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Article pubs.acs.org/JPCA

Vibrating H2+(2Σ+g, JM = 00) Ion as a Pulsating Quantum Bubble in the Laboratory Frame Jörn Manz,†,‡ Jhon Fredy Pérez-Torres,‡ and Yonggang Yang*,† †

State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, 92 Wucheng Road, Taiyuan 030006, China ‡ Institut für Chemie and Biochemie, Freie Universitat Berlin, Takustraße 3, 14195 Berlin, Germany ABSTRACT: We present quantum dynamics simulations of the concerted nuclear and electronic densities and flux densities of the vibrating H2+ ion with quantum numbers 2Σ+g , JM = 00 corresponding to the electronic and rotational ground state, in the laboratory frame. The underlying theory is derived using the nonrelativistic and Born−Oppenheimer approximations. It is well-known that the nuclear density of the nonrotating ion (JM = 00) is isotropic. We show that the electronic density is isotropic as well, confirming intuition. As a consequence, the nuclear and electronic flux densities have radial symmetry. They are related to the corresponding densities by radial continuity equations with proper boundary conditions. The time evolutions of all four observables, i.e., the nuclear and electronic densities and flux densities, are illustrated by means of characteristic snapshots. As an example, we consider the scenario with initial condition corresponding to preparation of H2+ by near-resonant weak field one-photon-photoionization of the H2 molecule in its ground state, 1Σ+g , vJM = 000. Accordingly, the vibrating, nonrotating H2+ ion appears as pulsating quantum bubble in the laboratory frame, quite different from traditional considerations of vibrating H2+ in the molecular frame, or of the familiar alternative scenario of aligned vibrating H2+ in the laboratory frame.

1. INTRODUCTION Joint information about four concerted quantum observables, i.e., the nuclear and electronic densities and flux densities, ρn, ρe, jn, je, may reveal fascinating quantum phenomena during adiabatic intramolecular processes, such as vibrations or reactions in the electronic ground state. It is, therefore, a challenge to determine these four properties by means of quantum dynamics simulations. The state-of-the-art of this emerging topic is, however, just in its beginnings: Till today, full quantum dynamics simulations of the time evolutions of all four properties have been performed for just a single system, and for just a single scenario, i.e., for the vibrating, aligned, nonrotating H2+(2Σ+g ).1−5 For this example, analyses of the time evolutions of ρn, ρe, jn, je show that by and large, the electronic density and flux density follow the nuclear ones, but there are also some significant differences. For example, the electronic flux density that travels with the nuclei is much weaker than the nuclear one, corresponding to the flux of just about 0.17 of an electron compared to the two protons.1 More specificly, for this scenario of aligned vibrating H2+(2Σ+g ), the nuclear densities and flux densities evolve along the direction of the internuclear axis which, by convention, is oriented along the Z axis in the laboratory frame, whereas the corresponding electronic densities and flux densities adapt to the nuclear ones, with corresponding cylindrical symmetries. The electronic flux density may then be decomposed into a major axial component parallel to Z, plus a minor, radial one, perpendicular to Z.4 A fascinating quantum effect has been discovered at the turn from bond stretch to compression, or vice versa. For reference, classical molecular © 2014 American Chemical Society

dynamics simulations would suggest that this event occurs instantaneously, as demonstrated by classical trajectories which stand still at the instant when they hit the turning points. In contrast, the corresponding turn of the representative quantum wavepackets takes time, typically in the domain of few femtoseconds.1,3−5 This is a consequence of their dispersions, which account for the delocalization of the nuclei, and even larger delocalization of the electron, such that their “heads” turn directions of motions before the “tails”. Amazingly, the nuclear flux density needs a little more time (1.4 fs) to change direction than the electronic one (1.0 fs).5 Within these partially overlapping periods, there is even an ultrashort time window of about 0.7 fs when the nuclei and electrons flow toward opposite directions.1,6 These examples of new quantum effects in vibrating aligned H2+(2Σ+g ) discovered by joint information on ρn, ρe, jn, je, motivate the present extended quantum dynamics simulations of the vibrating H2+ in an alternative scenario, that is, the nonaligned, nonrotating H2+(2Σ+g ), with corresponding orbital angular momentum quantum numbers JM = 00, in the laboratory frame. As a prerequisite, this requires the development of the underlying quantum theory, followed by the corresponding quantum dynamics simulations. To simplify this demanding task, Special Issue: A. W. Castleman, Jr. Festschrift Received: February 18, 2014 Revised: April 4, 2014 Published: April 7, 2014 8411

