Translational Effects on Electronic and Nuclear Ring Currents - The

Article pubs.acs.org/JPCA

Translational Effects on Electronic and Nuclear Ring Currents Ingo Barth* Max-Born-Institut, Max-Born-Strasse 2A, 12489 Berlin, Germany ABSTRACT: In previous works, it was predicted that electronic and nuclear ring currents in degenerate excited states of atomic and molecular systems persist after the end of driven circularly polarized atto- or femtosecond laser pulses on relatively long time scales, often on pico- or nanosecond time scales, before spontaneous emission occurs. Although this conclusion is true in the center of mass frame, it is not true in the laboratory frame, 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 shows that the ring currents and the corresponding induced magnetic fields in the laboratory frame persist on shorter timecales due to spreading of translational wavepackets. The electronic ring currents in 2p± orbitals of the hydrogen-like systems decay on the femtosecond time scale, but the corresponding nuclear ring currents with giant induced magnetic fields (for example up to 0.54 MT for 7Li2+) and very small mean ring current radii on the femtometer scale decay on the very short, zeptosecond time scale, according to the Heisenberg uncertainty principle. The theory is also applied to ring currents in many-electron atoms and ions as well as to nuclear ring currents in pseudorotating molecules. For example, in the first triply degenerate pseudorotational states |v1l±1⟩ of the tetrahedral molecule OsH4, the ring currents of the heavy central nucleus Os decay on the attosecond time scale.

1. INTRODUCTION Nonvanishing electronic and nuclear current densities occur not only in nonstationary states, corresponding to electron circulations,1−5 molecular vibrations,6 motors,7 pseudorotations,8,9 rotations, torsions,10 isomerizations, proton transfers,11,12 pericyclic rearrangements,13,14 or chemical reactions15−17 but also in stationary states.18,19 The existence of stationary current densities is guaranteed in electronic degenerate states of atoms, ions,20,21 and linear22,23 or ring-shaped molecules24 or in nuclear degenerate states of pseudorotating molecules.25,26 In these degenerate states, the electronic or nuclear wave functions are not purely real or imaginary but complex, e.g., presenting linear combinations of two real wave functions, |Ψ±⟩ = |Ψx⟩ ± i|Ψy⟩. If the ground state is nondegenerate, left or right circularly polarized ultrashort (atto- or femtosecond) laser pulses with suitable laser parameters can induce population transfer 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 lifetimes but they persist on relatively long (pico- or nanosecond) time scales, before spontaneous emission occurs. Therefore, it was predicted that the properties of the ring currents, including electric ring currents, mean angular momenta, mean ring current radii, and induced magnetic fields, do not change significantly on these timescales, even after the end of the circularly polarized laser pulses. This prediction is true only in the center of mass frame. In the laboratory frame, the translational wave function corresponding to the center of mass motion has to be included © XXXX American Chemical Society

in the total wave function. Though the mean angular momenta are conserved, the remaining properties of the ring currents, i.e., electric ring currents, mean ring current radii, and induced magnetic fields in the laboratory frame, depend on translational wavepacket width. I will show that due to center of mass motion the electronic and nuclear ring currents decay on shorter (zepto-, atto-, and femtosecond) time scales, depending on the size of the mean ring current radii. In section 2, the analytic formulas for the ring current densities (azimuthal components of the current densities), electric ring currents, mean angular momenta, mean ring current radii, and induced magnetic fields are derived. These formulas are defined in section 2.1 for the electronic and nuclear ring currents in orbitals of hydrogen-like atoms. Section 2.2 considers the ring currents in the center of mass frame. Then, I use spherically Gaussian-distributed translational wavepacket in section 2.3 to obtain analytical formulas for 2p± orbitals in the laboratory frame, depending on the translational wavepacket width. These formulas can be then generalized for application to ring currents in many-electron atoms and ions (section 2.4) as well as to nuclear ring currents in pseudorotating molecules (section 2.5). Section 3 discusses the results and section 4 summarizes this work. Special Issue: Jörn Manz Festschrift Received: May 31, 2012 Revised: August 8, 2012

A

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

The Journal of Physical Chemistry A

Article

2. THEORY 2.1. Basic Equations. In this work, I extend the theory of electronic ring currents in atomic orbitals20 to include nuclear ring currents and translational effects on ring currents. However, the relativistic effects of the electronic and nuclear spins on ring currents are not considered. In degenerate atomic orbitals carrying nonzero angular momenta, not only the electrons but also the nucleus circulate about the center of mass. For the hydrogen atom or one-electron ions with atomic number A , the total electronic and nuclear wave function is written as Ψ(R,r,t ) = ψtr(R,t ) ψnlm(r)

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

R ki ,nlm(t )

(1)

=

μ0 Aie 4π

r0(t ) =

(5)

i ρnlm (ri) =

i

=

i (ri,t ) dri ∭ ri × jnlm

i (ri,t ) ·d Si ∬ jnlm

(11)

(12)

δ(R)

(13)

∭ mi 3 μ3

⎛ μr μrj ⎞ ⎟⎟|ψnlm(r)|2 drj δ ⎜⎜ i + m m ⎝ j i⎠

⎛m r ⎞ ψnlm⎜ i i ⎟ ⎝ μ ⎠

(14)

2

(15)

The corresponding current densities (6) are evaluated as i jnlm (ri)

(16)

⎛ μr μrj ⎞ iℏ ⎟ = δ ⎜⎜ i + 2m i m i ⎟⎠ ⎝ mj * (r) − ψ * (r)∇r ψ (r)) drj × (ψnlm(r)∇ri ψnlm nlm i nlm

∭

(7)

and the electric ring currents18,20 i Inlm (t ) = Aie

dr′

(6)

one can calculate various physical properties of the ring currents, including the mean angular momenta18,26 ⟨Li(t )⟩nlm = m i

|r0(t ) − r′|3

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

∭ (Ψ(R,r,t )∇r Ψ*(R,r,t )

− Ψ*(R,r,t )∇ri Ψ(R,r,t )) drj ≠ i

∭

i jnlm (r′,t ) × (r0(t ) − r′)

∭ R |ψtr(R,t )|2 dR

ψtr(R) =

With the electronic (i = e) and nuclear (i = n) current densities iℏ 2m i

(10)

2.2. Center of Mass Frame. In this section, I consider the ring currents in the center of mass frame (atomic or molecular frame) or equivalently in the laboratory frame without translation of atoms or ions. In this case, the square-integrable timeindependent wave function for the translation can be defined as18,26

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

i jnlm (ri,t ) =

⎞1/ k dS i ⎟ ⎠

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

(2)

where aμ = 4πε0ℏ2/(μe2) is the Bohr radius for the reduced mass μ = memn/M, me is the electron mass, mn is the nuclear mass, and M = me + mn is the total mass. Because ring currents in atomic orbitals are toroidal,18,20 I also use cylindrical coordinates (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 relative motion r are connected to the ones of the electron re and of the nucleus rn by mr m r R= ee + n n (3) M M r = re − rn (4)

∭ |Ψ(R,r,t )|2 drj≠i

∬

i ρik jnlm (ri,t )

i Bnlm (ri=r0(t ),t )

l

i ρnlm (ri,t ) =

⎛ Ae =⎜ i i ⎝ Inlm(t )

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

⎛ 2A ⎞3 (n − l − 1)! ⎜⎜ ⎟⎟ e−Ar / naμ ⎝ naμ ⎠ 2n(n + l)! ⎛ 2Ar ⎞ ⎛ 2Ar ⎞ ⎟⎟ Ln2−l +l −1 1⎜⎜ ⎟⎟Ylm(θ ,ϕ) × ⎜⎜ ⎝ naμ ⎠ ⎝ naμ ⎠

(9)

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

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

Aie i Inlm (t )

i Tnlm (t ) =

=

(8)

with Ae = −1 and An = A . Here, I consider only ring currents with its axis of symmetry as z-axis. Hence, Si is the half plane perpendicular to the x/y-plane at fixed arbitrary azimuthal angle ϕi with the domains ρi ∈ [0,∞) and zi ∈ (−∞, ∞) or with the domains ri ∈ [0, ∞) and θi ∈ [0, π], thus dSi = dρi dzi

iℏm i 2 * (r) (ψ (r)∇ri ψnlm 2μ3 nlm

(17)

* (r)∇r ψ (r)) − ψnlm i nlm m jrj=−mi ri

=

mi 2 mℏ |ψnlm(r)|2 ∇ri ϕ μ3 m r =−m r j j

B

(18)

i i



Article

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 the relative angle ϕ with respect to re and rn are ⎛ x ∇re ϕ = sgn(y)∇re arccos⎜⎜ 2 2 ⎝ x +y x2 + y2

=−

y

∇re

x + y ⎛⎜ ∂ =− ⎜ ∂x y ⎝ e 2

=

2

⎞ ⎟ ⎟ ⎠

and

x

(20) ⎞ x e⎟ 2 2 y⎟ x +y ⎠

x

∂ ex + ∂ye x2 + y2

− y ex + x ey

tot ⟨L⟩nlm = ⟨Le⟩nlm + ⟨Ln⟩nlm = mℏez

(21)

and by analogy y ex − x ey ∇rn ϕ = 2 x + y2

i Inlm =

μ −yi ex + x i ey μ eϕ(ϕi) = = 2 2 m i x i + yi m i ρi

∬ jnlm(r) dS

i Inlm = −AiInlm

n Inlm = −AInlm = sgn(m)

⟨Li⟩nlm =

mi mℏ μ2

⎛m r ⎞ ψnlm⎜ i i ⎟ ⎝ μ ⎠

mi 2 mℏ μ2

⎛m r ⎞ ψnlm⎜ i i ⎟ ⎝ μ ⎠

2

=

∭ ∭

μ = mℏ mi μ = mℏez mi

∭ ρnlmi (ri) driez

×

ri × eϕ(ϕi)

driez

ρi

dri

⟨L ⟩nlm

μ = mℏez me

(40)

(41)

i.e., the nuclear electric ring current is exactly A times larger than the electronic one and they are opposite. It also confirms that the mean periods for the electron and for the nucleus (9) i Tnlm = |(

∬ jnlm(r) dS)−1| =

e |Inlm|

(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 in the center of mass frame are

(28)

(29)

R ki ,nlm

(30) (31)

⎛ e m2 i = ⎜⎜ − 2 I μ ⎝ nlm

∬

⎞1/ k ⎛ m i ri ⎞ ρi jnlm ⎜ ⎟ ·d Si ⎟⎟ ⎝ μ ⎠ ⎠ k

(43)

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

hence e

eℏ A3 2πμaμ 2 n3

n Inlm = −A e Inlm

(27)

2

(39)

thus the ratio of nuclear and electronic electric ring currents is

2.2.2. Mean Angular Momenta. The electronic (i = e) and nuclear (i = n) mean angular momenta (7) are 2

eℏ A2 2πμaμ 2 n3

For the nucleus, I get

I obtain the important relations for the electron (i = e) and nuclear (i = n) current densities μ

(38)

Hence, the electric ring current for the electron Ienlm is exactly equal to the one for the reduced mass Inlm, i.e.20

(26)

⎛ m i ri ⎞ ⎟ ⎝ μ ⎠

(37)

it becomes (24)

(25)

2 nlm ⎜

(36) 20

Inlm = −e

e Inlm = Inlm = −sgn(m)

mℏ jnlm (r) = |ψ (r)|2 eϕ(ϕi) μρ nlm

j

(35)

With the definition for the reduced mass

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

mi 2

⎛m r ⎞ jnlm ⎜ i i ⎟ dSi ⎝ μ ⎠

∬ jnlm(r) dS

2

⎛m r ⎞ m mℏ = 2i ψnlm⎜ i i ⎟ eϕ(ϕi) ⎝ μ ⎠ μ ρi

i jnlm (ri) =

∬

i Inlm = Aie

(23)

where in the last step ρi2 = xi2 + yi2 and −yiex + xiey = ρi(−sin ϕi ex + cos ϕi ey) = ρieϕ(ϕi) were used. With eqs 18 and 24, the electronic (i = e) and nuclear (i = n) current densities (6) i jnlm (ri)

mi 2 Aie μ2

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

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) ∇ri ϕ|mjrj=−mi ri

(34)

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

(22)

x2 + y2

(33)

In eq 29, the relation ri × eϕ(ϕi) = (ρieρ(ϕi) + ziez) × eϕ(ϕi) = ρiez − zieρ(ϕi) was used and the ρ-component does not contribute to the integral because of ϕi-oscillations in eρ(ϕi) = cos ϕiex+ sin ϕiey. As expected, the total angular momentum is

(19)

x2 + y2

μ mℏez mn

⟨Ln⟩nlm =

R ki ,nlm

(32) C

μ⎛ e = ⎜− m i ⎝ Inlm

∬

⎞1/ k ρ jnlm (r) dS⎟ ⎠ k

(44)



Article

me Bnlm(r=0) μ μ eme A3mℏ = − 02 3 3 ez πμ aμ n l(2l + 1)(2l + 2)

