sin e

21;2;

sin2 ( )

k PP

r a

r m t

P

k

,21 1i i

5

21 1i

i2 5 2

0

4 /

0

3 sin /2 ( )

11

2 2 2

i2 2

1/

P2 2 2

i2 2

(129)

The corresponding final result of the electronic (i = e) and nuclear(i = n) mean ring current radii (129) for k > −2 and k ≠ 0 is

μ= Γ + +

− + −μ

±

⎡⎣⎢⎢

⎤⎦⎥⎥R t

a c

mc k U k c

c c c( )

2 2 ( /2 1) (3, /2 3, )

1 [1 e Ei( )]k c

k

,21 1i

i

4 2

2 2 2

1/

2

(130)

(Appendix C). This function increases monotonically with c orwith σP(t) = σ(t) ∝ c or with time t, again due to the spreading ofthe translational wavepacket. Because c ∝ mi, the nuclear meanring current radius increases faster than the electronic one. In thelimit c → ∞, the mean ring current radii are infinite, whereas inthe limit c = 0, they are equal to the mean ring current radii in thecenter of mass frame, i.e.

μ μ= = Γ + Γ +μ± ⎜ ⎟ ⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠⎤⎦⎥R

mR

m

a k k2

21

22k k nlm

k

,21 1i

i,

i

1/

(131)

for k > −2 and k ≠ 0 (cf. eq 46). In particular, Rk=−1,21±1i = 4μaμ/

(πmi ) and Rk=1,21±1i = 3πμaμ/(4 mi ).

2.3.5. Induced Magnetic Fields. Now I consider the magneticfields induced by electronic (i = e) and nuclear (i = n) ringcurrents at the ring center ri = r0(t), because the magnetic fieldsat this center are strongest. The time-dependent position of thisring center is determined by the mean position of the centerof mass (see eq 12). With eqs 63 and 64 and substitutionsP′ = P/(21/2σP(t)) and Z′ = (Z − pZt/M)/(21/2σZ(t)), the ringcenter is located at

∫ ∫∫

∫∫

∫∫

∫∫

ψ ψ

π ψ

ψ

π σ σ

π

σ

= | | | |

× Φ Φ +

= | |

× | |

=

×

= ′ ′

× ′ ′ +

=

π

ρ

σ

σ

∞

−∞

∞

∞

−∞

∞

∞−

−∞

∞− −

∞− ′

−∞

∞− ′⎛

⎝⎜⎞⎠⎟

t P P P t Z Z t

P Z

P P P t

Z Z Z t

t tP P

Z Z

P P

Z t Zp t

M

p t

M

r

e e

e

e

e

e

( ) d ( , ) d ( , )

d ( ( ) )

(132)

2 d ( , )

d ( , )

(133)

12 ( ) ( )

d e

d e

(134)

2d e

d 2 ( ) e

(135)

(136)

P Z

z

P

Zz

P Z

P t

Z p t M tz

P

ZZ Z

z

Zz

00 tr

2tr

2

0

2

0 tr2

tr2

2 0

/2 ( )

( / ) /2 ( )

0

P

Z Z

2 2

2 2

2

2

wehre the integrals ∫ 0∞du ue−u

2

= 1/2, ∫ −∞∞ du ue−u

2

= 0, and

∫ −∞∞ du e−u

2

= π1/2 were used (cf. eq 108). Therefore, the ringcenter moves with the constant velocity pZ/M along the z-axis. For ring currents with cylindrical symmetry, theinduced magnetic fields along the z-axis do not depend onthe ρ- and z-components of the current densities and only itsz-components do not vanish;18 thus they depend only on thering current densities (ϕ-component). At the ring center, theinduced magnetic fields according to the Biot−Savart law(11) are

∫∫μ

ρ ρ

ρ

ρ

=

= ′ ′

× ′′ ′

′ + ′ −ϕ

∞

−∞

∞

⎡⎣ ⎤⎦

t te

zj z t

z p t M

B r r

e

( ( ), )

2d

d( , , )

( / )

nlm

nlm

Z

z

ii 0

0 i

0

2

,i

2 2 3/2

(137)

With the general formula for the ring current densities (83) andthe substitution z = z′ − pZt/M, the magnetic fields at the ringcenter (137)

∫ ∫

∫

∫

μπ σ σ

ρ ρρ

ρ μρ ρσ

ψ ρ ϕ

= =ℏ

× ′ ′′ +

× ″ ′ ″

× ″ | ″ ″ ″ |

ρ σ

μ ρ σ

μ σ

∞− ′

−∞

∞

∞− ″

−∞

∞− − ″

⎛⎝⎜

⎞⎠⎟

t tem

m t t

zz

Im t

z z

B r r

e

( ( ), )2 2 ( ) ( )

d e d1

( )

d e( )

d e ( , , )

nlmP Z

t

m t

P

z z m tnlm z

ii 0

0 i

i2

0

2 /2 ( )2 2 3/2

0

/2 ( )1

i2

( / ) /2 ( ) 2

P

P

Z

2 2

2 2i2 2

i2 2

(138)

are independent of the translational momentum pZ. In general,they decrease with increasing wavepacket widths σP(t) andσZ(t). For the special case σ = σP = σZ, i.e., σ(t) = σP(t) = σZ(t),the integrals in the expression (138) are then rewritten inspherical coordinates as

∫

∫ ∫

∫

μπ σ

θ ψ θ ϕ

θ θ μ θ θσ

= =ℏ

″ ″

× ″ | ″ ″ ″ | ′

× ′ ′ ′ ″ ′ ″

μ σ

πσ

πμ θ θ σ

∞− ″

∞− ′

′ ″ ′ ″⎛⎝⎜

⎞⎠⎟

t tem

m tr r

r r

Ir r

m t

B r r

e

( ( ), )2 2 ( )

d e

d ( , , ) d e

d sin esin sin

( )

nlmr m t

nlmr t

r r m tz

ii 0

0 i

i3 0

/2 ( )

0

2

0

/2 ( )

0

2 cos cos / ( )1

i2

2 2i2 2

2 2

i2

(139)

Evaluating r′- and θ′-integrals yields

∫

∫

μμ

μσ

μπ σ

θ θ θ

= =

× ″ ″

− ′′

× ″ ″ ″ ″

μ σ

π

ϕ

∞

− ″

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎥

t tem

rr

m t

rm t

j r

B r r

e

( ( ), )2

d erf2 ( )

2( )

e

d sin ( , )

nlm

r m t

nlm z

ii 0

0 i i

0 i

i

/2 ( )

0

2,

2 2i2 2

(140)

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(Appendix D). In the limit σ(t) = 0, the first term of ther″-integrand tends to unity and the corresponding second termtends to zero. Therefore, in this limit, the induced magneticfields at the ring center are equal to the ones in the center ofmass frame (cf. eq 18 of ref 20) and they are maximal. In theopposite limit σ(t) → ∞, both terms of the r″-integrand tendto zero and the induced magnetic fields are suppressed asexpected, because the corresponding electric ring currents andthe mean ring current radii tend to zero and infinity (seesections 2.3.3 and 2.3.4), respectively.For the electronic (i = e) and nuclear (i = n) ring currents in

2p± orbitals, I get the analytical expression for the inducedmagnetic fields at the ring center

π

= = = × − +

× + +

± ±⎧⎨⎩

⎫⎬⎭

t t c c

cc

c

B r r B r 0( ( ), ) ( ) [1 4 ( 1)]e

erfc( )2

(2 1)

c21 1i

i 0 21 1i

i2 2

2

2

(141)

(Appendix E). The scalar function in eq 141 is in the rangefrom 1 for c = 0 to 0 for c → ∞. Hence, the induced magneticfields decrease with increasing translational wavepacket widthσP(t) = σZ(t) = σ(t). In the limit c = 0, the magnetic fields in thelaboratory frame are equal to the ones in the center of massframe. Furthermore, the magnetic fields induced by nuclear ringcurrents decrease faster than the ones induced by electronicring currents because c ∝ mi.2.4. Application to Ring Currents in Many-Electron

Atoms and Ions. The formulas derived in the previoussections can be generalized for electronic and nuclear ringcurrents in degenerate electronic states |Ψ±⟩ of N-electronatoms and ions in the laboratory frame. The electronic (i = e)and nuclear (i = n) ring current densities in the center of massframe are defined similarly as in eq 27 where the reduced massμ is replaced with the generalized one μ = memn/M, where me isnow the total mass of N electrons, i.e.