dx.doi.org/10.1021/jp5017246 | J. Phys. Chem. A 2014, 118, 8411−8425

The Journal of Physical Chemistry A

Article

Figure 1. (a) Laboratory frame (eX, eY, eZ), nuclear center of mass frame (ex, ey, ez), and molecular frame (ex̃, eỹ, ez̃). (b) Nuclear center of mass frame (ex, ey, ez) and electronic frame (ex̂, eŷ, eẑ). The translational vector Rcom is from the origin of the laboratory to the center of mass. The internuclear vector R = R̃ = R̂ is from proton b to a. The vector r = r̃ = r̂ is from the nuclear center of mass to the electron e.

(protons) and electrons. Approximation i allows us to derive the properties ρn, ρe, jn, je from the system’s wave function, which is obtained as a solution of the time-dependent Schrödinger equation. Furthermore, we may neglect all spin-dependent terms (such as spin−orbit coupling) of the system’s Hamiltonian. Approximation ii allows us to write the system’s wave function as a product of an electronic times a nuclear wave function, and to employ state-of-the-art ab initio methods of quantum chemistry for the evaluation of the electronic wave function and the potential energy curve V(R) for the electronic ground state 2Σ+g depending on the internuclear distance R, in the molecular frame. 2.1. Laboratory, Nuclear Center of Mass, Molecular, and Electron Frames. From the outset, let us emphasize the little caveat that has just been raised as a consequence of the BOA (ii): quantum chemical evaluations of electronic properties are always carried out in the molecular frame. Because it is our goal to evaluate the concerted ρn, ρe, jn, je in the laboratory frame, we will have to make corresponding transformations between the frames. This derivation starts, therefore, with a concise definition of various frames that are invoked in the derivation, in particular, the laboratory frame, the nuclear center of mass (ncom) frame, the molecular frame, and the so-called “electron frame”; cf. Figure 1. Here the electron and the two protons of the H2+ ion are labeled e and a, b, respectively. Concerning the notations, the coordinates, observables, etc. for the molecular and electron frames will be marked by tilde “∼” and by hat “ ”̂ , respectively; e.g., ρe denotes the electronic density in the laboratory or ncom frames, whereas ρ̃e and ρ̂e are used for the molecular and electron frames. Moreover, electronic and nuclear coordinates will be written with small and Capital letters, respectively. For simplicity of the notation, we shall use the same symbols for observables irrespective of their dependence on either Cartesian (e.g., x, y, z) or spherical (e.g., r, θ, ϕ) coordinates; for example, the time (t)-dependent electron density in the ncom/laboratory frame will be written as ρe(x,y,z,t) or ρe(r,θ,ϕ,t). As further simplification of the notation, the list of coordinates will occasionally be reduced to the relevant ones, provided that it has been shown that the observable under consideration does not depend on the other variables. For example, we shall show below that the electronic density is isotropic; i.e., ρe does not depend on θ and ϕ. This allows the short-hand notation ρe(r,θ,ϕ,t) = ρe(r,t). The laboratory (l) frame is characterized by a set of Cartesian unit vectors eX, eY, eZ that are attached to its origin. The vectors from the origin to the electron and to the nuclei a and b are rle, Rla,

we shall invoke the nonrelativistic and Born−Oppenheimer approximations. Before starting the new task, let us first give credit to some related progresses that have been achieved by other groups, with supplementary References to some of our previous work. All of those publications have in common that they present results for part of the four quantum observables ρn, ρe, jn, je, e.g., just for the electronic ones ρe, je, or for the nuclear ones ρn, jn, but not for all of them together. Thus, quantum dynamics simulations of electronic densities and flux densities accompanying adiabatic transitions during classical nuclear motions have been pioneered by Takatsuka et al.7−13 and by Patchkovskii.14 Electronic densities and flux densities of electronic ring currents in electronic excited degenerate states of rigid molecules were investigated in refs 15−17; see also ref 18. Analogous nuclear densities and flux densities of nuclear ring currents in pseudorotating molecules are presented in refs 19−21. These results are complementary to nuclear densities and flux densities during bimolecular reactions22 or inelastic collisions,23 or unimolecular processes such as photodissociations,24 or vibrations and adiabatic reactions, for example during tunneling isomerizations.25−28 In particular, ref 29 has the first nuclear flux densities that have been derived from experimental pump−probe spectroscopy of vibrating diatomic molecules30 or molecular ions,31 with spatial and temporal resolutions down to 5 pm and 200 as. The rest of this paper is organized as follows: section 2 has the theory, supplemented by an Appendix; the results and discussions are in section 3, followed by the conclusions in section 4.