With the definition for the reduced mass (cf. refs 18 and 20) R k ,nlm

⎛ e = ⎜− ⎝ Inlm

∬

⎞1/ k ρ jnlm (r) dS⎟ ⎠

Benlm(re=0) =

k

(45)

I get

and

μ R k ,nlm mi

(46)

and in particular for the electron μ R k ,nlm R ke,nlm = me

(47)

R ki ,nlm =

Bzn,nlm(rn=0) Bze,nlm(re=0)

(48)

m = e mn

(49)

∭ Rδ(R) dR = 0

=−

=−

μ0 Aie 4π

∭

mi 2 μ0 Aie × μ2 4π mi μ0 Aie μ 4π

r′

|r′|

∭ ∭

dr′

jnlm (mi r′/μ) × r′ |r′|3

jnlm (r) × r 3

|r|

dr

μ0 e 4π

∭

jnlm (r) × r 3

|r|

(52)

−1 σP ⎛ 2 iℏt ⎞⎟ −P 2 /4(σP 2 + iℏt /(2M)) ⎜σ e P + 2π ⎝ 2M ⎠

ψtrP(P ,t ) =

(59)

(60)

(61)

(53)

and −1/2 ⎛ σ ⎞1/2 ⎛ iℏt ⎞⎟ ψtrZ(Z ,t ) = ⎜ Z ⎟ ⎜σZ 2 + ⎝ 2π ⎠ ⎝ 2M ⎠ 2

× e−(σZ pZ

dr

2

/ ℏ2) − [(Z − 2iσZ 2pZ / ℏ)2 /4(σZ 2 + iℏt /(2M ))]

(62)

(54)

The corresponding densities are

I obtain for l > 0 i Bnlm (ri=0) = −

(58)

With the mean momenta pP = 0 and pZ and the wavepacket widths σP and σZ at the initial time t = 0, the Gaussian wavepackets are defined as29

(51)

dr′

Amn me

ψtr(R,t ) = ψtrP(P ,t ) ψtrZ(Z ,t )

where eq 27 and the substitution r = (mi/μ)r′ were used. With the definition for the reduced mass20 Bnlm(r=0) =

(57)

is dominated by the magnetic field induced by the nuclear ring current. 2.3. Translational Effects. In this section, I consider the system (atom or ion) in the laboratory frame where the Gaussian-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 Gaussian wavepacket along the z-axis and the 2D Gaussian wavepacket perpendicular to the z-axis, i.e.

(50)

i jnlm (r′) × 3

=−

tot e n Bnlm (r=0) = Bnlm (re=0) + Bnlm (rn=0) m − Amn = e Bnlm(r=0) μ

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

ez

Compared to the magnetic field induced by the electronic ring current, the nuclear ring current with nuclear charge A and very small ring current radius induces, in fact, very strong magnetic field at the ring center with ratio of A/mnme . Therefore, the total induced magnetic field at the ring center (r = r0 = 0)

and it is obvious that the condition of the center of mass is satisfied for all orders k ≠ 0. The analytic expression for Rk=−1,nlm is found in eq 30 of ref 20 and equal to 4aμ/(πA ) for 2p± orbitals. The corresponding general formula for 2p± orbitals for orders k > −2 and k ≠ 0 is derived in section 2.3.4 (see eq 131). Thus, for k = 1, it is equal to 3πaμ/(4A ) for 2p± orbitals. In this work, the nucleus is considered as a point charge; therefore, the mean ring current radius for the nucleus must be much larger than the nuclear radius Rn ≈ r0A1/3 where r0 ≈ 1 fm =10−15 m is the 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 r0 =

πμ2 aμ3 n3l(2l + 1)(2l + 2)

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

The corresponding ratio is

R ke,nlm

Amn Bnlm(r=0) μ μ0 emn A 4m ℏ

n Bnlm (rn=0) = −

=

and for the nucleus μ R k ,nlm R kn,nlm = mn

R kn,nlm

(56)

Aim i Bnlm(r=0) μ

|ψtrP(P ,t )|2 = (55)

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

2 2 1 e−P /2σP(t ) 2 2πσP(t )

(63)

and |ψtrZ(Z ,t )|2 = D

2 2 1 e−(Z − pZ t / M) /2σZ(t ) 2π σZ(t )

(64)



Article

where σP(t) and σZ(t) are the time-dependent wavepacket widths defined as ⎛ ℏt ⎞2 ⎟ 1+⎜ 2 ⎝ 2MσP ⎠

σP(t ) = σP

Using the substitutions r′i = ri − R, x′i = ρ′i cos ϕ′i , y′i = ρ′i sin ϕ′i , and −sin ϕ′i ex + cos ϕ′i ey = eϕ(ϕ′i ), the ring current densities are rewritten as jϕi ,nlm (ri,t ) =

(65)

and

mi μ2

∭

|ψtr(ri−r′i,t )|2 2

⎛ ℏt ⎞2 ⎟ σZ(t ) = σZ 1 + ⎜ 2 ⎝ 2MσZ ⎠

(66)

i (ri,t ) jnlm

iℏ = 2m i

∭

⎛ m r′ ⎞ eϕ(ϕi′) ·eϕ(ϕi) dr′i × ψnlm⎜ i i ⎟ ρi′ ⎝ μ ⎠

(72)

i (r′i) ·eϕ(ϕi) dr′i ∭ |ψtr(ri−r′i,t )|2 jnlm

(73)

or using eq 25

Therefore, the electronic and nuclear ring currents depend on the Gaussian widths σP(t) and σZ(t) and on the translational momentum pZ. 2.3.1. Ring Current Densities. It is indeed possible to obtain an analytic formula for the electronic and nuclear probability densities (5) but the main propose of this work is to evaluate integrals for electronic (i = e) and nuclear (i = n) current densities (6) (i ≠ j)

jϕi ,nlm (ri,t ) =

i.e., the ring current densities in the laboratory frame are obtained by averaging the ring current densities in the center of mass frame over the translational motion. Then, using the substitution r′ = (mi/μ)r′i and the relation eϕ(ϕi′) ·eϕ(ϕi) = ( −sin ϕi′ex + cos ϕi′ey)

2

* (r) |ψtr(R,t )| (ψnlm(r)∇ri ψnlm

+

iℏ 2m i

∭ |ψnlm(r)|2 (ψtr(R,t )∇r ψtr*(R,t ) i

− ψtr*(R,t )∇ri ψtr(R,t )) drj

jϕi ,nlm (ri,t )

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

=

2μ

(68)

=

mi 2 μ3

=

* (r) ∭ |ψtr(R,t )| (ψnlm(r)∇r ψnlm i

eϕ(ϕi)

mjrj= M R − m i ri

i

mjrj= M R − m i ri

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) ∇ri ϕ|mjrj= M R − mi ri

∭

cos(ϕ′ − ϕi)

2

dr′

ρ′

(77)

⎛ μρ′ ⎞2 2μρi ρ′ cos(ϕ′ − ϕi) ρi 2 + ⎜ ⎟ − mi ⎝ mi ⎠

(79)

μz′ mi

jϕi ,nlm (ri,t ) =

(70)

dR

μ −(yi − Y )ex + (x i − X )ey = m i (x i − X )2 + (yi − Y )2

⎛ μr′ ⎞ ψtr ⎜ri − ,t ⎟ mi ⎠ ⎝

(80)

dr′ = ρ′ dρ′ dz′ dϕ′, and the fact that |ψnlm(r′)|2 does not depend on the azimuthal angle ϕ′, the ring current densities are expressed as

dR

∭ |ψtr(R,t )|2 |ψnlm(r)|2 (∇r ϕ)·

mℏ

mℏ = mi

(78)

Z = zi −

2

* (r)∇r ψ (r)) ·eϕ(ϕ ) − ψnlm i i nlm

(76)

2 2 ⎛ ⎛ μy′ ⎞ μx′ ⎞ ⎟ ⎟ + ⎜yi − ⎜x i − mi ⎠ mi ⎠ ⎝ ⎝

(69) 3

= cos(ϕi′ − ϕi)

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

(cf. eq 3), the ϕ-components of the current densities (denoted as ring current densities) are evaluated as iℏm i 2

(75)

× |ψnlm(r′)|2

P=

jϕi ,nlm (ri,t )

= sin ϕi′ sin ϕi + cos ϕi′ cos ϕi

I obtain from eq 72 (67)

mr MR − i i mj mj

(74)

× ( −sin ϕi ex + cos ϕi ey)

* (r)∇r ψ (r)) drj − ψnlm i nlm

rj =

mℏ

3/2

(2π )

∞

2 2 mℏ e−ρi /2σP(t ) 2 m i σP(t ) σZ(t ) 2

2

2

2

×

∫0

×

∫ dz′ e−(z −m ′/m −p t /M) /2σ (t)

dρ′ e−μ ρ ′ /2mi σP(t )

2

i

z

i

2

Z

Z

× |ψnlm(ρ′,z′,ϕ′)|2 ×

∫0

2π

× cos(ϕ′ − ϕi)

(71) E

2

dϕ′ e[μρi ρ ′ / miσP(t ) ]cos(ϕ ′− ϕi) (81)



Article

Using the substitution ϕ = ϕ′ − ϕi and the integral representation of the modified Bessel function of the first kind30 1 I1(z) = 2π

∫0

2π

dϕ e

z cos ϕ

cos ϕ

c=

Am i σ(t ) 2 μaμ

(88)

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

(82)

it yields jϕi ,21 ± 1 (ri,t ) = ±

2 2 mℏ jϕi ,nlm (ri,t ) = e−ρi /2σP(t ) 2 2π m i σP(t ) σZ(t ) ∞ μρi ρ′ ⎞ 2 2 2 2 ⎛ ⎟ dρ′ e−μ ρ ′ /2mi σP(t ) I1⎜ × 2 0 ⎝ m i σP(t ) ⎠

∞

∫−∞

2

dz′ e−(z i − μz ′ / mi − pZ t / M)

(83)

Therefore, the ring current densities do not depend on the azimuthal angle ϕi; i.e., they have cylindrical symmetry and vanish for m = 0. Because |ψnlm(ρ′,z′,ϕ′)|2 has the exponential function exp(−2A (ρ′2 + z′2)1/2/(naμ)), the z′- and ρ′-integrals have 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 the center of mass is zero pP = pZ = 0 and the translational wave function 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

jϕi ,21 ± 1 (ri) = ±

= ×

∫0

π

∫0

∞

dr′ r′e

−μ2 r ′2 /2m i 2σ (t )2

(84)

× ×

∫0 ∫0

dθ′ sin θ′e

⟨Li(t )⟩nlm = m i

sin θi

(90)

(91)

(jρi ,nlm (ri,t ) eρ(ϕi) + jzi ,nlm (ri,t )ez jϕi ,nlm (ri,t ) eϕ(ϕi))

∭ ρi jϕi ,nlm (ri,t ) driez

(93)

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

∭

⟨Li(t )⟩nlm = mℏ

(86)

With abbreviations σ (t ) b= 2 ri

−Am iri / μaμ

and the fact that all components of the current densities do not depend on the azimuthal angle ϕi, the mean angular momenta (7) are rewritten as

μrir ′ cos θi cos θ ′ / m iσ (t )2

⎛ μr r′ sin θ sin θ′ ⎞ i ⎟ × I1⎜ i m i σ(t )2 ⎠ ⎝

re 5 i

+ [z ijρi ,nlm (ri,t ) − ρi jzi ,nlm (ri,t )]eϕ(ϕi)

(85)

3 −[μ2 r ′2 /2m i2σ(t )2 ] − (Ar ′ / aμ) 2

(89)

= −z ijϕi ,nlm (ri,t ) eρ(ϕi) + ρi jϕi ,nlm (ri,t )ez (92)

dr ′ r ′ e π

64πμ aμ

+

2 2 ℏA5 e−ri /2σ(t ) 3/2 5 3 32(2π ) m i aμ σ(t )

∞

4

×

and the corresponding ring current densities (84) are jϕi ,21 ± 1 (ri,t ) = ±

ℏA5m i 3

i (ri,t ) = (ρi eρ(ϕi) + z i ez) ri × jnlm

Although the θ′- and r′-integrals are analytically executable for all 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 5/2 1 ⎛A⎞ ⎜⎜ ⎟⎟ r e−Ar /2aμ sin θ e±iϕ ψ21 ± 1(r ,θ ,ϕ) = 8 π ⎝ aμ ⎠

2

(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 mean angular momenta (7) because these quantities are conserved in quantum mechanics. With

2 ⎛ μr r ′ sin θ sin θ ′ ⎞ i ⎟ dθ′ e μrir ′ cos θicos θ ′ / miσ(t ) I1⎜ i m i σ(t )2 ⎝ ⎠