μ μ=Ψ Ψ± ±

⎛⎝⎜

⎞⎠⎟

m mj r j

r( )i

ii2

2i i

(142)

The formulas for the current densities for the reduced massjΨ±(r) in different many-electron systems are found in ref 18.With eqs 26, 84, and 142 and the substitution r = (μ/mi)r′, thecorresponding ring current densities in the laboratory frame inthe case of σ(t) = σP(t) = σZ(t) are

∫

∫

θπ σ

θ θθ θ

σθ

=

×

ϕσ σ

πθ θ σ

ϕ

Ψ−

∞−

Ψ

±

±

⎛⎝⎜

⎞⎠⎟

j r tt

r r

Ir r

tj r

( , , )1

2 ( )e d e

d sin esin sin

( )( , )

r t r t

r r t

,i

i i 3/2 ( )

0

2 /2 ( )

0

cos cos / ( )1

i i2 ,

i

i2 2 2 2

i i2

(143)

Because the mean angular momenta are not affected by thetranslation (see section 2.3.2), the corresponding electronic andnuclear mean angular momenta

μ⟨ ⟩ = ⟨ ⟩ = ⟨ ⟩Ψ Ψ Ψ± ± ±t

mL L L( )i i

i (144)

are conserved and equal to the ones in the center of mass frame(cf. eq 102). Using the electronic and nuclear ring currentdensities (142) in spherical coordinates and the substitutionr = (μ/mi)r′, the formulas for the electric ring currents (116),mean ring current radii (128), and induced magnetic fields at

the ring center (140) with the translational wavepacket widthσ(t) = σP(t) = σZ(t) are then generalized to

∫ ∫ θ θ= −π

θ σϕΨ Ψ

∞−

Ψ± ± ±I t I e r r j r( ) d d e ( , )r ti i

i0 0

sin /2 ( ),

i2 2 2

(145)

where IΨ±

i = − IΨ±(cf. eq 38),

∫

∫

σσ

θ θ

θσ

θ

=Γ +

×

× +

πθ σ

ϕ

ΨΨ

∞

−

Ψ

±±

±

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

R t te kI t t

r r

Fk r

tj r

( ) 2 ( )( /2 1)

2 ( ) ( )d

d sin e

21;2;

sin2 ( )

( , )

k

r t

k

,i i

i 2 0

3

0

2 sin /2 ( )

11

2 2

2 ,i

1/

2 2 2

(146)

for k > 2, k ≠ 0, and

∫

∫

μσ

π σ

θ θ θ

= =

−

×

σ

π

ϕ

Ψ

∞

−

Ψ

±

±

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

t te

rr

t

rt

j r

B r r

e

( ( ), )2

d erf2 ( )

2( )

e

d sin ( , )

r t

z

ii 0

0 i

0

/2 ( )

0

2,

i

2 2

(147)

respectively.2.5. Application to Pseudorotating Molecules. How-

ever, the general formulas (143)−(147) are applied not only toring currents in atoms and ions but also to pseudorotatingmolecules, i.e., to nuclear ring currents in degenerate vibrationalstates of molecules, such as linear CdH2 and FHF− ortetrahedral OsH4 molecules (see refs 8, 18, 25, and 26). Inpseudorotating molecules, all nuclei circulate about itsindividual equilibrium positions. For the symmetric moleculeshaving a central nucleus, the center of the ring current of thecentral nucleus is located at the center of mass of the molecule.Therefore, I focus on the ring currents of the central nucleus inpseudorotating molecules. Here, I take the tetrahedral moleculeAB4 as an example and derive analytic formulas for the centralnuclear ring currents in the first triply degenerate pseudorota-tional states |v1l±1⟩ that belong to the triply degenerate bends orantisymmetric stretches.26

The corresponding nuclear (i = A) ring current densities inthe center of mass frame and in the harmonic approximationare26

θ π

θ

= ± ℏ

×

ϕ

π− ℏ

±j r ) cM m

m T

r

( ,2

e sin

v l

Mm r m T

,A

A A

3A

3

3B

3A

5

A/2

A

1 1

A A2

B A (148)

where c is the so-called pseudorotational weight (for examplec = 0.64 and c = 0.36 for triply degenerate bends andantisymmetric stretches of the OsH4 molecule, respectively),mA is the mass of the central nucleus A, mB is the mass of theperipheral nucleus B, M = mA + 4mB is the total mass of themolecule AB4, and TA is the mean pseudorotational period ofthe central nucleus; for details, see ref 26. With the nuclear ringcurrent densities in the center of mass frame (148), the formulafor the corresponding ring current densities in the laboratory

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frame (143) is then rewritten as

∫

∫

θσ

π

θ θθ θ

σ

= ± ℏ

×

×

ϕσ

π σ

πθ θ σ

−

∞− ℏ −

±

⎛⎝⎜

⎞⎠⎟

j r tc

tM mm T

r r

Ir r

t

( , , )2 ( )

e

d e

d sin esin sin

( )

v lr t

Mm r m T r t

r r t

,A

A A 3

3A

3

3B

3A

5/2 ( )

0

3 ( /2 ) [ /2 ( ) ]

0

2 cos cos / ( )1

A A2

1 1A

2 2

A2

B A2 2

A A2

(149)

Using the substitution u = rAr/σ(t)2, the θ-integral (169), the

abbreviations (87) and

σπ

=ℏ

d tMmm T

( ) A

B A (150)

I obtain

∫

θθ

π= ±

× −

ϕ

−

−

∞+

±j r tcb d

T r

u u u u u

( , , )2 sin

e

d ( cosh sinh )e

v lb

b d u

,A

A A

2 3A

A A2

1/4

0

(1 )

1 1

2

2 2 2

(151)

The integrals in eq 151 are evaluated as31

∫π= + +

+

∞− +

+

u u u

b db d

d cosh e

(2 (1 ) 1)8 (1 )

e

b d u

b d

0

2 (1 )

2 2

5 2 5/21/4 (1 )

2 2 2

2 2

(152)

and

∫ π=+

∞− + +u u u

b dd sinh e

4 (1 )eb d u b d

0

(1 )3 2 3/2

1/4 (1 )2 2 2 2 2

(153)

hence

∫π

−

=+

∞− +

+

u u u u u

b d

d ( cosh sinh )e

8 (1 )e

b d u

b d

0

(1 )

5 2 5/21/4 (1 )

2 2 2

2 2

(154)

Therefore, the ring current densities of the central nucleus inthe first pseudorotational states |v1l±1⟩ of the AB4 molecule inthe laboratory frame, using eqs 87 and 150, are

θ π

θ

= ± + ℏ

×

ϕ

π− ℏ +

±j r tc

dM m

m T

r

( , , )(1 ) 2

e sin

v l

Mm r m T d

,A

A A 2 5/2

3A

3

3B

3A

5

A/2 (1 )

A

1 1

A A2

B A2

(155)

or, using eq 148, in the simple form

θ θ=+ +ϕ ϕ± ±

⎛⎝⎜⎜

⎞⎠⎟⎟j r t

dj

r

d( , , )

1(1 ) 1

,v l v l,A

A A 2 2 ,A A

2 A1 1 1 1

(156)

Thus, in the limit d = 0, the nuclear ring current densities in thelaboratory frame and in the center of mass frame are equal.The corresponding mean angular momenta of the central

nucleus26

⟨ ⟩ = ⟨ ⟩ = ± ℏ± ±t cmM

L L e( )4

v l v l zA A B

1 1 1 1(157)

are conserved and unaffected by the molecular translation (seesection 2.3.2 and eq 144). Because the analytic expression ofthe ring current densities (156) is very simple, the extraevaluation of the integrals for electric ring currents (145), meanring current radii (146), and induced magnetic fields (147) isnot necessary. Using the simple substitution r = rA/(1 + d2)1/2,the definitions (8), (10), and (11), and the expressions for theelectric ring currents, mean ring current radii (k = 1), andinduced magnetic fields at the ring center in the center of massframe26

= ± ±I ce

Tv lA A

A1 1

(158)

=ℏ

±Rm TMm2v l1,

A B A

A1 1

(159)

μ= = ±

ℏ

±

c e Mmm T

B r 0 e( )3

2v l zA

AA 0 A

B A31 1

(160)

I obtain the corresponding formulas in the first pseudorota-tional states |v1l±1⟩ of the AB4 molecule in the laboratory frame

=+

±

±I t

I

d( )

1v lv lAA

21 1

1 1

(161)

= +± ±R t R d( ) 1v l v l1,A

1,A 2

1 1 1 1 (162)

= ==

+±

±t t

dB r r

B r 0( ( ), )

( )

(1 )v lv lA

A 0

AA2 3/21 1

1 1

(163)

accoring to the Biot−Savart law ∝ I/R ∝ 1/(1 + d2)·1/(1 + d2)1/2. As already discussed in the case of ring currents inatomic orbitals, the electric ring currents and induced magneticfields in the laboratory frame decrease with increasingtranslational wavepacket width σ(t) ∝ d, whereas the meanring current radii increase with σ(t) ∝ d. In the limit d = 0, allvalues in the laboratory frame are equal to the ones in thecenter of mass frame. In the opposite limit d → ∞, the electricring currents and induced magnetic fields tend to zero, whereasthe mean ring current radii tend to infinity. Furthermore, thetranslational effects are considerable for heavy central nucleibecause d ∝ (MmA)

1/2.