2. THEORY The purpose of this section is to derive the quantum theory for the electronic density and flux density of the vibrating H2+(2Σ+g ,JM=00) ion, in the laboratory frame, in a coherent manner together with the nuclear density and flux density. We aim at rather simple working equations for quantum dynamics evaluations of all four observables, ρn, ρe, jn, je. For this purpose, we shall invoke two standart approximations: (i) the derivation will be nonrelativistic, and (ii) the adiabatic quantum dynamics will be described using the Born−Oppenheimer approximation (BOA). Approximation i is valid because the resulting absolute mean values of the electronic and nuclear velocities are well below the velocity of light. Approximation ii is a well-known consequence of the large ratio of the masses mp/me of the nuclei 8412



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and Rlb, respectively. The nuclear com is at Rn = (Rla + Rlb)/2, close to the total center of mass Rcom = (mpRla + mpRlb + merle)/ (2mp + me); see Figure 1a. The nuclear com frame is characterized by corresponding unit vectors ex, ey, ez which are attached to the ncom; cf. Figure 1a,b. By definition, these are parallel to the unit vectors in the laboratory frame, ex = eX ey = eY ez = eZ (1)

molecular frames. The reverse transformation from the unit vectors in the molecular frame to the ncom/laboratory frame is just the inverse of eq 6,

This definition (1) allows us to refer to the quantities which are expressed with reference to (ex, ey, ez) as “in the ncom/laboratory frame”; the “translation by Rcom” goes without saying. Next it is convenient to employ the internuclear vector,

The transformation of the unit vectors (6) implies the same transformation for the corresponding Cartesian coordinates, from the ncom/laboratory frame to the molecular frame. Consider an arbitrary vector a = a ̃ with corresponding Cartesian coordinates ax, ay, az and ãx, ãy, ãz,

R = R la − R lb

⎛ ex̃ ⎞ ⎛ ex̃ ⎞ ⎛ ex̃ ⎞ ⎛ ex ⎞ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜e ⎟ T T T −1 −1 ⎜ y ⎟ = +Φ +Θ ⎜ eỹ ⎟ ≡ +Φ+Θ⎜ eỹ ⎟ ≡ +ΘΦ⎜ eỹ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ez ⎠ ⎝ ez̃ ⎠ ⎝ ez̃ ⎠ ⎝ ez̃ ⎠

(2)

⎛ ax̃ ⎞ ⎛ ax ⎞ ⎜ ⎟ ⎜ ⎟ T T ã = ( ex̃ eỹ ez̃ )·⎜ aỹ ⎟ = ( ex ey ez )+Φ +Θ+Θ+Φ⎜ ay ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ az ⎠ ⎝ az̃ ⎠ ⎛ ax ⎞ ⎛ ax ⎞ ⎜ ⎟ ⎜ ⎟ = ( ex ey ez )+Φ−1+Θ−1+Θ+Φ⎜ ay ⎟ = ( ex ey ez )·⎜ ay ⎟ = a ⎜ ⎟ ⎜ ⎟ ⎝ az ⎠ ⎝ az ⎠

and to express its orientation in terms of either Cartesian coordinates X, Y, Z or equivalent spherical coordinates R, Θ, Φ (Figure 1a), ⎛ R sin Θ cos Φ ⎞ ⎛X⎞ ⎟ ⎜ ⎜ ⎟ R = (exeyez) ·⎜ Y ⎟ = (exeyez) ·⎜ R sin Θ sin Φ ⎟ ⎟ ⎜ ⎝Z ⎠ ⎠ ⎝ R cos Θ

(3)

R 2

(8)

hence ⎛ ax̃ ⎞ ⎛ ax ⎞ ⎛ ax ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ aỹ ⎟ = +Θ+Φ⎜ ay ⎟ = +ΘΦ⎜ ay ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ az ⎠ ⎝ az ⎠ ⎝ az̃ ⎠

The vectors from the nuclear center of mass to the nuclei a or b may be expressed in terms of the internuclear vector, R a = −R b =

(7)

(4)

Likewise, the vector from the nuclear center of mass to the electron in the ncom frame is

(9)