× |ψnlm(r′,θ′,ϕ′)|2

ec

(Appendix A). Because of the appearance of the function sin θi in eq 89, the electronic and nuclear ring currents in 2p± orbitals in the laboratory frame are toroidal. In the limit of zero translational wavepacket width (σ(t) = 0), i.e., b = 0, c = 0, c/b = A miri/(μaμ), the second term as well as the third term of eq 89 vanish, because erfc(∞) = 0 and exp(−∞) = 0. With erfc(−∞) = 2, the current densities of the 2p± orbitals in the laboratory frame in the limit σ(t) = 0 are then equal to the ones in the center of mass frame

jϕi ,nlm (ri,t ) 2 2 mℏ e−ri /2σ(t ) × 2π m i σ(t )3

m i 2ri4

2 ⎞ ⎛1⎛ ⎛ 1 ⎞⎟ 1 ⎞⎟ c / b e + ⎜ ⎜c + − c ⎟ erfc⎜c + ⎝ ⎝ ⎠ 2b 2b ⎠ ⎠ ⎝b 2c −c 2 − (1/4b2)⎤ ⎥ e − ⎥⎦ πb

/2σZ(t )2

× |ψnlm(ρ′,z′,ϕ′)|2

2

32πb 2 ⎡⎛ 1 ⎛ ⎞ ⎛ 1 ⎞⎟ 1 ⎞⎟ −c / b e × ⎢⎜ ⎜c − + c ⎟ erfc⎜c − ⎝ ⎢⎣⎝ b ⎝ 2b ⎠ 2b ⎠ ⎠

∫

×

ℏc 5μ sin θi

× (87) F

∭

|ψnlm(r′)|2 ρ′

dr′

2

⎛ μr′ ⎞ ,t ⎟ cos(ϕ′ − ϕi) driez ρi ψtr ⎜ri − mi ⎠ ⎝

(94)



Article

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

1 (x′x i + y′yi ) ρ′ρi

(96)

μρ′2 ⎞ 1 ⎛ ⎜x′X + y′Y + ⎟ mi ⎠ ρ′ρi ⎝

(98)

=

μρ′ ⎞ 1⎛ ⎜P(cos ϕ′ cos Φ + sin ϕ′ sin Φ) + ⎟ mi ⎠ ρi ⎝

(99)

(100)

∭ |ψnlm(r′)|2 dr′∭ |ψtr(R,t )|2 dRez

Γ(j) =

=

2

2

∞

×

∫−∞ dz i e

×

/2σP(t )2

2


× =

⎛ μρ ρ′ ⎞ i ⎟ I1⎜ 2 m σ ⎝ i P (t ) ⎠

= (104)

2

=

∞

= =

2 σZ(t )

∫−∞ dz e

2π σZ(t )

Z

Z

−z

(106)

(107)

∞

du u 2j − 1e−u

2

(108)

/2σP(t )2

⎛ μρ ρ′ ⎞ i ⎟ I1⎜ 2 ⎝ m i σP(t ) ⎠

(110)

1 j !Γ(j + 2)

∫0

∞

⎛ μρ ρ′ ⎞2j + 1 2 2 i ⎟ e−ρi /2σP(t ) dρi ⎜ 2 ⎝ 2m i σP(t ) ⎠

∫0

∞

σP(t ) 2

j=0

⎞ 2j + 1 ⎛ μρ′ 1 ⎟ ⎜ j !Γ(j + 2) ⎝ 2 m i σP(t ) ⎠

du u 2j + 1e−u ∞

∑ j=0

m i σP(t )2 μρ′

(111)

2

⎞ 2j + 1 Γ(j + 1) ⎛ μρ′ ⎟ ⎜ j !Γ(j + 2) ⎝ 2 m i σP(t ) ⎠

(112)

⎛ μ2 ρ′2 ⎞ j + 1 1 ⎟ ⎜ (j + 1)! ⎝ 2m i 2σP(t )2 ⎠

(113)

∞

∑ j=0

m i σP(t )2 μ2 ρ ′2 /2mi 2σP(t )2 (e − 1) μρ′

(114)

hence

2

∫−∞ dz i e−(z −μz′/m −p t /M) /2σ (t) i

2

∞

−(z i − μz ′ / m i − pZ t / M )2 /2σZ(t )2

∞

∫0

du u j − 1e−u = 2

2 σP(t ) ∑

=

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

∑ j=0

∫−∞ dz′ |ψnlm(ρ′,z′,ϕ′)|2 2

⎛ μρ ρ′ ⎞ i ⎟ I1⎜ 2 ⎝ m i σP(t ) ⎠

(109)

dρi e−ρi ∞

(103)

2

∞

Z

dρi e−ρi

/2σP(t )2

the ρi-integral in eq 105 is evaluated as

∞

∞

2

uj j!

∑ j=0

∫0

n

∫0

dρi e−ρi

⎛ u ⎞ 2j + 1 1 ⎜ ⎟ j !Γ(j + 2) ⎝ 2 ⎠

∞

∫0

∞

eu =

(cf. eq 34). Therefore, the mean angular momenta in the center of mass frame and in the laboratory frame are equal, conserved, and independent of the translational wavepacket widths and mean translational momenta. 2.3.3. Electric Ring Currents. With the formula for the ring current densities (83), the electric ring currents (8) are expressed as

×

∞

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

(101)

= ⟨L ⟩nlm + ⟨L ⟩nlm = mℏez

×

∫0

the substitution u = ρi/(2 σP(t)), the definition of the Gamma function30

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

i P

×

j=0

With the fact that the wave functions ψtr(R,t) and ψnlm(r) are normalized, it yields μ ⟨Li(t )⟩nlm = mℏez mi (102)

∞

ρ ′2 /2m i 2σP(t )2

1/2

⟨L (t )⟩nlm μ = mℏ mi

∫ 2π m σ (t )2 σ (t ) 0

2

∫−∞ dz′ |ψnlm(ρ′,z′,ϕ′)|2

∞

i

Aiemℏ

d ρ ′ e −μ

×

∑

I1(u) =

μρ′ ⎞ 1⎛ ⎜P cos(ϕ′ − Φ) + ⎟ ρi ⎝ mi ⎠

i Inlm (t ) =

∞

The electric ring currents as the integral of the ring current densities over the half-plane along the axis of symmetry (z-axis) are independent of the distribution of the translational wavepacket along the z-axis, in particular they are independent of the translational wavepacket width σZ(t) and the mean momentum pZ. With the Taylor series of the modified Bessel function30

(97)

=

e

∫0

∞

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

tot ⟨L⟩nlm

Aiemℏ m i σP(t )2

i Inlm (t ) =

(95)

⎛ μy′ ⎞⎞ μx′ ⎞ 1 ⎛ ⎛ ⎜⎜x′⎜X + = ⎟ + y′⎜Y + ⎟⎟⎟ m i ⎠⎠ mi ⎠ ρ′ρi ⎝ ⎝ ⎝

=

thus

i Inlm (t ) =

2

Aiemℏ μ

∫0

∞

dρ ′

2 2 2 2 1 (1 − e−μ ρ ′ /2mi σP(t ) ) ρ′

∞

×

(105) G

∫−∞ dz′ |ψnlm(ρ′,z′,ϕ′)|2

(115)



Article

⎡ Aiemℏ R ki ,nlm(t ) = ⎢⎢ i 2 I ( ⎣ nlm t )m i σP(t )

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

∫0

i i Inlm (t ) = Inlm − Aie

∞

2

2

2

2


∫−∞ dz′ jϕ,nlm (ρ′,z′)

(116)

ℏA5 r e−Ar / aμ sin θ 64πμaμ5

i i I21 ± 1(t ) = I21 ± 1 ∓

×

∫0

eℏAiA 64πμaμ

5

∫0

π

dθ′ sin θ′e

∞

∫0

∫−∞ dz′ |ψnlm(ρ′,z′,ϕ′)|2

×

∫0

∑ j=0

×

dρi ρik e−ρi

2

/2σP(t )2

ρ ′2 /2m i2σP(t )2

2

/2σP(t )2

1/ k ⎛ μρ ρ′ ⎞⎤ i ⎥ ⎟ I1⎜ 2 ⎝ m i σP(t ) ⎠⎥⎦

⎛ μρ ρ′ ⎞ i ⎟ I1⎜ 2 ⎝ miσP(t ) ⎠

(121)

∫0

∞

⎛ μρ ρ′ ⎞2j + k + 1 2 2 i ⎟ dρi ⎜ e−ρi /2σP(t ) 2 ⎝ 2miσP(t ) ⎠ j=0

1 j !Γ(j + 2)

⎛ ⎞2j + 1 μρ′ ×⎜ ⎟ ⎝ 2 miσP(t ) ⎠ 1 = ( 2 σP(t ))k + 1 2

∫0

(117)

∞

×

(122)

⎛ μρ′ ⎞−k 1 ⎜ ⎟ j !Γ(j + 2) ⎝ 2miσP(t )2 ⎠

∞

∑ j=0

∞

(123)

du u 2j + k + 1e−u

2

(124)

⎞ Γ(j + k /2 + 1) ⎛ μρ′ ⎜ ⎟ j !Γ(j + 2) ⎝ 2 miσP(t ) ⎠

2j + 1

⎛k ⎞ μρ′ ( 2 σP(t ))k Γ⎜ + 1⎟ = ⎝2 ⎠ 2mi ∞

−μ2 r ′2 sin 2 θ ′ /2m i 2σP(t )2

×

(118)

∑ j=0

(125)

j 1 Γ(j + k /2 + 1)Γ(2) ⎛ μ2 ρ′2 ⎞ ⎟ ×⎜ 2 2 j! Γ(k /2 + 1)Γ(j + 2) ⎝ 2mi σP(t ) ⎠

⎛k ⎞ μρ′ ( 2 σP(t ))k Γ⎜ + 1⎟ = ⎝2 ⎠ 2mi 2 2 ⎛k μ ρ′ ⎞ ⎟ × 1F1⎜ + 1;2; 2 2 2 2 m ⎝ i σP(t ) ⎠

(119)

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

(126)

for k > −2, where the confluent hypergeometric function of the first kind is defined as ∞ 1F1(a ;b ; u) =

∑ j=0

1 Γ(j + a)Γ(b) j u j! Γ(a)Γ(j + b)

(127)

Thus, with eq 26, I obtain from eq 121 ⎡ A eμ2 Γ(k /2 + 1) 2 σP(t )⎢ i i ⎢⎣ 2Inlm(t )m i 2σP(t )2

R ki ,nlm(t ) =

i Tnlm

× 2

1 − c 2[1 + c 2ec Ei( −c 2)]

∞

= ( 2 σP(t ))k + 1∑

2

=

dρi ρik e−ρi =

and have the simple analytical form as

i T21 ± 1(t )

∞

∞

dr′ r′2 e−Ar ′ / aμ

i i 2 2 c 2 I21 ± 1(t ) = I21 ± 1(1 − c [1 + c e Ei( − c )])

2

The mean ring current radii are also independent of the distribution of the translational wavepacket along the z-axis; i.e., the wavepacket width σZ(t) and the mean momentum pZ do not affect the mean ring current radii. Using eqs 107 and 108, the ρi-integral is evaluated in a similar way as in eqs 110−114, i.e.

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 5

d ρ ′ e −μ

∞

Thus, the electric ring currents in the laboratory frame are equal to the ones in the center of mass frame minus the ones weighted by the translational exponential function exp(−μ2ρ′2/ (2mi2σP(t)2)) perpendicular to the z-axis. In the limit σP(t) = 0; i.e., there is no translation perpendicular to the z-axis, the second term of eq 116 vanishes, and the electric ring currents in the laboratory frame and in the center of mass frame are equal. If σP tends to infinity, then the exponential function tends to unity and the second term is equal to Iinlm; therefore, the electric ring currents tend to zero because in the limit σP(t) → ∞ the individual ring currents with finite ring current radii are spread everywhere and the effect of the ring currents with respect to the z-axis vanishes. Now I focus on the strong ring currents in 2p± orbitals. Using eq 85, the corresponding ring current densities for the reduced mass (26) is jϕ ,21 ± 1 (r ,θ ) = ±

∞

×

∞

×

∫0

∫0

∞

2

2

2

2

dρ′ ρ′2 e−μ ρ ′ /2mi σP(t )

⎛k μ2 ρ′2 ⎞ ⎟ × 1F1⎜ + 1;2; 2m i 2σP(t )2 ⎠ ⎝2

(120)

increase with wavepacket width σP(t) = σ(t) and with time 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 similar way as in eqs 104−106, i.e.