3. RESULTS AND DISCUSSIONTable 1 lists magnitudes of the electric ring currents |I21±1

e |(eq 39), mean ring current radii R1,21±1

e (eq 47), and inducedmagnetic fields at the ring center |B21±1

e (0)| (eq 56) in thecenter of mass frame for electronic ring currents in 2p± orbitalsof nonrelativistic hydrogen-like systems with nuclear charges

n = 1, ..., 13 and masses mn of the stable isotopes; for dis-cussion of the nonrelativistic limit of n, see ref 20. Becausethe nucleus is much heavier than the electron, i.e., me ≈ μ anda0 ≈ aμ, the listed values for |I21±1

e | and |B21±1e (0)| are almost

identical to the ones in ref 20. Furthermore, for isotopes withthe same nuclear charges n, all values in the center of massframe do not change significantly, again due to large masses ofheavy nuclei. As already discussed in ref 20, the electric ringcurrents and induced magnetic fields increase with n as n

2

and n3 (cf. eqs 39 and 56), respectively, whereas the mean

ring current radii decrease with increasing n according to the

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Biot−Savart law as1/ n. Therefore, the induced magnetic fieldfor Al12+ is giant, i.e., 1146 T at the ring center.The corresponding properties of the nuclear ring currents in

the center of mass frame are listed in Table 2, i.e., magnitudesof the electric ring currents |I21±1

n | (eq 40), mean ring currentradii R1,21±1

n (eq 48), and induced magnetic fields at the ringcenter |B21±1

n (0)| (eq 57). Because the formula for the nuclearring currents derived in section 2 are applied only for R1,21±1

n ≫Rn and the nuclear radius Rn ≈ r0A

1/3 for A = 8 is about 2 fm,only the ring current properties for stable isotopes with A < 8or equivalently n = 1, 2, 3 are listed in Table 2. As forelectronic ring currents, the nuclear electric ring currents |I21±1

n |(eq 40) do not change significantly for isotopes with the samenuclear charges n because me ≈ μ. However, for nuclear ringcurrents, the electric ring currents and induced magnetic fieldsincrease with n as n

3 and n4 (cf. eqs 40 and 57),

respectively, whereas the mean ring current radii decrease withincreasing n as 1/ n. In addition, the induced magnetic fieldsincrease with mn as mn and, hence, the mean ring current radiidecrease with increasing mn according to the Biot−Savart lawas 1/mn. Therefore, the mean ring current radii are very smalland only a few times larger than the nuclear radii, i.e., on thefemtometer scale. For such very small radii, the inducedmagnetic fields at the ring center are giant, i.e., 0.96 kT for 1Hand 0.54 MT for 7Li2+.Table 2 also lists the properties of the ring currents of the

central nucleus Os in the first triply degenerate peseuorota-

tional states |v1l±1⟩ of the tetrahedral molecule OsH4, usingeqs 158, 159, and 160 for electric ring currents, mean ringcurrent radii, and induced magnetic fields, respectively. Thesevalues and other molecular parameters for the first triplydegenerate antisymmetric stretches (a) and bends (b) of theOsH4 molecule are adopted from ref 26.To illustrate the translational effects with zero mean

momenta pP = pZ = 0 on electronic and nuclear ring currents,the magnitudes of the electronic (i = e) and nuclear (i = n, A)ring current densities |jϕ,Ψ±

i |(ρi,θi=π/2,t=0)| depending on theradius ρi and on the initial translational wavepacket width σ =σP = σZ in units of R1,Ψ±

i are shown in Figure 1. Figure 1a showsthe ring currents in 2p± orbitals with identical distributions forelectron and nucleus. Figure 1b shows the ring currents of thecentral nucleus A in the first triply degenerate pseudorotationalstates |v1l±1⟩ of the tetrahedral molecule AB4 with identicaldistributions for antisymmetric stretches and bends (cf. ref 26).Futhermore, these unique scaled distributions are independentof the molecular properties of the tetrahedral molecule AB4.The distributions for 2p± orbitals and for the first pseudorota-tional states of the AB4 molecule are similar and represent thetoroidal structure of the ring currents. The black curvescorrespond to the ring current densities in the center of massframe or equivalently in the laboratory frame with translationalwavepacket width σ = 0. As expected, the maxima of the ringcurrent densities decrease with increasing translational wave-packet widths σ and its positions are shifted to larger radii,according to the spreading of the translational wavepacket.

Table 1. Properties of the Electronic Ring Currents in 2p± Orbitals of Hydrogen-Like Systems

system na mn

a (u) |IΨ±

e |b (mA) R1,Ψ±

e b (a0) |BΨ±

e (0)|b (T) |BΨ±

e,max(0, 100 fs)|c (T) |BΨ±

e,max(0, 1 ps)|c (T)1H 1 1.008 0.13 2.36 0.52 0.19 0.022H 1 2.014 0.13 2.36 0.52 0.27 0.053He+ 2 3.016 0.53 1.18 4.17 1.26 0.124He+ 2 4.002 0.53 1.18 4.17 1.52 0.186Li2+ 3 6.015 1.19 0.79 14.1 3.92 0.367Li2+ 3 7.016 1.19 0.79 14.1 4.37 0.449Be3+ 4 9.012 2.11 0.59 33.4 8.17 0.6810B4+ 5 10.01 3.29 0.47 65.2 12.1 0.8411B4+ 5 11.01 3.29 0.47 65.2 13.1 0.9612C5+ 6 12.00 4.74 0.39 113 17.7 1.1313C5+ 6 13.00 4.74 0.39 113 19.0 1.2714N6+ 7 14.00 6.46 0.34 179 24.3 1.4515N6+ 7 15.00 6.46 0.34 179 25.9 1.6016O7+ 8 15.99 8.43 0.29 267 31.7 1.7917O7+ 8 17.00 8.43 0.29 267 33.7 1.9618O7+ 8 18.00 8.43 0.29 267 35.7 2.1219F8+ 9 19.00 10.7 0.26 380 42.4 2.3420Ne9+ 10 19.99 13.2 0.24 522 49.2 2.5521Ne9+ 10 20.99 13.2 0.24 522 51.8 2.7422Ne9+ 10 21.99 13.2 0.24 522 54.5 2.9323Na10+ 11 22.99 15.9 0.21 694 62.0 3.1624 Mg11+ 12 23.99 19.0 0.20 901 69.7 3.4025 Mg11+ 12 24.99 19.0 0.20 901 72.9 3.6026 Mg11+ 12 25.98 19.0 0.20 901 76.1 3.8127Al12+ 13 26.98 22.3 0.18 1146 84.5 4.06

aOnly stable isotopes with nuclear charges n = 1, ..., 13 and nuclear masses mn are considered.bIn the center of mass frame, magnitudes of the

electric ring currents |IΨ±

e | (eq 39)), mean ring current radii R1,Ψ±

e (eq 47), and magnetic fields at the ring center |BΨ±

e (0)| (eq 56) induced byelectronic ring currents in 2p± orbitals of hydrogen-like systems are listed. cIn the laboratory frame, the maxima of the induced magnetic fields attimes t = 100 fs and t = 1 ps for individual optimal initial translational wavepacket widths σ = σP = σZ are listed.