Application to the internuclear vector R with coordinates (X, Y, Z) = (R sin Θ cos Φ, R sin Θ sin Φ, R cos Θ) in the ncom/ laboratory frame yields the familiar coordinates (X̃ , Ỹ, Z̃ ) = (0, 0, R̃ ) in the molecular frame, with conservation of the value of the internuclear distance, R̃ = R. Likewise, the vector r from the nuclear com to the position of the electron is transformed from coordinates (x, y, z) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) in the ncom/laboratory frame to

⎛ r sin θ cos ϕ ⎞ ⎛x⎞ ⎜ ⎟ ⎜ ⎟ r = rle − R n = (exeyez) · y = (exeyez) ·⎜ r sin θ sin ϕ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝z⎠ ⎝ r cos θ ⎠ (5)

The vectors Rcom, R, and r are a well-known set of Jacobi vectors; the associated masses are the total mass M = 2mp + me, the reduced nuclear mass μn = mp/2, and the reduced mass of the electron and the two nuclei, μe = me2mp/M, respectively; see, e.g., ref 2. The molecule-fixed Cartesian unit vectors ex̃ , eỹ , ez̃ are also attached to the nuclear center of mass, such that ez̃ = R/R is along the internuclear axis, pointing to proton a, eỹ is perpendicular to the projection of the internuclear axis (again pointing to a) on the ex/ey plane, and ex̃ is perpendicular to eỹ and ez̃ . Using the orientation angles Θ and Φ for the internuclear axis, the unit vectors for the ncom/laboratory frame may be transformed to those of the molecular frame by first rotating (ex, ey, ez) about the axis along ez by Φ into unit vectors (ex′, ey′, ez′ = ez) and then rotating (ex′, ey′, ez′ = ez) about the axis along ey′ by Θ, thus

⎛ r sin θ cos ϕ ⎞ ⎛ x ̃ ⎞ ⎛⎜ r ̃ sin θ ̃ cos ϕ ̃⎞⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜⎜ y ̃⎟⎟ = ⎜ r ̃ sin θ ̃ sin ϕ ̃ ⎟⎟ = +Θ+Φ⎜ r sin θ sin ϕ ⎟ ⎜ ⎟ ⎟ ⎝ z ̃ ⎠ ⎜⎝ ⎝ r cos θ ⎠ ⎠ r ̃ cos θ ̃ ⎛ r sin θ cos Θ cos(ϕ − Φ) − r cos θ sin Θ ⎞ ⎟ ⎜ ⎟ = ⎜ r sin θ sin(ϕ − Φ) − r cos θ sin Θ ⎟⎟ ⎜⎜ ⎝ r sin θ sin Θ cos(ϕ − Φ) + r cos θ cos Θ ⎠ (10)

in the molecular frame, with conservation of the distance r̃ = r. Again, the reverse transformation from the coordinates in the molecular frame to those in the ncom/laboratory frame is obtained by the inverse of eq 10, using the transformation T T T +Θ = +ΘΦ matrices +Φ−1+Θ−1 = +ΘΦ−1 = +Φ instead of +Θ+Φ. In particular, for the special case when the internuclear axis of the ion is oriented along the Z-direction of the laboratory frame (Θ = 0, Φ = 0), eq 10 reduces to the expected simple expression in terms of spherical coordinates,

⎛ ex̃ ⎞ ⎛ ⎛ ex ⎞ ⎞⎛ ⎞⎛ ex ⎞ ⎜ ⎟ ⎜ cos Θ 0 −sin Θ ⎟⎜ cos Φ sin Φ 0 ⎟⎜ ⎟ ey ≡ + + ⎜ ey ⎟ ⎜ eỹ ⎟ = ⎜ 0 1 0 − Φ Φ sin cos 0 Θ Φ⎜ ⎟ ⎟⎟⎜⎜ ⎟⎟⎜⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎝ 0 1 ⎠⎝ ez ⎠ ⎝ ez ⎠ sin Θ 0 cos Θ ⎠⎝ 0 ⎝ ez̃ ⎠ ⎛ ex ⎞ ⎛ cos Θ cos Φ cos Θ sin Φ −sin Θ ⎞⎛ ex ⎞ ⎜e ⎟ ⎜ ⎟⎜ e ⎟ y = ⎜− sin Φ ≡ + cos Φ 0 ΘΦ⎜ y ⎟ ⎟⎟⎜⎜ ⎟⎟ ⎜ ⎜ ⎟ ⎝ sin Θ cos Φ sin Θ sin Φ cos Θ ⎠⎝ ez ⎠ ⎝ ez ⎠ (6)