∞

×

⎤1/ k

∫−∞ dz′ jϕ,nlm (ρ′,z′)⎥⎦

(128)

for k > −2 and k ≠ 0. H



Article 2

2

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

⎡ eℏAiA5μ Γ(k /2 + 1) 2 σP(t )⎢⎢ ± i 2 5 2 ⎣ 128πI21 ± 1(t )m i aμ σP(t )

R ki ,21 ± 1(t ) = ×

∫0

×

∫0

∞

π

dr′ r′4 e−Ar ′ / aμ

dθ′sin 3 θ′e−μ

2 2

r ′ sin 2 θ ′ /2m i 2σP(t )2

1/ k ⎛k μ2 r′2 sin 2 θ′ ⎞⎤ ⎥ ⎟ × 1F1⎜ + 1;2; 2m i 2σP(t )2 ⎠⎥⎦ ⎝2

i Bnlm (ri=r0(t ),t ) μ Aie ∞ dρ′ ρ′2 = 0 0 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 1/ k 2μaμc ⎡ 2c 4 Γ(k /2+1)U (3,k /2+3,c 2) ⎤ ⎥ ⎢ Am i ⎢⎣ 1 − c 2[1 + c 2ec 2Ei(− c 2)] ⎥⎦

R ki ,21 ± 1(t ) =

∞

1/ k ⎞ ⎛k ⎞⎤ μ μ 2aμ ⎡ ⎛ k R k ,nlm = ⎢Γ⎝⎜ + 1⎠⎟Γ⎝⎜ + 2⎠⎟⎥ ⎦ mi mi A ⎣ 2 2

∞

i Bnlm (ri=r0(t ),t ) =

(131)

1 2π σP(t )2 σZ(t )

=

∫0

∞

×

Z

∫0 ∞

=

2

∞

dP′ P′e ⎛

∫−∞ dZ′ ⎜⎝

pZ t M

ez

2 σZ(t )Z′ +

μ0 Aiemℏ 2 2π m i σP(t )2 σZ(t ) 2

2

∞

∫−∞ dz (ρ′2 +1 z 2)3/2

×

∫0

×

∫0

×

∫−∞ dz″e−(z−μz″/m ) /2σ (t) |ψnlm(ρ″,z″,ϕ″)|2 ez

∞

dρ′ ρ′2 e−ρ ′ /2σP(t )

2 2 2 2 ⎛ μρ′ρ″ ⎞ ⎟ dρ″ e−μ ρ ″ /2mi σP(t ) I1⎜ 2 ⎝ m i σP(t ) ⎠ 2

2

Z

i

(138)

×

∫0

×

∫0

π

π

μ0 Aiemℏ 2

∫ 2π m σ(t ) 0

∞

3

d r ″ r ″ e −μ

2 2

r ″ /2m i 2σ (t )2

i

dθ″ |ψnlm(r″,θ″,ϕ″)|2

∫0

∞

2

2

dr′ e−r ′ /2σ(t )

2 ⎛ μr ′r ″ sin θ ′ sin θ ″ ⎞ ⎟ez dθ′ sin 2 θ′e μr ′r ″cos θ ′ cos θ ″/ miσ(t ) I1⎜ m iσ(t )2 ⎝ ⎠ (139)

Evaluating r′- and θ′-integrals yields (134) i Bnlm (ri=r0(t ),t ) =

2

Z

−P ′2

∞


(132)

∫−∞ dZ Ze−(Z−p t /M) /2σ (t) ez

2 = π ×

dP P e

−P 2 /2σP(t )2

(137)

are independent of the translational momentum pZ. In general, they decrease with increasing wavepacket widths σP(t) and σZ(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 in spherical coordinates as

(133)

∞

e 3/2 z

⎡⎣ρ′2 + (z′ − pZ t /M )2 ⎤⎦

∞

∞

∫0 dP P |ψtrP(P ,t )|2 ∞ × ∫ dZ Z |ψtrZ(Z ,t )|2 ez −∞

= 2π

jϕi ,nlm (ρ′,z′,t )

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

∞

∫0 dP P |ψtrP(P ,t )|2 ∫−∞ dZ |ψtrZ(Z ,t )|2 2π × ∫ dΦ (P eρ(Φ) + Z ez) 0

d z′

−∞

(130)

i for k > −2 and k ≠ 0 (cf. eq 46). In particular, Rk=−1,21±1 = 4μaμ/ i (πmiA ) and Rk=1,21±1 = 3πμaμ/(4 miA ). 2.3.5. Induced Magnetic Fields. Now I consider the magnetic fields induced by electronic (i = e) and nuclear (i = n) ring currents at the ring center ri = r0(t), because the magnetic fields at this center are strongest. The time-dependent position of this ring center is determined by the mean position of the center of mass (see eq 12). With eqs 63 and 64 and substitutions P′ = P/(21/2σP(t)) and Z′ = (Z − pZt/M)/(21/2σZ(t)), the ring center is located at

r0(t ) =

∫

×

(Appendix C). This function increases monotonically with c or with σP(t) = σ(t) ∝ c or with time t, again due to the spreading of the translational wavepacket. Because c ∝ mi, the nuclear mean ring current radius increases faster than the electronic one. In the limit c → ∞, the mean ring current radii are infinite, whereas in the limit c = 0, they are equal to the mean ring current radii in the center of mass frame, i.e. R ki ,21 ± 1 =

2

−u −u wehre the integrals ∫ ∞ = 1/2, ∫ ∞ = 0, and 0 du ue −∞du ue

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

μ0 Aiem i 2μ ×

(135) pZ t ⎞ −Z ′2 ⎟e ez M⎠

− (136)

× I

∫0

∞

⎡ ⎛ μr″ ⎞ dr″⎢erf⎜ ⎟ ⎢⎣ ⎝ 2 m i σ(t ) ⎠

2 μr′′ −μ2 r ″2 /2mi 2σ(t )2⎤ ⎥ e ⎥⎦ π m i σ (t )

∫0

π

dθ″ sin 2 θ″jϕ ,nlm (r″,θ″)ez

(140)



Article

(Appendix D). In the limit σ(t) = 0, the first term of the r″-integrand tends to unity and the corresponding second term tends to zero. Therefore, in this limit, the induced magnetic fields at the ring center are equal to the ones in the center of mass frame (cf. eq 18 of ref 20) and they are maximal. In the opposite limit σ(t) → ∞, both terms of the r″-integrand tend to zero and the induced magnetic fields are suppressed as expected, because the corresponding electric ring currents and the mean ring current radii tend to zero and infinity (see sections 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 induced magnetic fields at the ring center

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

∫0

IΨi ±(t ) = IΨi ± − Aie

⎫ 2c (2c 2 + 1)⎬ ⎭ π

±

mi 2 μ

⎛ m i ri ⎞ ⎟ ⎝ μ ⎠

2 Ψ± ⎜

j

dθ e −r

⎡ A e Γ(k /2 + 1) 2 σ (t )⎢ i i ⎢⎣ 2IΨ (t ) σ(t )2 ±

R ki ,Ψ±(t ) = ×

2

sin 2 θ /2σ(t )2 i jϕ ,Ψ (r ,θ ) ±

∫0

π

dθ sin 2 θ e−r

2

∫0

∞

dr r 3

sin 2 θ /2σ(t )2

⎤1/ k ⎛k r 2 sin 2 θ ⎞ i × 1F1⎜ +1;2; ⎟j (r ,θ )⎥ ⎥⎦ 2σ(t )2 ⎠ ϕ ,Ψ± ⎝2

(146)

for k > 2, k ≠ 0, and

(141)

i BΨ (ri=r0(t ),t ) = ±

μ0 Aie 2 − ×

∫0

∞

⎡ ⎛ r ⎞ dr ⎢erf⎜ ⎟ ⎣ ⎝ 2 σ (t ) ⎠

2 r −r 2 /2σ(t )2⎤ ⎥ e π σ (t ) ⎦

∫0

π

dθ sin 2 θjϕi ,Ψ (r ,θ )ez ±

(147)

respectively. 2.5. Application to Pseudorotating Molecules. However, the general formulas (143)−(147) are applied not only to ring currents in atoms and ions but also to pseudorotating molecules, i.e., to nuclear ring currents in degenerate vibrational states of molecules, such as linear CdH2 and FHF− or tetrahedral OsH4 molecules (see refs 8, 18, 25, and 26). In pseudorotating molecules, all nuclei circulate about its individual equilibrium positions. For the symmetric molecules having a central nucleus, the center of the ring current of the central nucleus is located at the center of mass of the molecule. Therefore, I focus on the ring currents of the central nucleus in pseudorotating molecules. Here, I take the tetrahedral molecule AB4 as an example and derive analytic formulas for the central nuclear ring currents in the first triply degenerate pseudorotational states |v1l±1⟩ that belong to the triply degenerate bends or antisymmetric stretches.26 The corresponding nuclear (i = A) ring current densities in the center of mass frame and in the harmonic approximation are26

(142)

The formulas for the current densities for the reduced mass jΨ±(r) in different many-electron systems are found in ref 18. With eqs 26, 84, and 142 and the substitution r = (μ/mi)r′, the corresponding ring current densities in the laboratory frame in the case of σ(t) = σP(t) = σZ(t) are ∞ 2 2 2 2 1 e−ri /2σ(t ) dr r 2e−r /2σ(t ) 3 ± 0 2π σ(t ) π 2 ⎛ r r sin θ sin θ ⎞ i × dθ sin θ erir cos θicos θ / σ(t ) I1⎜ i ⎟jϕi ,Ψ (r ,θ ) 0 σ(t )2 ⎝ ⎠ ±

jϕi ,Ψ (ri ,θi ,t ) =

π

where IiΨ± = −A IΨ± (cf. eq 38),

(Appendix E). The scalar function in eq 141 is in the range from 1 for c = 0 to 0 for c → ∞. Hence, the induced magnetic fields decrease with increasing translational wavepacket width σP(t) = σZ(t) = σ(t). In the limit c = 0, the magnetic fields in the laboratory frame are equal to the ones in the center of mass frame. Furthermore, the magnetic fields induced by nuclear ring currents decrease faster than the ones induced by electronic ring currents because c ∝ mi. 2.4. Application to Ring Currents in Many-Electron Atoms and Ions. The formulas derived in the previous sections can be generalized for electronic and nuclear ring currents in degenerate electronic states |Ψ±⟩ of N-electron atoms and ions in the laboratory frame. The electronic (i = e) and nuclear (i = n) ring current densities in the center of mass frame are defined similarly as in eq 27 where the reduced mass μ is replaced with the generalized one μ = memn/M, where me is now the total mass of N electrons, i.e. jiΨ (ri) =

∫0

dr r

(145)

⎧ i i 2 2 c2 B21 ± 1(ri=r0(t ),t ) = B21 ± 1(ri=0) × ⎨[1 − 4c (c + 1)]e ⎩ × erfc(c) +

∞

∫

∫

jϕA,v1l±1 (rA ,θA) = ±c π̃

(143)

Because the mean angular momenta are not affected by the translation (see section 2.3.2), the corresponding electronic and nuclear mean angular momenta μ ⟨Li(t )⟩Ψ± = ⟨Li⟩Ψ± = ⟨L⟩Ψ± mi (144)

M3mA 3 2ℏ3mB3TA 5

× rA e−πMmA rA

2

/2ℏmBTA

sin θA

(148)

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

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



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frame (143) is then rewritten as c̃ 2σ(t )3

jϕA,v1l±1 (rA ,θA ,t ) = ± ×

∫0

×

∫0

∞

π

dr r 3e−(πMmA r

2

πM3mA 3 3

3

ℏ mB TA

e−rA

5

2

are conserved and unaffected by the molecular translation (see section 2.3.2 and eq 144). Because the analytic expression of the ring current densities (156) is very simple, the extra evaluation of the integrals for electric ring currents (145), mean ring current radii (146), and induced magnetic fields (147) is not necessary. Using the simple substitution r = rA/(1 + d2)1/2, the definitions (8), (10), and (11), and the expressions for the electric ring currents, mean ring current radii (k = 1), and induced magnetic fields at the ring center in the center of mass frame26

/2σ (t )2

/2ℏmBTA ) − [r 2 /2σ(t )2 ]

2 ⎛ r r sin θ sin θ ⎞ A dθ sin 2 θ erAr cos θA cos θ / σ(t ) I1⎜ A ⎟ σ(t )2 ⎝ ⎠

(149)

Using the substitution u = rAr/σ(t) , the θ-integral (169), the abbreviations (87) and 2

πMmA ℏmBTA

d = σ (t )

IvA1l±1 = ±c ̃ R1,Av1l±1 =

(150)

AAe TA ℏmBTA 2MmA

I obtain jϕA,v1l±1 (rA ,θA ,t ) = ± ×

∫0

∞

2cb̃ 2d3 sin θA πTArA 2

e−1/4b

BvA1l±1(rA=0) = ±

2

2

2

du u(u cosh u − sinh u)e−b (1 + d )u

∫0

2

du u cosh ue

(151)

=

IvA1l±1(t ) =

−b2(1 + d 2)u 2

2

2

π (2b (1 + d ) + 1) 8b5(1 + d 2)5/2

∞

2

2

=

(152)

2

2

du u(u cosh u − sinh u)e−b (1 + d )u 2 2 π e1/4b (1 + d ) 2 5/2 8b (1 + d )

(154)

Therefore, the ring current densities of the central nucleus in the first pseudorotational states |v1l±1⟩ of the AB4 molecule in the laboratory frame, using eqs 87 and 150, are c π̃ (1 + d 2)5/2

× rA e−πMmA rA

2

3

(160)