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The σ-dependent electronic and nuclear ring currentdensities in the laboratory frame shown in Figure 1 possessthe electric ring currents |IΨ±

i (t=0)| (eqs 119 and 161), themean ring current radii R1,Ψ±

i (t=0) (eqs 130 and 162), and theinduced magnetic fields at the ring center |BΨ±

i (0,t=0)| (eqs 141and 163). These properties depending on the translational wave-packet widths σ in units of R1,Ψ±

i are shown in Figure 2. Dueto the spreading of the ring current densities by translation, theelectric ring currents and induced magnetic fields decrease withincreasing σ and the corresponding mean ring current radiiincrease with σ, according to the Biot−Savart law. Interestingly,

these properties for ring currents in 2p± orbitals and in the firsttriply degenerate pseudorotational states |v1l±1⟩ of the tetrahedralmolecule AB4 are similar, supported by the similar properties forthe corresponding ring current densities shown in Figure 1.However, there is little difference between 2p± orbitals andpseudorotational states |v1l±1⟩: The decrease of the electric ringcurrents and induced magnetic fields is a little faster for 2p±orbitals, because the position of the maximum of the ring currentdensities for 2p± orbitals and σ = 0 is a little shifted to smallvalues of the radius compared to the ones for|v1l±1⟩ states (Figure 1) and because the translational effectsare stronger for smaller radius.Using the time-dependent spherically Gaussian-distributed

translational wavepacket with mean momenta pP = pZ = 0 andtime-dependent wavepacket widths σ(t) = σP(t) = σZ(t), eqs60−66, the time-dependent properties of the ring currents forselected initial translational wavepacket widths σ = σP = σZ inunits of R1,Ψ±

i , i.e., electric ring currents |IΨ±

i (t)| (eq 119 and

161), mean ring current radii R1,Ψ±

i (t) (eq 130 and 162), and

Table 2. Properties of the Nuclear Ring Currents in 2p± Orbitals of Hydrogen-Like Systems and in the First Triply-DegeneratePseudorotational States |v1l±1⟩ of the Tetrahedral Molecule OsH4

system na mn

a (u) |IΨ±

n |b (mA) R1,Ψ±

n b (fm) |BΨ±

n (0)|b (T) |BΨ±

n,max(0, 1 as)|c (T) |BΨ±

n,max(0, 1 fs)|c (T)1H 1 1.008 0.13 68 958 8.27 d2H 1 2.014 0.13 34 1914 6.17 d3He+ 2 3.016 1.05 11 45869 10.6 d4He+ 2 4.002 1.05 8.5 60869 9.21 d6Li2+ 3 6.015 3.56 3.8 463152 11.3 d7Li2+ 3 7.016 3.56 3.2 540236 10.5 d

OsH4(a)e 76 192.0 0.14 164 351 341 4.00

OsH4(b)e 76 192.0 0.25 164 624 607 6.85

aOnly stable isotopes with nuclear charges n = 1, 2, 3 and nuclear masses mn are considered.bIn the center of mass frame, magnitudes of the electric

ring currents |IΨ±

n | (eqs 40 and 158), mean ring current radii R1,Ψ±

n (eqs 48 and 159), and magnetic fields at the ring center |BΨ±

n (0)| (eqs 57 and 160)induced by nuclear ring currents in 2p± orbitals of hydrogen-like systems and in the first triply degenerate pseudorotational states |v1l±1⟩ of thetetrahedral molecule OsH4 are listed.

cIn the laboratory frame, the maxima of the induced magnetic fields at times t = 1 as and t = 1 fs for individualoptimal initial translational wavepacket widths σ = σP = σZ are listed.

dThe induced magnetic fields at t = 1 fs are strongly suppressed for nuclear ringcurrents in hydrogen-like systems. eOnly the nuclear ring currents of the central nucleus Os are considered. The molecular properties of the OsH4molecule in the first triply degenerate pseudorotational states assigned to antisymmetric stretches (a) and bends (b) are adopted from ref 26.

Figure 1. Electronic (i = e) and nuclear (i = n, A) ring currentdensities on the x/y-plane |jϕ,Ψ±

i (ρi,θi=π/2,t=0)| of the 2p± orbitals (eq89, panel a) and of the central nucleus A in the first triply degeneratepseudorotational states |v1l±1⟩ of the tetrahedral molecule AB4 (eq 156,panel b), normalized to its maxima in the center of mass frame, versuselectronic/nuclear radius ρi in units of mean ring current radii in thecenter of mass frame R1,Ψ±

i (eqs 131 and 159). The colored curvescorrespond to different initial translational wavepacket widths σ = σP =σZ in units of R1,Ψ±

i , i.e., 0.0 (black), 0.2 (red), 0.4 (green), 0.6 (blue),

0.8 (magenta), and 1.0 (orange). The ring current densities for σ = 0(black) correspond to the ones in the center of mass frame (eqs 90and 148). For 2p± orbitals (panel a), the curves for the normalizedelectronic and nuclear ring current densities are identical due to scalingof the radii. For the AB4 molecule (panel b), the curves for thenormalized ring currents densities of the central nucleus A do notdepend on the molecular properties of the AB4 molecule again due toscaling of the radii. Furthermore, these curves are identical for the firsttriply degenerate antisymmetric stretches and bends (see also ref 26).

Figure 2. Electric ring currents |IΨ±

i (t=0)| (eqs 119 and 161, panel a),

mean ring current radii R1,Ψ±

i (t=0) (eqs 130 and 162, panel b), and

induced magnetic fields at the ring center |BΨ±

i (0,t=0)| (eqs 141 and

163, panel c) of the 2p± orbitals (i = e, n, black) and of the centralnucleus A in the first triply degenerate pseudorotational states |v1l±1⟩of the tetrahedral molecule AB4 (i = A, red), normalized to its values inthe center of mass frame σ = 0, versus initial translational wavepacketwidths σ = σP = σZ in units of mean ring current radii in the center ofmass frame R1,Ψ±

i (eqs 131 and 159).

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induced magnetic fields at the ring center |BΨ±

i (0,t)| (eq 141and 163), are shown in Figures 3−5. Exemplarily, Figures 3−5show the properties for the electronic and nuclear ring currentsin 2p± orbitals of the hydrogen atom 1H, and for the ringcurrents of the central nucleus Os in the first triply degeneratepseudorotational states (assigned to bends) |v1l±1⟩ of thetetrahedral molecule OsH4, respectively. At the initial time t =0, the corresponding values depending on the initial transla-tional wavepacket widths σ are already shown in Figure 2.Because the wavepacket width σ(t) increases with time t, thegeneral trend in Figure 2 is similar to the one in Figures 3−5;i.e., the electric ring currents and induced magnetic fieldsdecrease with increasing t and the mean ring current radiiincrease with t, as expected. Furthermore, for small initialwavepacket widths σ, the electric ring currents and induced

magnetic fields are large at t = 0 but they decrease fast. Theincrease effects on mean ring current radii for small σ are alsoobserved in Figures 3−5.It is recognized in Figure 3 that the decrease of the electronic

ring currents in 2p± orbitals is on the femtosecond time scale,whereas the corresponding nuclear ring currents decrease muchfaster, i.e., on the zeptosecond time scale (Figure 4). It is due tothe Heisenberg uncertainty principle applied to translation.Because the nuclear mean ring current radius is very small (on thefemtometer scale), the very small translational wavepacket widthσ (in units of the ring current radius) leads to the rapid spreadingof the translational wavepacket and of the nuclear ring currentdensity on the zeptosecond time scale. For ring currents ofthe central nucleus Os in the pseudorotating molecule OsH4(Figure 5), the decrease of these ring currents is on the atto-second time scale. This decrease is thus slower than for nuclearring currents in 2p± orbitals of 1H, because the total mass M andthe mean ring current radius for OsH4 are larger (Table 2),leading to slower spreading of the translational wavepacket.For the electronic ring currents in 2p± orbitals of 1H