⎛ x ̃ ⎞ ⎛ r sin θ cos ϕ ⎞ ⎟ ⎜ ⎟ ⎜ ⎜⎜ y ̃⎟⎟ = ⎜ r sin θ sin ϕ ⎟ ⎜ ⎟ ⎝ z ̃ ⎠ ⎝ r cos θ ⎠

with short-hand notations + = +Θ, +Φ, or +ΘΦ for the rotation matrices. They are all unitary, +−1 = +T, thus the mapping (6) conserves the lengths of vectors as well as the angles between two vectors in the different nuclear com/laboratory and

for Θ = 0, Φ = 0 (11)

thus θ̃ = θ and ϕ̃ = ϕ. 8413



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one of them has nuclear spin up whereas the other one has nuclear spin down; i.e., we consider the vibrating paraH2+(2Σ+g ,JM=00). The assumption concerning the electron spin is motivated in the Appendix. The nuclear spin function will be rationalized below. The total wave function Ψ(x,y,z,X,Y,Z,t) is obtained as solution of the time-dependent Schrödinger equation (TDSE) in the laboratory frame, in the nonrelativistic limit (approximation i),

For the subsequent derivation, it is helpful to introduce still another frame, called the “electron frame” (indicated by “ ”̂ ); see Figure 1b. Its unit vectors ex̃ , eỹ , ez̃ are also attached to the nuclear center of mass. Their definition is entirely analogous to the unit vectors ex̃ , eỹ , ez̃ for the molecular frame, except that êz = r/r is oriented along the vector from the nuclear center of mass to the electron, r = (ex, ey, ez)·(r sin θ cos ϕ, r sin θ sin ϕ, r cos θ)T, whereas ẽz = R/R is oriented along the internuclear vector R. Moreover, the transformation from ex, ey, ez in the nuclear com/ laboratory frame to êx, êy, êz employs the orientation angles θ, ϕ for the electron, whereas the orientation angles Θ, Φ are used for entirely analogous expressions for the transformation from ex, ey, ez to ex̃ , eỹ , ez̃ , thus ⎛ ex̂ ⎞ ⎛ ⎛ cos ϕ sin ϕ 0 ⎞⎛ ex ⎞ ⎜ ⎟ ⎜ cos θ 0 −sin θ ⎞⎟⎜ ⎟⎜ ⎟ ⎜ eŷ ⎟ = ⎜ 0 1 0 ⎜−sin ϕ cos ϕ 0 ⎟⎜ ey ⎟ ⎟ ⎟⎜ ⎜⎜ ⎟⎟ ⎜ ⎟⎜ ⎟ 0 1 ⎠⎝ ez ⎠ ⎝ eẑ ⎠ ⎝ sin θ 0 cos θ ⎠⎝ 0 ⎛ ex ⎞ ⎛ ex ⎞ ⎜e ⎟ ⎜ ⎟ ≡ +θ+ϕ⎜ y ⎟ ≡ +θϕ⎜ ey ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ez ⎠ ⎝ ez ⎠

iℏ

(14)

where PR = −iℏ∇R and pr = −iℏ∇r denote the nuclear and electronic momentum operators in the ncom/laboratory frame, respectively. The Hamiltonian / may be written as sum of the operator for the nuclear kinetic energy, plus the electronic Hamilton operator, / = PR2/2 μn + /e . In the BOA approximation (ii), the total wave function of the vibrating H2+ ion in the electronic and rotational ground state 2Σ+g , JM = 00 is written as a product of the electronic times the nuclear wave functions,

(12)

Compare with eq 6. Likewise, the definitions of the coordinates of the internuclear and electron vectors in the electron frame, as well as their transformations to the nuclear com/laboratory frame, are entirely analogous to the previous definitions and transformations of the molecular frame, eqs 10 and 11. In particular, application to the electron vector r with coordinates (x, y, z) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) in the ncom/ laboratory frame yields the expected coordinates (x̂, ŷ, ẑ) = (0, 0, r̂) in the electron frame, with conservation of the value of the distance of the electron from the nuclear com, r̂ = r. Because the unitarity of the transformations between the ncom/laboratory, molecular, and electron frames implies conservations of length and angles, the distances between the electron and the nuclei a or b are the same in all frames,

ΨBOA(x ,y ,z ,X ,Y ,Z ,t ) = ψe(x ,y ,z ;X ,Y ,Z) Ψnuc,JM = 00(X ,Y ,Z ,t ) (15)