IvA1l±1 1 + d2

(161) (162)

M mA

2ℏ3mB3TA 5

/2ℏmBTA(1 + d 2)

sin θA

⎛ r ⎞ 1 A jA ⎜ ,θA ⎟⎟ 2 2 ϕ ,v1l ±1 ⎜ (1 + d ) ⎝ 1+d 2 ⎠

(155)

(156)

Thus, in the limit d = 0, the nuclear ring current densities in the laboratory frame and in the center of mass frame are equal. The corresponding mean angular momenta of the central nucleus26 4mB ℏez M

BvA1l±1(rA=0) (1 + d 2)3/2

(163)

3. RESULTS AND DISCUSSION Table 1 lists magnitudes of the electric ring currents |Ie21±1| (eq 39), mean ring current radii Re1,21±1 (eq 47), and induced magnetic fields at the ring center |Be21±1(0)| (eq 56) in the center of mass frame for electronic ring currents in 2p± orbitals of nonrelativistic hydrogen-like systems with nuclear charges An = 1, ..., 13 and masses mn of the stable isotopes; for discussion of the nonrelativistic limit of An , see ref 20. Because the nucleus is much heavier than the electron, i.e., me ≈ μ and a0 ≈ aμ, the listed values for |Ie21±1| and |Be21±1(0)| are almost identical to the ones in ref 20. Furthermore, for isotopes with the same nuclear charges An , all values in the center of mass frame do not change significantly, again due to large masses of heavy nuclei. As already discussed in ref 20, the electric ring currents and induced magnetic fields increase with An as An 2 and An 3 (cf. eqs 39 and 56), respectively, whereas the mean ring current radii decrease with increasing An according to the

3

or, using eq 148, in the simple form

⟨LA(t )⟩v1l±1 = ⟨LA⟩v1l±1 = ±c ̃

ez

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 in atomic orbitals, the electric ring currents and induced magnetic fields in the laboratory frame decrease with increasing translational wavepacket width σ(t) ∝ d, whereas the mean ring current radii increase with σ(t) ∝ d. In the limit d = 0, all values in the laboratory frame are equal to the ones in the center of mass frame. In the opposite limit d → ∞, the electric ring currents and induced magnetic fields tend to zero, whereas the mean ring current radii tend to infinity. Furthermore, the translational effects are considerable for heavy central nuclei because d ∝ (MmA)1/2.

2

5

jϕA,v1l±1 (rA ,θA ,t ) =

ℏmBTA 3

BvA1l±1(rA=r0(t ),t ) =

2 2 π e1/4b (1 + d ) 3 2 3/2 4b (1 + d )

2

du u sinh ue−b (1 + d )u =

jϕA,v1l±1 (rA ,θA ,t ) = ±

3

e1/4b (1 + d )

(153) ∞

2MmA

2

hence

∫0

c ̃AAeμ0

R1,Av1l±1(t ) = R1,Av1l±1 1 + d 2 2

and

∫0

(159)

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

2

The integrals in eq 151 are evaluated as31 ∞

(158)

(157) K



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

An a

mna (u)

|IeΨ±|b (mA)

Re1,Ψ± b (a0)

|BeΨ±(0)|b (T)

c |Be,max Ψ± (0, 100 fs)| (T)

c |Be,max Ψ± (0, 1 ps)| (T)

H H 3 He+ 4 He+ 6 2+ Li 7 2+ Li 9 3+ Be 10 4+ B 11 4+ B 12 5+ C 13 5+ C 14 6+ N 15 6+ N 16 7+ O 17 7+ O 18 7+ O 19 8+ F 20 Ne9+ 21 Ne9+ 22 Ne9+ 23 Na10+ 24 Mg11+ 25 Mg11+ 26 Mg11+ 27 12+ Al

1 1 2 2 3 3 4 5 5 6 6 7 7 8 8 8 9 10 10 10 11 12 12 12 13

1.008 2.014 3.016 4.002 6.015 7.016 9.012 10.01 11.01 12.00 13.00 14.00 15.00 15.99 17.00 18.00 19.00 19.99 20.99 21.99 22.99 23.99 24.99 25.98 26.98

0.13 0.13 0.53 0.53 1.19 1.19 2.11 3.29 3.29 4.74 4.74 6.46 6.46 8.43 8.43 8.43 10.7 13.2 13.2 13.2 15.9 19.0 19.0 19.0 22.3

2.36 2.36 1.18 1.18 0.79 0.79 0.59 0.47 0.47 0.39 0.39 0.34 0.34 0.29 0.29 0.29 0.26 0.24 0.24 0.24 0.21 0.20 0.20 0.20 0.18

0.52 0.52 4.17 4.17 14.1 14.1 33.4 65.2 65.2 113 113 179 179 267 267 267 380 522 522 522 694 901 901 901 1146

0.19 0.27 1.26 1.52 3.92 4.37 8.17 12.1 13.1 17.7 19.0 24.3 25.9 31.7 33.7 35.7 42.4 49.2 51.8 54.5 62.0 69.7 72.9 76.1 84.5

0.02 0.05 0.12 0.18 0.36 0.44 0.68 0.84 0.96 1.13 1.27 1.45 1.60 1.79 1.96 2.12 2.34 2.55 2.74 2.93 3.16 3.40 3.60 3.81 4.06

1 2

a Only stable isotopes with nuclear charges An = 1, ..., 13 and nuclear masses mn are considered. bIn the center of mass frame, magnitudes of the electric ring currents |IeΨ±| (eq 39)), mean ring current radii Re1,Ψ± (eq 47), and magnetic fields at the ring center |BeΨ±(0)| (eq 56) induced by electronic ring currents in 2p± orbitals of hydrogen-like systems are listed. cIn the laboratory frame, the maxima of the induced magnetic fields at times t = 100 fs and t = 1 ps for individual optimal initial translational wavepacket widths σ = σP = σZ are listed.

tional states |v1l±1⟩ of the tetrahedral molecule OsH4, using eqs 158, 159, and 160 for electric ring currents, mean ring current radii, and induced magnetic fields, respectively. These values and other molecular parameters for the first triply degenerate antisymmetric stretches (a) and bends (b) of the OsH4 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 |jiϕ,Ψ±|(ρi,θi=π/2,t=0)| depending on the radius ρi and on the initial translational wavepacket width σ = σP = σZ in units of Ri1,Ψ± are shown in Figure 1. Figure 1a shows the ring currents in 2p± orbitals with identical distributions for electron and nucleus. Figure 1b shows the ring currents of the central nucleus A in the first triply degenerate pseudorotational states |v1l±1⟩ of the tetrahedral molecule AB4 with identical distributions for antisymmetric stretches and bends (cf. ref 26). Futhermore, these unique scaled distributions are independent of the molecular properties of the tetrahedral molecule AB4. The distributions for 2p± orbitals and for the first pseudorotational states of the AB4 molecule are similar and represent the toroidal structure of the ring currents. The black curves correspond to the ring current densities in the center of mass frame or equivalently in the laboratory frame with translational wavepacket width σ = 0. As expected, the maxima of the ring current densities decrease with increasing translational wavepacket widths σ and its positions are shifted to larger radii, according to the spreading of the translational wavepacket.

Biot−Savart law as 1/An . Therefore, the induced magnetic field for 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., magnitudes of the electric ring currents |In21±1| (eq 40), mean ring current radii Rn1,21±1 (eq 48), and induced magnetic fields at the ring center |Bn21±1(0)| (eq 57). Because the formula for the nuclear ring currents derived in section 2 are applied only for Rn1,21±1 ≫ Rn and the nuclear radius Rn ≈ r0A1/3 for A = 8 is about 2 fm, only the ring current properties for stable isotopes with A < 8 or equivalently An = 1, 2, 3 are listed in Table 2. As for electronic ring currents, the nuclear electric ring currents |In21±1| (eq 40) do not change significantly for isotopes with the same nuclear charges An because me ≈ μ. However, for nuclear ring currents, the electric ring currents and induced magnetic fields increase with An as An 3 and An 4 (cf. eqs 40 and 57), respectively, whereas the mean ring current radii decrease with increasing An as 1/An . In addition, the induced magnetic fields increase with mn as mn and, hence, the mean ring current radii decrease with increasing mn according to the Biot−Savart law as 1/mn. Therefore, the mean ring current radii are very small and only a few times larger than the nuclear radii, i.e., on the femtometer scale. For such very small radii, the induced magnetic fields at the ring center are giant, i.e., 0.96 kT for 1H and 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 peseuorotaL



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

An a

mna (u)

|InΨ±|b (mA)

Rn1,Ψ± b (fm)

|BnΨ±(0)|b (T)

c |Bn,max Ψ± (0, 1 as)| (T)

c |Bn,max Ψ± (0, 1 fs)| (T)

H H 3 He+ 4 He+ 6 2+ Li 7 2+ Li OsH4(a)e OsH4(b)e

1 1 2 2 3 3 76 76

1.008 2.014 3.016 4.002 6.015 7.016 192.0 192.0

0.13 0.13 1.05 1.05 3.56 3.56 0.14 0.25

68 34 11 8.5 3.8 3.2 164 164

958 1914 45869 60869 463152 540236 351 624

8.27 6.17 10.6 9.21 11.3 10.5 341 607

d d d d d d 4.00 6.85

1 2

a

Only stable isotopes with nuclear charges An = 1, 2, 3 and nuclear masses mn are considered. bIn the center of mass frame, magnitudes of the electric ring currents |InΨ±| (eqs 40 and 158), mean ring current radii Rn1,Ψ± (eqs 48 and 159), and magnetic fields at the ring center |BnΨ±(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 the tetrahedral 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 individual optimal initial translational wavepacket widths σ = σP = σZ are listed. dThe induced magnetic fields at t = 1 fs are strongly suppressed for nuclear ring currents in hydrogen-like systems. eOnly the nuclear ring currents of the central nucleus Os are considered. The molecular properties of the OsH4 molecule in the first triply degenerate pseudorotational states assigned to antisymmetric stretches (a) and bends (b) are adopted from ref 26.

Figure 2. Electric ring currents |IiΨ±(t=0)| (eqs 119 and 161, panel a), mean ring current radii Ri1,Ψ±(t=0) (eqs 130 and 162, panel b), and induced magnetic fields at the ring center |BiΨ±(0,t=0)| (eqs 141 and 163, panel c) of the 2p± orbitals (i = e, n, black) and of the central nucleus A in the first triply degenerate pseudorotational states |v1l±1⟩ of the tetrahedral molecule AB4 (i = A, red), normalized to its values in the center of mass frame σ = 0, versus initial translational wavepacket widths σ = σP = σZ in units of mean ring current radii in the center of mass frame Ri1,Ψ± (eqs 131 and 159).

Figure 1. Electronic (i = e) and nuclear (i = n, A) ring current densities on the x/y-plane |jiϕ,Ψ±(ρi,θi=π/2,t=0)| of the 2p± orbitals (eq 89, panel a) and of the central nucleus A in the first triply degenerate pseudorotational states |v1l±1⟩ of the tetrahedral molecule AB4 (eq 156, panel b), normalized to its maxima in the center of mass frame, versus electronic/nuclear radius ρi in units of mean ring current radii in the center of mass frame Ri1,Ψ± (eqs 131 and 159). The colored curves correspond to different initial translational wavepacket widths σ = σP = σZ in units of Ri1,Ψ±, 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 90 and 148). For 2p± orbitals (panel a), the curves for the normalized electronic and nuclear ring current densities are identical due to scaling of the radii. For the AB4 molecule (panel b), the curves for the normalized ring currents densities of the central nucleus A do not depend on the molecular properties of the AB4 molecule again due to scaling of the radii. Furthermore, these curves are identical for the first triply degenerate antisymmetric stretches and bends (see also ref 26).

these properties for ring currents in 2p± orbitals and in the first triply degenerate pseudorotational states |v1l±1⟩ of the tetrahedral molecule AB4 are similar, supported by the similar properties for the corresponding ring current densities shown in Figure 1. However, there is little difference between 2p± orbitals and pseudorotational states |v1l±1⟩: The decrease of the electric ring currents and induced magnetic fields is a little faster for 2p± orbitals, because the position of the maximum of the ring current densities for 2p± orbitals and σ = 0 is a little shifted to small values of the radius compared to the ones for |v1l±1⟩ states (Figure 1) and because the translational effects are stronger for smaller radius. Using the time-dependent spherically Gaussian-distributed translational wavepacket with mean momenta pP = pZ = 0 and time-dependent wavepacket widths σ(t) = σP(t) = σZ(t), eqs 60−66, the time-dependent properties of the ring currents for selected initial translational wavepacket widths σ = σP = σZ in i units of R1,Ψ , i.e., electric ring currents |IiΨ±(t)| (eq 119 and ± i (t) (eq 130 and 162), and 161), mean ring current radii R1,Ψ ±

The σ-dependent electronic and nuclear ring current densities in the laboratory frame shown in Figure 1 possess the electric ring currents |IiΨ±(t=0)| (eqs 119 and 161), the i mean ring current radii R1,Ψ (t=0) (eqs 130 and 162), and the ± induced magnetic fields at the ring center |BiΨ±(0,t=0)| (eqs 141 and 163). These properties depending on the translational wavei packet widths σ in units of R1,Ψ are shown in Figure 2. Due ± to the spreading of the ring current densities by translation, the electric ring currents and induced magnetic fields decrease with increasing σ and the corresponding mean ring current radii increase with σ, according to the Biot−Savart law. Interestingly, M



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Figure 3. Electric ring currents |Ie21±1(t)| (eq 119, panel a), mean ring current radii Re1,21±1(t) (eq 130, panel b), and magnetic fields at the ring center |Be21±1(0,t)| (eq 141, panel c) induced by electronic ring currents in 2p± orbitals of the hydrogen atom 1H versus time on the femtosecond time scale. The colored thick curves correspond to different initial translational wavepacket widths σ = σP = σZ in units of Re1,21±1 (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), and 2.5 (green).