(Figure 3), the induced magnetic fields at 100 fs and 1 pscannot be larger than 0.19 and 0.02 T, respectively, which are,of course, much smaller than the ones at t = 0 with themaximum of 0.52 T (corresponding to the one in center ofmass frame). The maxima of the induced magnetic fields at t =100 fs and t = 1 ps for electronic ring currents in 2p± orbitals inhydrogen-like systems are listed in Table 1. For example, thegiant induced magnetic field for 12Al (1146 T) with properinitial translational wavepacket width is strongly reduced to86 T at t = 100 fs and to 4 T at t = 1 ps. However, the maximaof the induced magnetic fields at times t = 100 fs and t = 1 psincrease with the nuclear charge n and mass mn.In contrast to electronic ring currents, the nuclear ring

currents decrease much faster as discussed above. For 2p±orbitals of 1H (Figure 4), the induced magnetic fields at t = 1as with proper initial translational wavepacket width can bemaximized to only 8 T and they are very small compared to itsvalues at t = 0 with the maximum of 0.96 kT. The correspondingmaxima at t = 1 as for other hydrogen-like systems are listed in

Figure 3. Electric ring currents |I21±1e (t)| (eq 119, panel a), mean ring

current radii R1,21±1e (t) (eq 130, panel b), and magnetic fields at the

ring center |B21±1e (0,t)| (eq 141, panel c) induced by electronic ring

currents in 2p± orbitals of the hydrogen atom 1H versus time on thefemtosecond time scale. The colored thick curves correspond todifferent initial translational wavepacket widths σ = σP = σZ in units ofR1,21±1e (eq 131), i.e., 0.1 (black), 1.0 (red), 2.0 (green), and 3.0 (blue),

whereas the thin curves correspond to intermediate values of σ, i.e.,0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 (black), 1.2, 1.4, 1.6, 1.8 (red), and2.5 (green).

Figure 4. Electric ring currents |I21±1n (t)| (eq 119, panel a), mean ring

current radii R1,21±1n (t) (eq 130, panel b), and magnetic fields at the

ring center |B21±1n (0,t)| (eq 141, panel c) induced by nuclear ring

currents in 2p± orbitals of the hydrogen atom 1H versus time on thezeptosecond time scale. The colored thick curves correspond todifferent initial translational wavepacket widths σ = σP = σZ in units ofR1,21±1n (eq 131), i.e., 0.1 (black), 1.0 (red), 2.0 (green), and 3.0 (blue),

whereas the thin curves correspond to intermediate values of σ, i.e.,0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 (black), 1.2, 1.4, 1.6, 1.8 (red), and2.5 (green).

Figure 5. Electric ring currents |Iv1l±1Os (t)| (eq 161, panel a), mean ring

current radii R1,v1l±1Os (t) (eq 162, panel b), and magnetic fields at the ring

center |Bv1l±1Os (0,t)| (eq 163, panel c) induced by ring currents of the

central nucleus Os in the first triply degenerate pseudorotational states(assigned to bends) |v1l±1⟩ of the tetrahedral molecule OsH4 versustime on the attosecond time scale. The colored thick curvescorrespond to different initial translational wavepacket widths σ = σP= σZ in units of R1,v1l±1

Os (eq 159), i.e., 0.1 (black), 1.0 (red), 2.0 (green),and 3.0 (blue), whereas the thin curves correspond to intermediatevalues of σ, i.e., 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 (black), 1.2, 1.4, 1.6,1.8 (red), and 2.5 (green).

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Table 2. In particular, the induced magnetic fields for 7Li2+ at 1 asare smaller than 11 T; thus they are strongly decreased comparedto the ones at t = 0 with the maximum of 0.54 MT. Finally, forthe ring currents of the central nucleus Os in the pseudorota-tional states |v1l±1⟩ of OsH4, the maxima of the induced magneticfields at t = 1 as and t = 1 fs are also listed in Table 2. Forexample, for the triply degenerate bends of OsH4 (Figure 5), themaxima induced magnetic fields at t = 1 as and t = 1 fs are 607and 7 T, respectively. Thus, the nuclear ring currents are effectiveonly on atto- or zeptosecond time scales, whereas the electronicring currents are effective on the femtosecond time scale.

4. CONCLUSIONS

In this work, the theory of nonrelativistic electronic and nuclearring currents in degenerate excited states of free-moving atomicand molecular systems in the laboratory frame is established. Inparticular, the time-dependent wave function for the translationΨtr(R,t) is included in the total wave function (cf. eq 1) toinvestigate the effects of translation on electronic and nuclearring currents. Using the spherically zero-mean-velocity Gaussian-distributed translational wavepacket, depending on the time-dependent wavepacket width σ(t) and corresponding initialwidth σ = σ(t=0), the analytic formulas for the electronic andnuclear ring current densities (azimuthal components of thecurrent densities), mean angular momenta, electric ring currents,mean ring current radii, and induced magnetic fields at the ringcenter for 2p± orbitals of hydrogen-like systems are derived insection 2.3 (eqs 89, 102, 119, 130, and 141, respectively).Although the mean angular momenta are unaffected bytranslation, i.e., they are conserved, the electric ring currentsand induced magnetic fields decrease with increasing wavepacketwidth σ(t) or time t and the corresponding mean ring currentradii increase with σ(t) or t. As expected, in the limit of zerotranslation σ = 0, the corresponding formulas in the laboratoryframe are identical to the ones in the center of mass frame(section 2.2). Because me ≈ μ, the results for the electronic ringcurrents in 2p± orbitals coincide very well with the ones reportedin ref 20. Because the nucleus of the hydrogen-like system is notfixed and circulates about the center of mass, there are alsonuclear ring currents in atomic orbitals. Compared to theelectronic ring currents, the nuclear ring currents are stronger. Inthe center of mass frame, the nuclear (electronic) electric ringcurrents and corresponding induced magnetic fields at the ringcenter increase with nuclear charge n as n

3 ( n2) and n

4

( n3), respectively. The nuclear mean ring current radii decrease

with increasing nuclear mass mn as 1/mn and they are on thefemtometer scale (only few times larger than the nuclear radii).Therefore, the theory for point-nuclear ring currents in atomicorbitals is applied only to small nuclei of H, He, and Li.According to the Biot−Savart law, the induced magnetic fields atthe nuclear ring center increase linearly with nuclear mass mn.Thus, the induced magnetic fields at the ring center for 7Li2+ are0.54 MT in the center of mass frame and approach the hugemagnetic fields on neutron stars. In the laboratory frame,however, the electronic and nuclear ring currents in 2p± orbitalsdecay on femtosecond and zeptosecond time scales, respectively,due to fast spreading of the translational wavepacket, accordingto the Heisenberg uncertainty principle.The theory developed in this work is applied not only to ring

currents in hydrogen-like systems but also to ring currents inmany-electron atoms and ions or even to nuclear ring currentsin pseudorotating molecules. Using the expression for the ring

current densities in the center of mass frame (see, e.g., refs 18,20, and 26), and generalized formulas derived in section 2.4(see eqs 145−147), the electric ring currents, mean ring currentradii, and induced magnetic fields in the laboratory frame canbe evaluated analytically. For example, in the first triply de-generate pseudorotational states |v1l±1⟩ of the tetrahedralmolecule OsH4 (see ref 26), the ring currents of the centralnucleus Os in the laboratory frame (section 2.5) have propertiessimilar to those of the ring currents in 2p± orbitals but they decayon different time scales. In particular, the corresponding electricring currents and induced magnetic fields at the ring centerdecrease with increasing time t on the attosecond time scale.Hence, the decay of the ring currents of the heavy nucleus Os ismuch slower and faster than the one of the nuclear and electronicring currents in atomic orbitals, respectively.It was predicted in refs 8, 18, 20, 25, and 26 that electronic

and nuclear ring currents in degenerate excited states persist afterthe end of driven circularly polarized ultrashort (atto- orfemtosecond) laser pulses on long (pico- or nanosecond) timescales, before spontaneous emission occurs. It is indeed true forring currents in the center of mass frame but the strengths of thering currents in the laboratory frame decay on very short (zepto-,atto-, or femtosecond) time scales, due to fast spreading of thetranslational wavepacket with the initial wavepacket widthscomparable to small mean ring current radii. If the ring currentsshould persist on long (pico- or nanosecond) time scales, one hasto control translation of atomic and molecular systems, e.g., usingmagneto-optical traps for neutral atoms32 or Penning or Paultraps for charged ions.33,34 The atoms or molecules in these trapsare usually confined on nanometer spatial scale, which makesobservation of induced magnetic fields in the laboratory framevery challenging. To observe the translational effects in theexperiment, one could use the neutron beam that interacts withthe induced magnetic fields in the laboratory frame.19 Theneutron beam will be deflected and this deflection depends on thestrength and distribution of the induced magnetic fields in atomsand molecules. However, I note that the experimental observationof the induced magnetic fields in the center of mass frame may bepossible, on the basis of interactions of the induced magneticfields with spins of remaining electrons or nuclei in the same ring-current-carrying atomic or molecular system. Furthermore, it isalso a challenge for theoretists and experimentists to control smalltranslational wavepacket widths with resolution of small meanring current radii. However, for large mean ring current radii, e.g.,in degenerate high-lying np± (n≫ 2) atomic orbitals, for example4p± of Kr+ ion,35 or in electronic ring currents of large ring-shaped molecules, for example Mg−porphyrin,24 the relativelyweak electronic ring currents should persist on longer time scales.