The electronic ψe(x,y,z;X,Y,Z) represents the electronic state 2Σ+g of the H2+ ion, depending on the electronic coordinates (x, y, z), and parametrically on the nuclear ones, (X, Y, Z). The nuclear Ψnuc,JM=00(X,Y,Z,t) represents time-dependent vibrations in the electronic and rotational ground state 2Σ+g , JM = 00. Let us first consider the time-independent electronic wave function ψe(x,y,z;X,Y,Z) in the ncom/laboratory frame. It is evaluated in two steps. First, one obtains the solution ψ̃ e(x̃,ỹ,z̃;0,0,R̃ =R) of the time-independent electronic Schrödinger equation (TIESE) with electronic Hamiltonian /̃ e in the molecular frame,

ri = |r − R i| = |r ̃ − R̃ i| = |r ̂ − R̂ i| = rĩ = rî

for i = a, b

∂ Ψ(x ,y ,z ,X ,Y ,Z ,t ) = /Ψ(x ,y ,z ,X ,Y ,Z ,t ) ∂t ⎡P 2 p2 e2 e2 e2 ⎤⎥ =⎢ R + r − − + Ψ(x ,y ,z ,X ,Y ,Z ,t ) 2μe 4πε0ra 4πε0rb 4πε0R ⎥⎦ ⎢⎣ 2μn

/̃ eψẽ (x ̃,y ̃,z ;0,0, R̃ =R ) = V (R̃ =R ) ψẽ (x ̃,y ̃,z ;0,0, R̃ =R ) ̃ ̃

(13)

(16)

In eq 16, V(R̃ =R) denotes the electronic energy, which also serves as potential energy curve for the nuclear wave function, Ψnuc(X,Y,Z,t); see eq 23 below. The TIESE (16) is solved by means of the MOLPRO software,32 using the LCAO MO technique. The wave function is represented in the high level aug-cc-pVTZ basis set.33 The resulting electronic wave function is normalized according to

2.2. Wave Functions. Having defined the different ncom/ laboratory, molecular, and electron frames as well as the corresponding coordinates and the transformations between these frames, we are now ready to derive the four observables ρn, ρe, jn, je of the vibrating H2+(2Σ+g ,JM=00) ion in the ncom/ laboratory frame, starting from the system’s “total” wave function Ψ(x,y,z,X,Y,Z,t) which depends on the electronic (x, y, z) and nuclear (X, Y, Z) coordinates in the ncom/laboratory frame, as defined above, and on the time t. As explained above, we shall also express (x, y, z) and (X, Y, Z) in terms of the corresponding spherical coordinates (r, θ, ϕ) and (R, Θ, Φ), eqs 5 and 3 using the same symbol (e.g., “Ψ”) for the wave functions in Cartesian or spherical coordinates. The complementary, separable wave function describing translation Rcom of the total com is irrelevant and will not be considered. Also note that Ψ(x,y,z,X,Y,Z,t) is actually a short-hand notation; i.e., it is written without any explicit notations for the electronic or nuclear spin variables. Nuclear and electronic spins are taken into account implicitly, however; that means we shall assume, without further mentioning, that the electronic spin is either up (α-spin) or down (β-spin), with equal (=0.5) probabilities, and the two nuclear spins of the protons are in the singlet state, implying that

+∞

+∞

+∞

∫−∞ ∫−∞ ∫−∞

dx ̃ dy ̃ dz ̃ |ψẽ (x ̃ ,y ̃,z ;0,0, R̃ =R )|2 = 1 ̃ (17)

or equivalent expressions in terms of spherical coordinates r̃, θ,̃ ϕ̃ . Second, ψ̃ e(x̃,ỹ,z̃;0,0,R̃ =R) is transformed from (x̃, ỹ, z̃) = (r̃ sin θ̃ cos ϕ̃ , r̃ sin θ̃ sin ϕ̃ , cos θ̃) depending on (0, 0, R̃ = R) in the molecular frame to ψe(x,y,z;X,Y,Z) in the ncom/laboratory frame. The unitarity of the transformation implies that the normalization (17) also holds in this and in the other frames which have been introduced above. Next let us turn to the time-dependent nuclear wave function Ψnuc,JM=00(X,Y,Z,t) in the BOA ansatz (15), again in the ncom/ laboratory frame. Switching from Cartesian to spherical coordinates, the nuclear wave function may be written as a 8414



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product of a time-dependent radial function times the stationary angular eigenfunction for orbital quantum numbers JM = 00, Ψnuc,JM = 00(X ,Y ,Z ,t ) =