Figure 5. Electric ring currents |IOs v1l±1(t)| (eq 161, panel a), mean ring current radii ROs 1,v1l±1(t) (eq 162, panel b), and magnetic fields at the ring center |BOs v1l±1(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 versus time on the attosecond time scale. The colored thick curves correspond to different initial translational wavepacket widths σ = σP = σZ in units of ROs 1,v1l±1 (eq 159), 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), and 2.5 (green).

magnetic fields are large at t = 0 but they decrease fast. The increase effects on mean ring current radii for small σ are also observed 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 much faster, i.e., on the zeptosecond time scale (Figure 4). It is due to the Heisenberg uncertainty principle applied to translation. Because the nuclear mean ring current radius is very small (on the femtometer scale), the very small translational wavepacket width σ (in units of the ring current radius) leads to the rapid spreading of the translational wavepacket and of the nuclear ring current density on the zeptosecond time scale. For ring currents of the central nucleus Os in the pseudorotating molecule OsH4 (Figure 5), the decrease of these ring currents is on the attosecond time scale. This decrease is thus slower than for nuclear ring currents in 2p± orbitals of 1H, because the total mass M and the 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 ps cannot be larger than 0.19 and 0.02 T, respectively, which are, of course, much smaller than the ones at t = 0 with the maximum of 0.52 T (corresponding to the one in center of mass frame). The maxima of the induced magnetic fields at t = 100 fs and t = 1 ps for electronic ring currents in 2p± orbitals in hydrogen-like systems are listed in Table 1. For example, the giant induced magnetic field for 12Al (1146 T) with proper initial translational wavepacket width is strongly reduced to 86 T at t = 100 fs and to 4 T at t = 1 ps. However, the maxima of the induced magnetic fields at times t = 100 fs and t = 1 ps increase with the nuclear charge An 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 = 1 as with proper initial translational wavepacket width can be maximized to only 8 T and they are very small compared to its values at t = 0 with the maximum of 0.96 kT. The corresponding maxima at t = 1 as for other hydrogen-like systems are listed in

Figure 4. Electric ring currents |In21±1(t)| (eq 119, panel a), mean ring current radii Rn1,21±1(t) (eq 130, panel b), and magnetic fields at the ring center |Bn21±1(0,t)| (eq 141, panel c) induced by nuclear ring currents in 2p± orbitals of the hydrogen atom 1H versus time on the zeptosecond time scale. The colored thick curves correspond to different initial translational wavepacket widths σ = σP = σZ in units of Rn1,21±1 (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), and 2.5 (green).

induced magnetic fields at the ring center |BiΨ±(0,t)| (eq 141 and 163), are shown in Figures 3−5. Exemplarily, Figures 3−5 show the properties for the electronic and nuclear ring currents in 2p± orbitals of the hydrogen atom 1H, and for the ring currents of the central nucleus Os in the first triply degenerate pseudorotational states (assigned to bends) |v1l±1⟩ of the tetrahedral molecule OsH4, respectively. At the initial time t = 0, the corresponding values depending on the initial translational wavepacket widths σ are already shown in Figure 2. Because the wavepacket width σ(t) increases with time t, the general trend in Figure 2 is similar to the one in Figures 3−5; i.e., the electric ring currents and induced magnetic fields decrease with increasing t and the mean ring current radii increase with t, as expected. Furthermore, for small initial wavepacket widths σ, the electric ring currents and induced N



Article

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

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 current radii, and induced magnetic fields in the laboratory frame can be evaluated analytically. For example, in the first triply degenerate pseudorotational states |v1l±1⟩ of the tetrahedral molecule OsH4 (see ref 26), the ring currents of the central nucleus Os in the laboratory frame (section 2.5) have properties similar to those of the ring currents in 2p± orbitals but they decay on different time scales. In particular, the corresponding electric ring currents and induced magnetic fields at the ring center decrease with increasing time t on the attosecond time scale. Hence, the decay of the ring currents of the heavy nucleus Os is much slower and faster than the one of the nuclear and electronic ring 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 after the end of driven circularly polarized ultrashort (atto- or femtosecond) laser pulses on long (pico- or nanosecond) time scales, before spontaneous emission occurs. It is indeed true for ring currents in the center of mass frame but the strengths of the ring currents in the laboratory frame decay on very short (zepto-, atto-, or femtosecond) time scales, due to fast spreading of the translational wavepacket with the initial wavepacket widths comparable to small mean ring current radii. If the ring currents should persist on long (pico- or nanosecond) time scales, one has to control translation of atomic and molecular systems, e.g., using magneto-optical traps for neutral atoms32 or Penning or Paul traps for charged ions.33,34 The atoms or molecules in these traps are usually confined on nanometer spatial scale, which makes observation of induced magnetic fields in the laboratory frame very challenging. To observe the translational effects in the experiment, one could use the neutron beam that interacts with the induced magnetic fields in the laboratory frame.19 The neutron beam will be deflected and this deflection depends on the strength and distribution of the induced magnetic fields in atoms and molecules. However, I note that the experimental observation of the induced magnetic fields in the center of mass frame may be possible, on the basis of interactions of the induced magnetic fields with spins of remaining electrons or nuclei in the same ringcurrent-carrying atomic or molecular system. Furthermore, it is also a challenge for theoretists and experimentists to control small translational wavepacket widths with resolution of small mean ring current radii. However, for large mean ring current radii, e.g., in degenerate high-lying np± (n ≫ 2) atomic orbitals, for example 4p± of Kr+ ion,35 or in electronic ring currents of large ringshaped molecules, for example Mg−porphyrin,24 the relatively weak electronic ring currents should persist on longer time scales.

4. CONCLUSIONS In this work, the theory of nonrelativistic electronic and nuclear ring currents in degenerate excited states of free-moving atomic and molecular systems in the laboratory frame is established. In particular, the time-dependent wave function for the translation Ψtr(R,t) is included in the total wave function (cf. eq 1) to investigate the effects of translation on electronic and nuclear ring currents. Using the spherically zero-mean-velocity Gaussiandistributed translational wavepacket, depending on the timedependent wavepacket width σ(t) and corresponding initial width σ = σ(t=0), the analytic formulas for the electronic and nuclear ring current densities (azimuthal components of the current densities), mean angular momenta, electric ring currents, mean ring current radii, and induced magnetic fields at the ring center for 2p± orbitals of hydrogen-like systems are derived in section 2.3 (eqs 89, 102, 119, 130, and 141, respectively). Although the mean angular momenta are unaffected by translation, i.e., they are conserved, the electric ring currents and induced magnetic fields decrease with increasing wavepacket width σ(t) or time t and the corresponding mean ring current radii increase with σ(t) or t. As expected, in the limit of zero translation σ = 0, the corresponding formulas in the laboratory frame are identical to the ones in the center of mass frame (section 2.2). Because me ≈ μ, the results for the electronic ring currents in 2p± orbitals coincide very well with the ones reported in ref 20. Because the nucleus of the hydrogen-like system is not fixed and circulates about the center of mass, there are also nuclear ring currents in atomic orbitals. Compared to the electronic ring currents, the nuclear ring currents are stronger. In the center of mass frame, the nuclear (electronic) electric ring currents and corresponding induced magnetic fields at the ring center increase with nuclear charge An as An 3 (An 2 ) and An 4 (An 3), respectively. The nuclear mean ring current radii decrease with increasing nuclear mass mn as 1/mn and they are on the femtometer scale (only few times larger than the nuclear radii). Therefore, the theory for point-nuclear ring currents in atomic orbitals is applied only to small nuclei of H, He, and Li. According to the Biot−Savart law, the induced magnetic fields at the nuclear ring center increase linearly with nuclear mass mn. Thus, the induced magnetic fields at the ring center for 7Li2+ are 0.54 MT in the center of mass frame and approach the huge magnetic fields on neutron stars. In the laboratory frame, however, the electronic and nuclear ring currents in 2p± orbitals decay on femtosecond and zeptosecond time scales, respectively, due to fast spreading of the translational wavepacket, according to 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 in many-electron atoms and ions or even to nuclear ring currents in pseudorotating molecules. Using the expression for the ring

■

APPENDIX A

With the substitution u = μrir′/(miσ(t)2), I get from eq 86 jϕi ,21 ± 1 (ri,t ) = ±

ℏA5m i 3σ(t )5 3/2 4

32(2π )

×

∫0

×

∫0

∞

π

μ

aμ5ri 4

e−ri

2 2

du u3e−[σ(t ) u

2

/2σ(t )2

/2ri 2] − [Am iσ(t )2 u / μaμri]

dθ′ sin 2 θ′eucos θicos θ ′I1(u sin θi sin θ′) (164)

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



Article 2

where erf(v) = 2/π1/2∫ dv e−v is the error function. Then, using the integration by parts, I get from eq 173

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

∫0

π

dθ′ sin 2 θ′eucos θicos θ ′I1(u sin θi sin θ′) π

∫0

∫0

(165)

dθ′ sin θ′e−i(iu)cos θicos θ ′

=i

∞

−

= 2iP11(cos θi)j1 (iu)

(166)

= −2i sin θij1 (iu)

(167)

+

where j1(u) is the spherical Bessel function of the first kind sin u cos u − u u2

−

∞

⎛ 1 ⎞⎟⎤ ⎥ du (2u − 1) erf⎜bu + c − ⎝ 2b ⎠⎥⎦

∞ ⎛ 1 ⎞⎟ π −c 2 (c + (1/2b))2⎡ 2 ⎢(u + u) erf⎜bu + c + e e ⎝ ⎢⎣ 4b 2b ⎠ 0

∫0

∞

⎛ 1 ⎞⎟⎤ ⎥ du(2u + 1) erf⎜bu + c + ⎝ 2b ⎠⎥⎦

⎛

2

dθ′ sin θ′e

ucos θi cos θ′

I1(u sin θi sin θ′)

=

(169)

⎞

jϕi ,21 ± 1 (ri,t ) =±

ℏA m i σ(t ) sin θi 3/2 4

16(2π )

∫0

×

μ

∞ 2 2

8π

3/2

∫0

aμ5ri 4

e−ri

2

∞

m i 2ri 4

2

e−(1/4b ) + c

/2σ (t )2

= =

2

(171) 2

du u(u cosh u − sinh u)e−(bu + c)

2

du u(u cosh u − sinh u)e−(bu + c) 2 1 ∞ = du u(ueu + ue−u − eu + e−u)e−(bu + c) 2 0 2 2 1 2 ∞ du ((u 2 − u)e−b u − (2bc − 1)u = e −c 0 2 2 2 + (u 2 + u)e−b u − (2bc + 1)u)

=

∫

∫0

(172)

(173)

2 2

∫ du e−b u −(2bc±1)u 2

= e(c ± (1/2b))

2

2 1 (c ± (1/2b))2 e d v e −v b π (c ± (1/2b))2 = e erf(v) 2b

=

∫

∫ dv erf(v)

2v ⎛ 1 −v 2⎞ ⎜v erf(v) + e ⎟ ⎠ π b2 ⎝ 2 2 2 − 2 dv v erf(v) − dv e−v π b2 b 1 2(c ± 2b ) ⎛ 1 −v 2⎞ ⎜v erf(v) + − e ⎟ ⎝ ⎠ π b2 v − 2(c ± b2

(182)

∫

1 ) 2b ⎛

1 −v 2⎞ 1 e ⎟ − 2 erf(v) ⎠ 2b π

⎜v erf(v) + ⎝

(183)

∞

2


(184)

2 2 π = 2 e−c e(c − (1/2b)) 4b 2 ⎡⎛ 1 ⎛ ⎞ ⎛ 1 ⎟⎞ 1 ⎟⎞ × ⎢⎜ ⎜c − + c ⎟ × erf⎜bu + c − ⎝ 2b ⎠ 2b ⎠ ⎠ ⎣⎢⎝ b ⎝ ⎤∞ ⎞ 2 1 ⎛ 1 ⎜⎛ 1 ⎟⎞ ⎜ c− + + 1 − u⎟e−(bu + c − (1/2b)) ⎥ ⎠ ⎥⎦ 2b ⎠ π ⎝b⎝