■ APPENDIX A

With the substitution u = μrir′/(miσ(t)2), I get from eq 86

∫∫

σπ μ

θ θ θ θ

= ±ℏ

×

× ′ ′ ′ ′

ϕμ

σ

σ σ μ

πθ θ

±−

∞− − μ

j tm t

a r

u u

I u

r( , )( )

32(2 )e

d e

d sin e ( sin sin )

r t

t u r m t u a r

u

,21 1i

i

5i3 5

3/2 4 5i4

/2 ( )

0

3 [ ( ) /2 ] [ ( ) / ]

0

2 cos cos1 i

i2 2

2 2i2

i2

i

i

(164)

Using the representation of sin θ′ in the basis of the associatedLegendre polynomials sin θ′ = −P11(cos θ′) and the relation for

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the Bessel function I1(x) = −iJ1(ix), the θ′-integral with thehelp of ref 36 is evaluated as

∫∫

θ θ θ θ

θ θ

θ θ θ

θ

θ

′ ′ ′ ′

= ′ ′

× ′ ′

=

= −

πθ θ

πθ θ− ′

I u

P J u

P j u

j u

d sin e ( sin sin )

i d sin e

(cos ) (i sin sin )

(165)

2i (cos ) (i ) (166)

2i sin (i ) (167)

u

u0

2 cos cos1 i

0

i(i )cos cos

11

1 i

11

i 1

i 1

i

i

where j1(u) is the spherical Bessel function of the first kind

= −j uu

uu

u( )

sin cos1 2 (168)

hence

∫ θ θ θ θ

θ

′ ′ ′

= −

πθ θ′

⎜ ⎟⎛⎝

⎞⎠

I u

uu

uu

d sin e ( sin sin )

2 sincosh sinh

u

0

2 cos cos1 i

i 2

i

(169)

Then, the ring current densities (164) are

∫

∫

σ θπ μ

μ θπ

= ±ℏ

× −

×

= ±ℏ

× −

ϕ

μ

σ

σ σ μ

±

−

∞

− −

− +

∞− +

μ

j tm t

a r

u u u u u

cm r

u u u u u

r( , )( ) sin

16(2 )e

d ( cosh sinh )

e

(170)

sin8

e

d ( cosh sinh )e

(171)

r t

t u r m t u a r

b c

bu c

,21 1i

i5

i3 5

i3/2 4 5

i4

/2 ( )

0[ ( ) /2 ] [ ( ) / ]

5i

3/2i2

i4

(1/4 )

0

( )

i2 2

2 2i2

i2

i

2 2

2

where abbreviations (87) and (88) were used. Using sinh u =(eu − e−u)/2 and cosh u = (eu + e−u)/2, the integral in eq 171 isrewritten as

∫∫

∫

−

= + − +

= −

+ +

∞− +

∞− − − +

−∞

− − −

− − +

u u u u u

u u u u

u u u

u u

d ( cosh sinh )e12

d ( e e e e )e

(172)

12

e d (( )e

( )e )

(173)

bu c

u u u u bu c

c b u bc u

b u bc u

0

( )

0

( )

0

2 (2 1)

2 (2 1)

2

2

2 2 2

2 2

To evaluate this integral, I use the substitution v = bu + c ± (1/2b)and the indefinite integral

∫∫∫

∫π

=

=

=

=

− − ±

± − − ± − ±

± − + ±

± −

±

u

u

u

bv

bv

d e

e d e

(174)

e d e (175)

1e d e (176)

2e erf( ) (177)

b u bc u

c b b u b c b u c b

c b bu c b

c b v

c b

(2 1)

( (1/2 )) 2 ( (1/2 )) ( (1/2 ))

( (1/2 )) ( (1/2 ))

( (1/2 ))

( (1/2 ))

2 2

2 2 2 2

2 2

2 2

2

where erf(v) = 2/π1/2∫ dv e−v2

is the error function. Then, using theintegration by parts, I get from eq 173

∫

∫

∫

π

π

−

= − + −

− − + −

+ + + +

− + + +

∞− +

− −∞

∞

− +∞

∞

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎡⎣⎢⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠⎤⎦⎥⎥

⎡⎣⎢⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠⎤⎦⎥⎥

u u u u u

bu u bu c

b

u u bu cb

bu u bu c

b

u u bu cb

d ( cosh sinh )e

4e e ( ) erf

12

d (2 1) erf1

2

4e e ( ) erf

12

d (2 1) erf1

2

bu c

c c b

c c b

0

( )

( (1/2 )) 2

0

0

( (1/2 )) 2

0

0

2

2 2

2 2

(178)

Using the indefinite integrals involving the error function

∫∫

π

+ ±

=

= + −

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

u bu cb

bv v

bv v

d erf1

21

d erf( )

(179)

1erf( )

1e (180)v2

and

∫

∫ ∫

∫ ∫π

π

π

π

+ ±

= −±

= +

− −

−±

+

=− ±

+ −

−

−

−

−

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

u u bu cb

bv v v

c

bv v

vb

v v

bv v v

bv

c

bv v

v c

bv v

bv

d 2 erf1

22

d erf( )2( )

d erf( )

(181)

2erf( )

1e

2d erf( )

2d e

2( )erf( )

1e

(182)

2( )erf( )

1e

12

erf( ) (183)

b

v

v

b v

b v

2

12

2

2

2 2

12

2

12

2 2

2

2

2

2

the result of the integral (178) is

∫π

ππ

π

π

π

−

=

× − + × + −

+ − + −

+

× + − × + +

+ + − −

= − +

× − + + −

× + −

∞− +

− −

− + −∞

− +

− + +∞

−

−

+

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎡⎣⎢⎢⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎤⎦⎥⎥

⎡⎣⎢⎢⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎤⎦⎥⎥

⎡⎣⎢⎢⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠

⎤⎦⎥⎥

u u u u u

b

bc

bc bu c

b

bc

bu

b

bc

bc bu c

b

bc

bu

b bc

bc

cb b

cb

c

cb

cb

d ( cosh sinh )e

4e e

1 12

erf1

2

1 1 12

1 e

4e e

1 12

erf1

2

1 1 12

1 e

(184)

4e

1 12

erfc1

2e

1 12

erfc1

2e

2

(185)

bu c

c c b

bu c b

c c b

bu c b

c

c b

c b

0

( )

2( (1/2 ))

2

( (1/2 ))

0

2( (1/2 ))

2

( (1/2 ))

0

2

2

( (1/2 ))2

( (1/2 ))

2

2 2

2

2 2

2

2

2

2

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXP

where erfc(v) = 1 − erf(v) is the complementary error function.With this integral, eq 171 is then expressed as

μ θπ

π

= ±ℏ

− +

× − + + −

× + −

ϕ ±

−

− −

⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎡⎣⎢⎢⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎛⎝

⎞⎠

⎞⎠⎟

⎛⎝

⎞⎠

⎤⎦⎥⎥

j tc

b m r bc

bc

cb b

cb

c

cb

cb

r( , )sin

32e

1 12

erfc1

2e

1 12

erfc1

2e

2e

c

c b

c b c b

,21 1i

i

5i

2i2

i4

2

/2

/ (1/4 )

2

2 2

(186)

■ APPENDIX B

With the substitution u = μr′ cos θ′/(21/2miσP(t)), theθ′-integral of eq 118 is evaluated as

∫∫

∫

θ θ

θ θ

σμ

π σμ

μσ

′ ′

= ′ ′

=′

=′

′

πμ θ σ

μ σπ

μ θ σ

μ σ

μ σ

μ σ

μ σ

− ′ ′

− ′ ′ ′

− ′

− ′

′

− ′⎛⎝⎜

⎞⎠⎟

m tr

u

m tr

rm t

d sin e

e d sin e

(187)