Ψn(R ,t ) YJM = 00(Θ,Φ) R

(18)

with isotropic spherical harmonic YJM=00(Θ,Φ) = 1/(4π)1/2. The short-hand notation of the nuclear wave function (18) is thus Ψnuc,JM = 00(R ,t ) =

Ψn(R ,t ) R 4π

(19)

Thus, in the frame of the Born−Oppenheimer approximation, the isotropy of the nuclear wave function (18) and (19) supports the fact that for the rotational ground state, JM = 00, there are no effects of Coriolis coupling. As explained above, the spatial nuclear wave function (18) and (19) is multiplied by the nuclear spin function, but this is not written explicitly to simplify the notations. In any case, the indistinguishabilty of the two protons a and b with nuclear spins ℏ/2 (two Fermions) implies that the total nuclear wave function must be antisymmetric with respect to exchange of the nuclei a and b; because the spatial part (18) is symmetric, the nuclear spin function must be antisymmetric; hence it represents a nuclear singlet state. The normalizations of the radial and angular wave functions

∫0

+∞

∫0

dR |Ψn(R ,t )|2 = 1

π

dΘ sin Θ

∫0

2π

Figure 2. Model scenario for the preparation of the initial state of the vibrating H2+(2Σ+g ,JM=00) by single photon photoionization of the precursor molecule H2(1Σ+g ,vJM=000). The lower part of the figure shows the nuclear wave function ΨH2(1Σ+g ,v=0),n(R) representing the vibrational ground state of H2(1Σ+g ,vJM=0) embedded in its potential curve VH2(1Σg+)(R). The upper part shows the nuclear wave function Ψn0(R) representing the initial nuclear wave function Ψn0(R) of the vibrating H2+(2Σ+g ,JM=00) embedded in its potential curve V(R), defined by eqs 65 and 66. The vertical arrow symbolizes the Franck− Condon type photoionization of H2(1Σg+,vJM=000) that transfers ΨH2(1Σ+g ,v=0),n(R) into Ψn0(R). Also shown by horizontal lines embedded in V(R) are the vibrational eigenenergies Eν of H2+(2Σ+g ,JM=00).

ξν(R ) =

(20)

∫0

∞

dR R2 |Ψnuc,JM = 00(R ,t )|2 = 1

The solution of the TDNSE (23) is then obtained as

(21)

Ψn(R ,t ) =

⎡ ℏ2 ∂ 2 ⎤ ∂ ⎥Ψn(R ,t ) + V R iℏ Ψn(R ,t ) = ⎢ − ( ) ⎢⎣ 2μn ∂R2 ⎥⎦ ∂t

∑ cνξν(R)e−iE t /ℏ ν

(22)

2.3. Densities. The wave function (15) in the ncom/ laboratory frame yields the “total” density of all (electronic and nuclear) degrees of freedom ρ(x ,y ,z ,X ,Y ,Z ,t ) = |Ψ(x ,y ,z ,X ,Y ,Z ,t )|2

(23)

+∞

ρe (x ,y ,z ,t ) =

∫−∞

∑ cνξν(R)

+∞

dX

∫−∞

+∞

(24)

ρn (X ,Y ,Z ,t ) =

∫−∞

+∞

dY

∫−∞

+∞

dx

∫−∞

dZ ρ(x ,y ,z ,X ,Y ,Z ,t )

(30)

+∞

dy

where cν =

(29)

The normalizations of the electronic and nuclear wave functions (17) and (22) imply that ρ(x,y,z,X,Y,Z,t) is also normalized. It is reduced to the corresponding normalized time-dependent electron and nuclear densities by integrating over the complementary nuclear and electronic degrees of freedom, respectively, in the ncom/laboratory frame,

starting from the (normalized) radial initial state Ψn(R,t=0) = Ψn0(R) which will be specified in subsection 3.1; see also Figure 2 below. In practice, the normalized Ψn0(R) is expanded in terms of the bound nuclear vibrational eigenfunctions ξν(R), ν

(28)

ν

The normalized (20) time-dependent radial nuclear wave function Ψn(R,t) is obtained as solution of the radial time dependent nuclear Schrödinger equation (TDNSE) in the ncom/laboratory frame

Ψn0(R ) =

(27)

μ

dΦ |Y00|2 = 1

imply the normalization of the nuclear wave function 4π

∑ bμνBμ(R)

∫−∞

dz ρ(x ,y ,z ,X ,Y ,Z ,t )

(31)