0

−b2u 2 − 2b(c ± (1/2b))u − (c ± (1/2b))2

∫ du e−(bu+c±(1/2b))

b2

π −c 2 (c + (1/2b))2 e e 4b2 2 ⎡⎛ 1 ⎛ ⎞ ⎛ 1 ⎟⎞ 1 ⎟⎞ × ⎢⎜ ⎜c + − c ⎟ × erf⎜bu + c + ⎝ ⎢⎣⎝ b ⎝ 2b ⎠ 2 b⎠ ⎠ ⎤∞ ⎞ 2 1 ⎛ 1 ⎜⎛ 1 ⎟⎞ ⎜ c+ + − 1 − u⎟e−(bu + c + (1/2b)) ⎥ ⎠ ⎥⎦ 2b ⎠ π ⎝b⎝

(174)

∫ du e

1 ) 2b

+

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

=e

∫ dv v erf(v) −

2(c ±

the result of the integral (178) is

∫

(c ± (1/2b))2

2 b2

(181)

⎟

∫

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 is rewritten as ∞

(180)

⎞

⎜

/2ri 2] − [Am iσ (t )2 u / μaμri]

ℏc 5μ sin θi

×

⎛

du u(u cosh u − sinh u)

× e−[σ(t ) u =±

5

1⎛ 1 −v 2⎞ ⎜v erf(v) + e ⎟ ⎠ b⎝ π

∫ du 2u erf⎝bu + c ± 21b ⎠

(170) 3

(179)

⎟

and

Then, the ring current densities (164) are 5

(178)

∫ du erf⎝bu + c ± 21b ⎠ 1 = ∫ dv erf(v) b ⎜

π

⎛ cosh u sinh u ⎞ ⎟ = 2 sin θi⎜ − ⎝ u u2 ⎠

∫0

∫0

Using the indefinite integrals involving the error function

(168)

hence

∫0

∞ ⎛ 1 ⎞⎟ π −c 2 (c − (1/2b))2⎡ 2 ⎢(u − u) erf⎜bu + c − e e ⎝ ⎢⎣ 4b 2b ⎠ 0

=

× P11(cos θ′)J1(iu sin θi sin θ′)

j1 (u) =

2


0

2 ⎡ 2 ⎛1⎛ 1 ⎞⎟ π = 2 e −c ⎢ ⎜ ⎜ c − + ⎢⎣⎝ b ⎝ 2b ⎠ 4b ⎛ 1 ⎞⎟ (c − (1/2b))2 e × erfc⎜c − ⎝ 2b ⎠

(175) (176)

⎛ 1 ⎞⎟ (c + (1/2b))2 × erfc⎜c + e ⎝ 2b ⎠

(177) P

⎞ c⎟ ⎠

(185)

2 ⎛1⎛ ⎞ 1 ⎞⎟ + ⎜ ⎜c + − c⎟ ⎝ ⎠ 2b ⎝b ⎠ 2c ⎤ ⎥ − π b ⎥⎦



Article 2

where erfc(v) = 1 − erf(v) is the complementary error function. With this integral, eq 171 is then expressed as jϕi ,21 ± 1 (ri,t ) = ±

2 ⎞ ℏc 5μ sin θi c 2⎡⎛ 1 ⎛ 1 ⎞⎟ e ⎢⎜ ⎜c − + c⎟ 2 2 4 2b ⎠ 32πb mi ri ⎢⎣⎝ b ⎝ ⎠

∫0

∞

2

du ue−u − 2cu erfi(u) ∞ 1 du e−2cu − c = π 0

∫

2 ⎞ ⎛ 1 ⎞⎟ −c / b ⎛ 1 ⎛⎜ 1 ⎞⎟ e × erfc⎜c − +⎜ c+ − c⎟ ⎝ 2b ⎠ 2b ⎠ ⎠ ⎝b⎝ ⎤ ⎛ 1 ⎞⎟ c / b 2c −c 2− (1/4b2)⎥ e − e × erfc⎜c + ⎝ ⎥⎦ 2b ⎠ πb

■

Applying the L’Hôspital’s rule, d/du erfi(u) = (2/π1/2)eu , and erfi(0) = 0, the first term vanishes and the integral becomes (194)

∫0

∞

du e−u

2

− 2cu

erfi(u)

2 1 [1 + c 2ec Ei(− c 2)] 2c π

=

(195)

−u where in the last step ref 37 was used and Ei(v) = −∫ ∞ −vdu e /u is the exponential integral. Thus, eq 191 is then written as

(186)

2

i i 2 2 c 2 I21 ± 1(t ) = I21 ± 1(1 − c [1 + c e Ei( − c )])

APPENDIX B

■

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

∫0

π

dθ′ sin θ′e−μ =e

2 2

r ′ sin 2 θ ′ /2m i 2σP(t )2

−μ2 r ′2 /2m i 2σP(t )2

∫0

dθ′ sin θ′e

2 miσP(t ) −μ2 r ′2 /2mi 2σP(t )2 e μr′

=

1F1(a ;b ;u)

μ2 r ′2 cos2 θ ′ /2m i 2σP(t )2

∫−μr′/

= eu1F1(b−a;b;− u)

(197)

the θ′-integral of eq 129 is evaluated as

μr ′ / 2 m i σP(t )

du e u

2

(188)

∫0

2 m i σP(t )

⎛ ⎞ 2π miσP(t ) −μ2 r ′2 /2mi 2σP(t )2 μr′ e erfi⎜ ⎟ μr′ σ 2 m ( t ) ⎝ ⎠ i P

=

APPENDIX C With the substitution u = −μ2r′2 sin2 θ′/(2mi2σP(t)2) and Kummer’s transformation30

(187)

π

(196)

2

π

dθ′ sin 3 θ′e−μ

∫0

(189)

π /2

=2

2 2

r ′ sin 2 θ ′ /2mi 2σP(t )2

dθ′ sin 3 θ′e−μ

2 2

⎛k μ2 r′2 sin 2 θ′ ⎞ ⎟ × 1F1⎜ +1;2; 2mi 2σP(t )2 ⎠ ⎝2

2

2

(198)

2

r ′ sin θ ′ /2mi σP(t )

⎛k μ2 r′2 sin 2 θ′ ⎞ ⎟ × 1F1⎜ +1;2; 2mi 2σP(t )2 ⎠ ⎝2

2

where erfi(v) = −i erf(iv) = −2i/π1/2∫ iv0 du e−u = 2/π1/2∫ v0du eu = 2

1/π1/2∫ v−vdu eu is the imaginary error function. Then, eq 118 becomes i i I21 ± 1(t ) = I21 ± 1 ∓

×

∫0

⎛ 2 miσP(t ) ⎞3 = − i⎜ ⎟ μr′ ⎝ ⎠

eℏAiA5m i σP(t ) 32 2π μ2 aμ5

∞

dr′ r′e

×

−[μ2 r ′2 /2m i 2σP(t )2 ] − (Ar ′ / aμ)

⎞ ⎛ μr′ × erfi⎜ ⎟ ⎝ 2 m i σP(t ) ⎠

i i 3 I21 π ± 1(t ) = I21 ± 1(1 − 2c

∫0

− 2cu

2 2

erfi(u))

∫ du u e −u

∫0

=

2

− 2cu

∞

∫

−

μ2 r ′2 2mi 2σP(t )2

−u

μ2 r′2 ⎞ 4 ⎛ k 5 ⎟ 1F1⎜1− ; ;− 3 ⎝ 2 2 2mi 2σP(t )2 ⎠

(201)

R ki ,21 ± 1(t ) ⎡ eℏA A5μ Γ(k /2 + 1) i 2 σP(t ) ⎢ ± i 2 5 2 ⎢⎣ 96πI21 ± 1(t )m i aμ σP(t )

erfi(u)

1 2 = − e−u − 2cu erfi(u) 2 0 2⎛ d ⎞ 1 ∞ du e−u ⎜ e−2cv erfi(v)⎟ + ⎝ ⎠v = u 2 0 dv

∫

u

du

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

(192)

=

d u u e −u

2

⎛ k ⎞ × 1F1⎜1− ;2;u⎟ ⎝ 2 ⎠

and the integration by parts, the u-integral in eq 191 is rewritten as ∞

(200) 2

0

Using the indefinite integral 1 2 = − e −u 2

−u

−μ r ′ /2mi σP(t )

(191)

2

μ2 r ′2 2mi 2σP(t )2

⎛ 2 miσP(t ) ⎞3 = − i⎜ ⎟ μr′ ⎝ ⎠

×

d u u e −u

−

⎛k ⎞ × 1F1⎜ +1;2;− u⎟ ⎝2 ⎠

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

∫

ue u

du

0

(190)

∞

(199)

−μ2 r ′2 /2mi 2σP(t )2

×

∫0

∞

⎤1/ k ⎛ k 5 μ2 r′2 ⎞⎥ ⎟ dr′ r′4 e−Ar ′ / aμ 1F1⎜1− ; ;− 2 2 ⎝ 2 2 2m i σP(t ) ⎠⎥⎦

(202)

Using the substitution u = μr′/(2 miσ(t)) and eqs 38, 39, 88, and 119 (here again σ(t) = σP(t)), the expression for the mean 1/2

(193) Q



Article

ring current radii (202) is then simplified to (k > −2,k ≠ 0) R ki ,21 ± 1(t ) =

2μaμc ⎡ 8c 3 Γ(k /2 + 1) ⎢ Am i ⎢⎣ 3(1 − c 2[1 + c 2ec 2Ei(− c 2)]) 1/ k ∞ ⎛ k 5 ⎞⎤ × du u 4 e−2cu 1F1⎜1− ; ;− u 2⎟⎥ ⎝ 2 2 ⎠⎥⎦ 0

∫

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

∫0

∞

∞

=

(203)

Using eq 127, the derivative d 2 1F1(a ,b+1,− u ) du ∞ (− 1) j Γ(j + a) Γ(b + 1) 2j − 1 = 2∑ u j = 1 (j − 1)! Γ(a) Γ(j + b + 1) ∞

= − 2u ∑ j=0

=− =−

2au b+1

(− 1) j Γ(j+a+1) Γ(b+1) 2j u Γ(a) Γ(j+b+2) j! ∞

∑ j=0

(− 1) j Γ(j+a+1) Γ(b+2) 2j u j! Γ(a+1) Γ(j+b+2)

2au 2 1F1(a+1;b+2;− u ) b+1

1 5 −2cu ⎛ k 7 2⎞ ⎜ ⎟ ue 1F1 1− ; ;− u ⎝ 2 2 ⎠ 5 0 +

2c 5

∫0

∞

⎛ k 7 ⎞ du u5e−2cu 1F1⎜1− ; ;−u 2⎟ ⎝ 2 2 ⎠

(215)

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

(204)

∫0

(205)

∞

⎛ k 5 ⎞ du u 4 e−2cu 1F1⎜1− ; ;−u 2⎟ ⎝ 2 2 ⎠ ⎛ k 5 7 2⎞ 2c ∞ 5 −u 2 − 2cu ⎜ + ; ;u ⎟ du u e = 1F1 ⎝2 2 2 ⎠ 5 0

(216)

∫

(206) (207)

=

the recurrence relations30

⎞ 3c ⎛ k U ⎜3, +3,c 2⎟ ⎠ 4 ⎝ 2

(217)

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

a 1F1(a+1;b+2;−u 2)

⎛ k ⎞ ⎛ k Γ(k /2) 2⎞ ⎜ ⎟ U ⎜3, +3,c 2⎟ = − 1F1 3; +3;c ⎝ 2 ⎠ ⎝ ⎠ 2 Γ(k /2 + 3) ⎞ Γ(k /2 + 2) ⎛ k k 2⎟ ⎜ + 1F1 1− ;− −1;c k+4 ⎝ 2 2 ⎠ 2c

= (a − b − 1) 1F1(a;b+2;−u 2) + (b + 1) 1F1(a;b+1;−u 2)

⎛ k 5 ⎞ du u 4 e−2cu 1F1⎜1− ; ;−u 2⎟ ⎝ 2 2 ⎠

(208)

and

(218)

1F1(a ;b ;− u

2

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

2

)=

b−u 2 1F1(a ;b+1;− u ) b b−a+1 2 + u 1F1(a;b+2;−u 2) b(b + 1)

R ki ,21 ± 1(t ) =

(209)

1/ k 2μaμc ⎡ 2c 4 Γ(k /2 + 1) U (3,k /2+3,c 2) ⎤ ⎢ ⎥ 2 Am i ⎢⎣ 1 − c 2[1 + c 2ec Ei( −c 2)] ⎥⎦

(219)

I get d 1 2b u 1F1(a ,b+1,−u 2) du 2b = u 2b − 1 1F1(a ,b+1,−u 2) 1 2b d 2 u + 1F1(a ,b+1,− u ) 2b du = u 2b − 1 1F1(a ,b+1,−u 2) au 2b+1 2 − 1F1(a+1;b+2;− u ) b(b+1) b − u 2 2b − 1 2 u = 1F1(a ,b+1,− u ) b b − a + 1 2b + 1 2 u + 1F1(a ;b+2;− u ) b(b + 1)

■

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

(210)


(211)

(212)

×

∫0

×

∫0

π

πμ

2

∫0

∞

d v e −v

2

⎞ ⎛ 2 m i σ (t ) dθ″ sin θ″ ψnlm⎜ v ,θ″,ϕ″⎟ μ ⎠ ⎝

∞

du

2

⎛ cosh u sinh u ⎞ −u2 /4v 2 ⎟e ez − ⎝ u u2 ⎠

⎜

(220)

With the Taylor series of the hyperbolic functions30 ∞

= u 2b − 1 1F1(a;b;−u 2)

μ0 Aiemℏm i

(213)

sinh u =

∑ j=0

u 2j + 1 (2j + 1)!