2 ( )e d e (188)

2 ( )e erfi

2 ( )(189)

r m t

r m t r m t

P r m t

r m t

r m tu

P r m t

P

0

sin /2 ( )

/2 ( )

0

cos /2 ( )

i /2 ( )

/ 2 ( )

/ 2 ( )

i /2 ( )

i

P

P P

P

P

P

P

2 2 2i2 2

2 2i2 2 2 2 2

i2 2

2 2i2 2

i

i 2

2 2i2 2

where erfi(v) = −i erf(iv) = −2i/π1/2∫ 0ivdu e−u

2

= 2/π1/2∫ 0vdu eu

2

=

1/π1/2∫ −vv du eu

2

is the imaginary error function. Then, eq 118becomes

∫

σπ μ

μσ

= ∓ℏ

× ′ ′

× ′

μ

μ σ

± ±

∞− ′ − ′ μ

⎛⎝⎜

⎞⎠⎟

I t Ie m t

a

r r

rm t

( )( )

32 2

d e

erfi2 ( )

P

r m t r a

P

21 1i

21 1i i

5i2 5

0

[ /2 ( ) ] ( / )

i

P2 2

i2 2

(190)

Using the substitution u = μr′ /(21/2miσP(t)) and eqs 38, 39,and 88 (here σ(t) = σP(t)), it is simplified to

∫π= −± ±

∞− −I t I c u u u( ) (1 2 d e erfi( ))u cu

21 1i

21 1i 3

0

22

(191)

Using the indefinite integral

∫ = −− −u ud e12

eu u2 2

(192)

and the integration by parts, the u-integral in eq 191 isrewritten as

∫

∫

= −

+

∞− −

− −∞

∞− −

=

⎜ ⎟⎛⎝

⎞⎠

u u u

u

uv

v

d e erfi( )

12

e erfi( )

12

d ed

de erfi( )

u cu

u cu

u cv

v u

0

2

2

0

0

2

2

2

2

(193)

Applying the L’Hospital’s rule, d/du erfi(u) = (2/π1/2)eu2

, anderfi(0) = 0, the first term vanishes and the integral becomes

∫∫ ∫π

π

= −

= + −

∞− −

∞−

∞− −

u u u

u c u u

cc c

d e erfi( )1

d e d e erfi( )

(194)

12

[1 e Ei( )] (195)

u cu

cu u cu

c

0

2

0

2

0

2

2 2

2

2

2

where in the last step ref 37 was used and Ei(v) = −∫ −v∞du e−u/u

is the exponential integral. Thus, eq 191 is then written as

= − + −± ±I t I c c c( ) (1 [1 e Ei( )])c21 1i

21 1i 2 2 22

(196)

■ APPENDIX CWith the substitution u = −μ2r′2 sin2 θ′/(2mi

2σP(t)2) and

Kummer’s transformation30

= − −F a b u F b a b u( ; ; ) e ( ; ; )u11 11 (197)

the θ′-integral of eq 129 is evaluated as

∫

∫

∫∫

θ θ μ θσ

θ θ

μ θσ

σμ

σμ

μσ

′ ′ × + ′ ′

= ′ ′

× + ′ ′

= −′

×− −

× + −

= −′

×− −

× −

= − − ′

πμ θ σ

πμ θ σ

μ σ

μσ

μ σ

μσ

− ′ ′

− ′ ′

− ′

′

− ′

′

⎜ ⎟

⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

F k rm t

F k rm t

m tr

u u

u

F k u

m tr

u u

u

F k u

F k rm t

d sin e2

1;2;sin

2 ( )

2 d sin e

21;2;

sin2 ( )

(198)

i2 ( )

de

21;2;

(199)

i2 ( )

d

12

;2;

(200)

43

12

;52

;2 ( )

(201)

r m t

P

r m t

P

P

r m t

u

rm t

P

r m t

rm t

P

0

3 sin /2 ( )11

2 2 2

i2 2

0

/23 sin /2 ( )

11

2 2 2

i2 2

i3

0

/2 ( )

2 ( )

11

i3

0

/2 ( )

2 ( )

11

11

2 2

i2 2

P

P

P

P

P

P

2 2 2i2 2

2 2 2i2 2

2 2i2 2

2 2

i2 2

2 2i2 2

2 2

i2 2

(see ref 31). Hence, I get from eq 129 (k > −2,k ≠ 0)

∫

σμ

π σ

μσ

= ±ℏ Γ +

× ′ ′ − − ′

μ

±

±

∞− ′ μ

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎥

R t

te k

I t m a t

r r Fk r

m t

( )

2 ( )( /2 1)

96 ( ) ( )

d e 12

;52

;2 ( )

k

PP

r a

P

k

,21 1i

i5

21 1i

i2 5 2

0

4 /11

2 2

i2 2

1/

(202)

Using the substitution u = μr′/(21/2miσ(t)) and eqs 38, 39, 88,and 119 (here again σ(t) = σP(t)), the expression for the mean

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXQ

ring current radii (202) is then simplified to (k > −2,k ≠ 0)

∫

μ= Γ +

− + −

× − −

μ±

∞− ⎜ ⎟

⎡⎣⎢⎢

⎛⎝

⎞⎠⎤⎦⎥⎥

R ta c

mc k

c c c

u u Fk

u

( )2 8 ( /2 1)

3(1 [1 e Ei( )])

d e 12

;52

;

k c

cu

k

,21 1i

i

3

2 2 2

0

4 211

2

1/

2

(203)

Using eq 127, the derivative

∑

∑

∑

+ −

= −− !

Γ + Γ +Γ Γ + +

= − −!

Γ + + Γ +Γ Γ + +

= −+

−!

Γ + + Γ +Γ + Γ + +

= −+

+ + −

=

∞−

=

∞

=

∞

uF a b u

jj a ba j b

u

uj

j a ba j b

u

aub j

j a ba j b

u

aub

F a b u

dd

( , 1, )

2( 1)

( 1)( ) ( 1)( ) ( 1)

(204)

2( 1) ( 1) ( 1)

( ) ( 2)(205)

21

( 1) ( 1) ( 2)( 1) ( 2)

(206)

21

( 1; 2; ) (207)

j

jj

j

jj

j

jj

112

1

2 1

0

2

0

2

112

the recurrence relations30

+ + −

= − − + −

+ + + −

a F a b u

a b F a b u

b F a b u

( 1; 2; )

( 1) ( ; 2; )

( 1) ( ; 1; )

112

112

112

(208)

and

− = − + −

+ − ++

+ −

F a b ub u

bF a b u

b ab b

u F a b u

( ; ; ) ( ; 1; )

1( 1)

( ; 2; )

112

2

112

211

2

(209)

I get

+ −

= + −+ + −

= + −

−+

+ + −

= − + −

+ − ++

+ −

= −

−

−

+

−

+

−

u bu F a b u

u F a b u

bu

uF a b u

u F a b uau

b bF a b u

b ub

u F a b u

b ab b

u F a b u

u F a b u

dd

12

( , 1, )

( , 1, )1

2d

d( , 1, )

(210)

( , 1, )

( 1)( 1; 2; )

(211)

( , 1, )

1( 1)

( ; 2; )

(212)

( ; ; ) (213)

b

b

b

b

b

b

b

b

211

2

2 111

2

211

2

2 111

2

2 1

112

22 1

112

2 111

2

2 111

2

Thus, the corresponding indefinite integral is

∫ − = + −−u u F a b ub

u F a b ud ( ; ; )1

2( ; 1; )b b2 1

112 2

112

(214)

Using the integration by parts, the u-integral in eq 203 is thenrewritten as

∫

∫

− −

= − −

+ − −

∞−

−∞

∞−

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

u u Fk

u

u Fk

u

cu u F

ku

d e 12

;52

;

15

e 12

;72

;

25

d e 12

;72

;

cu

cu

cu

0

4 211

2

5 211

2

0

0

5 211

2

(215)

The first term vanishes in both limits u = 0 and u → ∞ and theintegral in the second term after Kummer’s transformation(197) is already known (see ref 31), hence

∫∫

− −

= +

= +

∞−

∞− −

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

u u Fk

u

cu u F

ku

cU

kc

d e 12

;52

;

25

d e2

52

;72

;

(216)

34

3,2

3, (217)

cu

u cu

0

4 211

2

0

5 211

2

2

2

where U(a,b,u) is the confluent hypergeometric function of thesecond kind,30 in particular

+ = − ΓΓ +

+

+ Γ + − − −+

⎜ ⎟ ⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝

⎞⎠

Uk

ck

kF

kc

kc

Fk k

c

3,2

3,( /2)

( /2 3)3;

23;

( /2 2)2

12

;2

1;k

211

2

4 112

(218)

Thus, eq 203 for k > −2 and k ≠ 0 is then expressed as

μ= Γ + +

− + −μ

±

⎡⎣⎢⎢

⎤⎦⎥⎥R t

a c

mc k U k c

c c c( )

2 2 ( /2 1) (3, /2 3, )

1 [1 e Ei( )]k c

k

,21 1i

i

4 2

2 2 2

1/

2

(219)

■ APPENDIX D

Using the substitutions u = μr′r″/(miσ(t)2), v = μr″/

(21/2miσ(t)) and the θ′-integral (169), I get from eq 139

∫

∫

∫

μπ μ

θ θ ψσ

μθ ϕ

= =ℏ

× ″ ″ ″ ″

× −

π

∞−

∞−⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

t tem m

v

m tv

uu

uu

u

B r r

e

( ( ), ) d e

d sin2 ( )

, ,

dcosh sinh

e

nlmv

nlm

u vz

ii 0

0 i i2 0

0

i2

0 2/4

2

2 2

(220)

With the Taylor series of the hyperbolic functions30

∑=+ !=

∞ +u

uj

sinh(2 1)j

j

0

2 1

(221)

∑=!=

∞

uu

jcosh

(2 )j

j

0

2

(222)

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXR

the substitution w = u/(2v), the definition (108), and the

duplication formula for the Gamma function30

πΓ = Γ Γ +

−⎜ ⎟⎛⎝

⎞⎠j j j(2 )

2( )

12

j2 1

(223)

Γ(j+1) = j! for integers j, Γ(3/2) = π1/2/2, the defintion for the

confluent hypergeometric function of the first kind (127), and

the corresponding relation to the error function30

π= − ⎜ ⎟

⎛⎝

⎞⎠u

uF uerf( )

2e 1;

32

;u11

22

(224)

the u-integral in eq 220 is then evaluated as

∫

∫

∫

∑

∑

∑

∑

π

π

−

=!

−+ !

×

=Γ +

=Γ +

=!

Γ + ΓΓ Γ +

−

= −

= −

∞−

=

∞

∞− −

=

∞ ∞− −

=

∞

=

∞

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢

⎤⎦⎥

⎛⎝

⎞⎠

uu

uu

u

j j

u u

j vj

w w

vj

jj

jv

F v

vv

dcosh sinh

e

1(2 )

1(2 1)

d e

(225)

2 (2 )(2 2)

d e (226)

2 ( 3/2)(227)

1 ( 1) (3/2)(1) ( 3/2)

1 (228)

1;32

; 1 (229)

2e erf( ) 1 (230)

u v

j

j u v

j

jj w

j

j

j

j

v

0 2/4

0

0

2 1 /4

1

2

0

2 1

1

2

0

2

112

2 2

2 2

2

2

hence

∫

∫

μμ π

θ θ ψσ

μθ ϕ

=

=ℏ

−

× ″ ″ ″ ″π

∞−⎡

⎣⎢⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

t tem m

vv

vv

m tv

B r r

e

( ( ), )

2d

1erf( )

2e

d sin2 ( )

, ,

nlm

v

nlm z

ii 0

0 i i2 0

0

i2

2

(231)

or

∫

∫

μμ

μσ

μπ σ

θ θ ψ θ ϕ

=

=ℏ

″″

″

− ″

× ″ ″| ″ ″ ″ |

μ σ

π

∞

− ″

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

t t

em mr

rr

m t

rm t

r

B r r

e

( ( ), )

2d

1erf

2 ( )

2( )

e

d sin ( , , )

nlm

r m t

nlm z

ii 0

0 i i2 0 i

i

/2 ( )

0

2

2 2i2 2

(232)

or with eq 26

∫

∫

μμ

μσ

μπ σ

θ θ θ

= = ″ ″

− ″

× ″ ″ ″ ″

μ σ

π

ϕ

∞

− ″

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

t tem

rr

m t

rm t

d j r

B r r

e

( ( ), )2

d erf2 ( )

2( )

e

sin ( , )

nlm

r m t

nlm z

ii 0

0 i i

0 i

i

/2 ( )

0

2,

2 2i2 2

(233)

■ APPENDIX E

With eq 117 and the θ″-integral31

∫ θ θ″ ″ =π

d sin430

3

(234)

I obtain from eq 140

∫

μπμ

μσ

μπ σ

= = ±ℏ

× ″ ″ ″

− ″

μ

μ σ

±

∞− ″

− ″

μ⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

t te m

a

r rr

m t

rm t

B r r

e

( ( ), )96

d e erf2 ( )

2( )

e

r a

r m tz

21 1i

i 00 i

5i

2 5

0

/

i

i

/2 ( )2 2i2 2

(235)

Using the substitution u = μr″/(21/2miσ(t)) and eqs 55−57 and

88, it is simplified to

∫ π

=

= = −

±

±

∞− −⎡

⎣⎢⎤⎦⎥

t t

c u u uu

B r r

B r 0

( ( ), )

4 ( ) d e erf( )2

ecu u

21 1i

i 0

221 1i

i0

2 2

(236)

Using the integration by parts and the indefinite integrals

∫ π= +− −u u cd e2

e erf( )u cu c22 2

(237)

∫ π+ = + + + − +u u c u c u cd erf( ) ( ) erf( )

1e u c( )2

(238)

∫

π

+

= − − + + − − +⎜ ⎟⎛⎝

⎞⎠

u u u c

u c u cu c

d 2 erf( )

12

erf( ) e u c2 2 ( )2

(239)

(cf. eqs 177, 180, and 183), the integral in eq 236 is evaluated

as

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp305318s | J. Phys. Chem. A XXXX, XXX, XXX−XXXS

∫∫

∫

∫∫

∫

∫

∫

∫

π

π

π

π

π

π

π

−

=

−

= −

+

+

− +

+ +

= −

+

+ +

− +

− +

+ +

= + − +

−

− + +

+ −

= − +

+ +

∞− −

∞−

∞− −

−∞

∞−

∞− −

∞

∞

−∞

∞− −

∞

∞

∞

∞

∞ ∞

− −∞

∞

− −∞

⎜ ⎟

⎡⎣⎢

⎤⎦⎥

⎛⎝

⎞⎠

u u uu

u u u

u u

cu u

cu u

cu u

u u c

u u u c

cu

cu

cu u c

cu u c

u u c

u u u c

cu c u c

c

c u c

u c

cc c c

cc

d e erf( )2

e

d e erf( )2

d e

(240)

12

e erf( )

12

d e erf( )1

d e

e erf( )

2e d erf( )

(241)

14

e erf( )

12

d e

12

e erf( )

12

e d erf( )

e erf( )

2e d erf( )

(242)

14

e erf( )12

e erf( )

12

e

12

e erf( )

e

(243)

14

[1 4 ( 1)]e erfc( )

2 12

(244)

cu u

cu

u cu

cu

cu

u cu

c

c

cu

u cu

c

c

c

c

c c

u cu

c

u cu

c

0

2

0

2

0

2 2

2

0

0

2

0

2

2

0

0

22

0

2 0

2

0

02

0

0

20 0

2

0

2

0

2

0

22 2

2

2

2

2

2

2

2

2

2

2

2

2 2

2

2

2

2

Thus, I get from eq 236

π

= = =

× − + + +

± ±

⎧⎨⎩⎫⎬⎭

t t

c c cc

c

B r r B r 0( ( ), ) ( )

[1 4 ( 1)]e erfc( )2

(2 1)c

21 1i

i 0 21 1i

i

2 2 22

(245)

■ AUTHOR INFORMATION

Corresponding Author*E-mail: barth@mbi-berlin.de.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

I thank Professor J. Manz (FU Berlin) for stimulating discussions.Financial support by the Deutsche Forschungsgemeinschaft(DFG, project Sm 292/2-1) is gratefully acknowledged.

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