(221)

u 2j (2j)!

(222)

Thus, the corresponding indefinite integral is ∞

∫ du u

2b − 1

1 2b u 1F1(a;b+1;−u 2) 1F1(a ;b ;− u ) = 2b

cosh u =

2

(214)

∑ j=0

R



Article

the substitution w = u/(2v), the definition (108), and the duplication formula for the Gamma function30 2j − 1

2

Γ(2j) =

π

or with eq 26 i Bnlm (ri=r0(t ),t ) =

⎛ 1⎞ Γ(j) Γ⎜j+ ⎟ ⎝ 2⎠

2u −u2 ⎛⎜ 3 2⎞⎟ e 1F1 1; ;u ⎝ 2 ⎠ π

×

■

(224)

the u-integral in eq 220 is then evaluated as ∞

⎛ cosh u sinh u ⎞ −u2 /4v 2 ⎜ ⎟e − ⎝ u u2 ⎠ ∞ ⎡ ⎤ 1 1 = ∑⎢ − ⎥ (2j + 1)! ⎦ j = 0 ⎣ (2j)!

∫0

×

∞

=

∞

π 2

∑ j=1

∞

=

j=0

π

dθ″ sin 3 θ″ =

∫0

∞

dw w 2j − 1e−w

2

× v 2j Γ(j+3/2)

96πμ2 aμ5

∫0

(227)

∞

⎡ ⎛ μr″ ⎞ dr″ r″e−Ar ″ / aμ⎢erf⎜ ⎟ ⎢⎣ ⎝ 2 m i σ(t ) ⎠

2 μr″ −μ2 r ″2 /2mi 2σ(t )2⎤ ⎥ez e ⎥⎦ π m i σ (t )

− (228)

(235)

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

(229)

88, it is simplified to

(230)

i B21 ± 1(ri=r0(t ),t ) i = 4c 2 B21 ± 1(ri=0)

∫0

μ0 eℏAiA5m i

(226)

hence

×

(233)

(234)

i B21 ± 1(ri=r0(t ),t ) = ±

π v e erf(v) − 1 2v

π

dθ″ sin 2 θ″jϕ ,nlm (r″,θ″)ez

4 3

e

2

i Bnlm (ri=r0(t ),t ) μ Aiemℏm i = 0 2μ2

π

I obtain from eq 140

⎛ 3 ⎞ = 1F1⎜1; ;v 2⎟ − 1 ⎝ 2 ⎠ =

∫0

APPENDIX E

∫0

(225)

1 Γ(j+1) Γ(3/2) 2j v −1 j! Γ(1) Γ(j+3/2)

∑

⎡ ⎛ μr″ ⎞ dr″ ⎢erf⎜ ⎟ ⎢⎣ ⎝ 2 m i σ(t ) ⎠

2j − 1 −u 2 /4v 2

2j(2v)2j Γ(2j+2)

∑ j=1

=

du u

∞

With eq 117 and the θ″-integral31

du

∞

∫0

2 μr″ −μ2 r ″2 /2mi 2σ(t )2⎤ ⎥ e ⎥⎦ π m i σ (t )

−

Γ(j+1) = j! for integers j, Γ(3/2) = π /2, the defintion for the confluent hypergeometric function of the first kind (127), and the corresponding relation to the error function30

∫0

2μ

(223) 1/2

erf(u) =

μ0 Aiem i

∫0

∞

⎡ 2u −u2⎤ du ue−2cu⎢erf(u) − e ⎥ ⎣ ⎦ π (236)

∫0

∞

dv

1⎡ 2v −v 2⎤ e ⎥ ⎢erf(v) − ⎦ v⎣ π

Using the integration by parts and the indefinite integrals 2

∫ du e−u −2cu =

2

⎞ ⎛ 2 m i σ (t ) dθ″ sin θ″ ψnlm⎜ v ,θ″,ϕ″⎟ ez μ ⎠ ⎝ (231)

π c2 e erf(u + c) 2

∫ du erf(u + c) = (u + c) erf(u + c) +

or

(237)

1 −(u + c)2 e π (238)

i Bnlm (ri=r0(t ),t )

=

μ0 Aiemℏm i 2μ − ×

2

∫0

∞

dr ″

μr″ ⎞ 1⎡ ⎛ ⎢erf⎜ ⎟ r″ ⎢⎣ ⎝ 2 m i σ(t ) ⎠

∫ du 2u erf(u + c) ⎛ 1⎞ u − c −(u + c)2 e = ⎜u 2 − c 2 − ⎟ erf(u + c) + ⎝ ⎠ π 2

2 μr″ −μ2 r ″2 /2mi 2σ(t )2⎤ ⎥ e ⎥⎦ π m i σ (t )

∫0

π

dθ″ sin θ″|ψnlm(r″,θ″,ϕ″)|2 ez

(239)

(cf. eqs 177, 180, and 183), the integral in eq 236 is evaluated (232)

as S



∫0

∞

⎡ 2u −u2⎤ du ue−2cu⎢erf(u) − e ⎥ ⎣ ⎦ π =

∫0

■

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.

(240)

∞

du ue−2cu erf(u) ∞ 2 2 − du u 2e−u − 2cu π 0

■

∫

∞

1 −2cu erf(u) ue 2c 0 ∞ 1 + du e−2cu erf(u) 2c 0 ∞ 2 1 + du ue−u − 2cu c π 0 ∞ 2 − u 2ec erf(u + c)

=−

(241)

∫

+ 2e

∫0

0

∞

du u erf(u + c) ∞

1 −2cu e erf(u) 4c 2 0 ∞ 2 1 du e−u − 2cu + 2 0 2c π ∞ 2 1 + uec erf(u + c) 2c 0 1 c2 ∞ du erf(u + c) − e 0 2c ∞ 2 − u 2ec erf(u + c)

=−

(242)

∫

∫

+ 2ec

2

∫0

0

∞

du u erf(u + c) ∞

=

=

∞ 1 c2 1 c2 e erf( ) e erf( ) + − + u c u c 2 4c 2 0 0 ∞ 1 −u2 − 2cu e − 2c π 0 ∞ ⎛ 2 1 ⎞ c2 ⎜ ⎟ e erf(u + c) − c + ⎝ 2⎠ 0 ∞ u − c −u2 − 2cu e + π 0 2 1 [1 − 4c 2(c 2 + 1)]ec erfc(c) 2 4c 2c 2 + 1 + 2c π

(243)

(244)

Thus, I get from eq 236 i i B21 ± 1(ri=r0(t ),t ) = B21 ± 1(ri=0)

⎫ ⎧ 2 2c × ⎨[1 − 4c 2(c 2 + 1)]ec erfc(c) + (2c 2 + 1)⎬ ⎩ ⎭ π (245)

■

REFERENCES

(1) Matos-Abiague, A.; Berakdar, J. Phys. Rev. Lett. 2005, 94, 166801. (2) Barth, I.; Manz, J. Angew. Chem., Int. Ed. 2006, 45, 2962. (3) Kanno, M.; Hoki, K.; Kono, H.; Fujimura, Y. J. Chem. Phys. 2007, 127, 204314. (4) Kanno, M.; Kono, H.; Fujimura, Y.; Lin, S. H. Phys. Rev. Lett. 2010, 104, 108302. (5) Rai, D.; Hod, O.; Nitzan, A. J. Phys. Chem. C 2010, 114, 20583. (6) Barth, I.; Hege, H.-C.; Ikeda, I.; Kenfack, A.; Koppitz, M.; Manz, J.; Marquardt, F.; Paramonov, G. K. Chem. Phys. Lett. 2009, 481, 118. (7) Yamaki, M.; Nakayama, S.; Hoki, K.; Kono, H.; Fujimura, Y. Phys. Chem. Chem. Phys. 2009, 11, 1662. (8) Barth, I.; Manz, J.; Sebald, P. Chem. Phys. 2008, 346, 89. (9) Abdel-Latif, M. K.; Kühn, O. J. Chem. Phys. 2011, 135, 084314. (10) Floß, J.; Grohmann, T.; Leibscher, M.; Seideman, T. J. Chem. Phys. 2012, 136, 084309. (11) Nagashima, K.; Takatsuka, K. J. Phys. Chem. A 2009, 113, 15240. (12) Accardi, A.; Barth, I.; Kühn, O.; Manz, J. J. Phys. Chem. A 2010, 114, 11252. (13) Hege, H.-C.; Manz, J.; Marquardt, F.; Paulus, B.; Schild, A. Chem. Phys. 2010, 376, 46. (14) Andrae, D.; Barth, I.; Bredtmann, T.; Hege, H.-C.; Manz, J.; Marquardt, F.; Paulus, B. J. Phys. Chem. B 2011, 115, 5476. (15) Miller, W. H. J. Phys. Chem. A 1998, 102, 793. (16) Okuyama, M.; Takatsuka, K. Chem. Phys. Lett. 2009, 476, 109. (17) Schiffel, G.; Manthe, U. J. Chem. Phys. 2010, 132, 084103. (18) Barth I. Quantum control of electron and nuclear circulations, ring currents, and induced magnetic fields in atoms, ions, and molecules by circularly polarized laser pulses; Ph.D. thesis, Freie Universität Berlin, Germany, 2009; http://www.diss.fu-berlin.de/diss/receive/FUDISS_ thesis_00000010586. (19) Barth, I.; Manz, J. In Progress in Ultrafast Intense Laser Science VI; Yamanouchi, K., Bandrauk, A. D., Gerber, G., Eds.; Springer: Berlin, 2010; p 21. (20) Barth, I.; Manz, J. Phys. Rev. A 2007, 75, 012510. (21) Räsänen, E.; Castro, A.; Werschnik, J.; Rubio, A.; Gross, E. K. U. Phys. Rev. Lett. 2007, 98, 157404. (22) Barth, I.; Manz, J.; Serrano-Andrés, L. Chem. Phys. 2008, 347, 263. (23) Barth, I.; Serrano-Andrés, L.; Seideman, T. J. Chem. Phys. 2008, 129, 164303. Barth, I.; Serrano-Andrés, L.; Seideman, T. Erratum. J. Chem. Phys. 2009, 130, 109901. (24) Barth, I.; Manz, J.; Shigeta, Y.; Yagi, K. J. Am. Chem. Soc. 2006, 128, 7043. (25) Barth, I.; Manz, J; Pérez-Hernandéz, G.; Sebald, P. Z. Phys. Chem. 2008, 222, 1311. (26) Barth, I.; Bressler, C.; Koseki, S.; Manz, J. Chem. Asian J. 2012, 6, 1261. (27) Tipler, P. A.; Llewellyn, R. A. Modern Physics; W. H. Freeman and Co.: New York, 2003. (28) Hinchliffe, A.; Munn, R. W. Molecular Electromagnetism; John Wiley and Sons: Chichester, U.K., 1985. (29) Tannor, D. J. Introduction to Quantum Mechanics: A TimeDependent Perspective; University Science Books: Sausalito, CA, 2007. (30) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1972. (31) Gradshteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series, and Products; Academic Press: Amsterdam, 2007. (32) Metcalf, H. J.; van der Straten, P. Laser Cooling and Trapping; Springer: New York, 1999. (33) Brown, L. S.; Gabrielse, G. Rev. Mod. Phys. 1986, 58, 233. (34) Leibfried, D.; Blatt, R.; Monroe, C.; Wineland, D. Rev. Mod. Phys. 2003, 75, 281.

∫

c2

Article

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Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. T



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(35) Barth, I.; Smirnova, O. Phys. Rev. A 2011, 84, 063415. Barth, I.; Smirnova, O. Erratum. Phys. Rev. A 2012, 85, 029906(E). Barth, I.; Smirnova, O. Erratum. Phys. Rev. A 2012, 85, 039903(E). (36) Neves, A. A. R.; Padilha, L. A.; Fontes, A.; Rodriguez, E.; Cruz, C. H. B.; Barbosa, L. C.; Cesar, C. L. J. Phys. A 2006, 39, L293. (37) McLachlan, N. W.; Humbert, P. Formulaire pour le calcul symbolique; L’Acad. des Sciences: Paris, 1950.

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