∫0

∞

Ψn0(R ) ξν(R ) dR

Using the BOA ansatz (15) together with the normalization of the electronic wave function (17) and eqs 18 and 19 for the isotropic nuclear wave function, we obtain the nuclear density in terms of spherical coordinates

(25)

The ξν(R) are obtained as solutions of the radial time independent nuclear Schrödinger equation (TINSE) ⎡ ℏ2 d2 ⎤ ⎢− + V (R )⎥ξν(R ) = Eνξν(R ) 2 ⎢⎣ 2μn dR ⎥⎦

ρn (R ,t ) = (26)

|Ψn(R ,t )|2 4πR2

(32)

which is normalized according to

The vibrational eigenenergies Eν embedded in the potential curve V(R) are illustrated in Figure 2. To solve eq 26, we expand the nuclear eigenfunctions in terms of B-spline functions,34,35

4π 8415

∫0

∞

dR R2ρn (R ,t ) = 1

(33)



Article

where we may consider ψe(0,0,r;X,Y,Z) again as the electronic wave function which is evaluated originally in the molecular frame and then transformed into the ncom/laboratory frame. Comparison of eqs 35 and 38 shows that the electronic density has the same value at (x, y, z) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) and at (0, 0, r). This equality holds for arbitrary angles θ and ϕ. Hence the electronic density is isotropicit depends only on the distance r of the electron from the nuclear center of mass. This allows us to drop the variables θ, ϕ in the notation of the electronic density in the ncom/laboratory frame. Inserting the result (32) for the isotropic nuclear density into eq 36, we obtain the final result

The related time derivative of the nuclear density is obtained as ⎤ ⎡ 1 ∂ ∂ 29⎢Ψ*n (R ,t ) Ψn(R ,t )⎥ ρn (R ) = 2 ⎦ ⎣ ∂t ∂t 4πR

(34)

The results (32) and (34) confirm the well-known fact that the nuclear density, as well as its time derivative, of a diatomic molecule or molecuar ion with orbital angular quantum numbers JM = 00 are isotropic. We shall now show that the related electron density is isotropic as well. The result will confirm intuition, but the derivation is somewhat tricky. First, the reduction (30) may be carried out in terms of spherical coordinates, in the ncom/laboratory frame. Using the isotropy of the nuclear wave function (JM = 00), eqs 18 and 19

ρe (r ,t ) =

∫0

=

∫0

ρe (x ,y ,z ,t ) |Ψ (R ,t )|2 dR n = 0 4π ∞ |Ψn(R ,t )|2 dR = 0 4π

∫

∞

∫

∫0 ∫0

π

dΘ sin Θ π

dΘ sin Θ

∫0 ∫0

2π

2

dΦ |ψe(x ,y ,z ;X ,Y ,Z)| 2π

=

dR

|Ψn(R ,t )|2 4π

∫0

π

dθ ̃ sin θ ̃

∫0

2π

4π

dR ̂

|Ψn(R̂ ,t )|2 4π

∫0

π

dΘ̂ sin Θ̂

∫0

∫0

dΦ̂ |ψê (0,0,r ;̂ X̂ ,Y ̂ ,Z)̂ |2

|Ψn(R ,t )|2 4π

∫0

π

dΘ sin Θ

∫0

2π

∫0

2π

̃ dϕ ̃ |ψẽ (r = R̃ =R )|2 ̃ r ,θ ̃,ϕ;0,0,

∞

dr r 2ρe (r ,t ) = 1

(41)

∫0

∞

dR R2ρe (r ,R )

∂ ρ (R ,t ) ∂t n

(42)

⎤⎞ Ψ(x ,y ,z ,X ,Y ,Z ,t )⎥ ⎟ me ⎦⎟ ⎟ ⎤⎟ PR Ψ(x ,y ,z ,X ,Y ,Z ,t )⎥ ⎟⎟ ⎥⎦ ⎠ μn pr

(43)

je (x ,y ,z ,t ) +∞

= dR

dθ ̃ sin θ ̃

The first three (x, y, z)-components describe electronic contributions to jtotal(x,y,z,X,Y,Z,t), the other three (X, Y, Z)components account for the nuclear contributions. It is reduced to the corresponding normalized time-dependent 3D electron and nuclear flux densities by integrating over the complementary nuclear and electronic degrees of freedom, respectively, in the ncom/laboratory frame

ρe (0,0,r ,t ) ∞

π

⎛ ⎡ ⎜ 9⎢Ψ*(x ,y ,z ,X ,Y ,Z ,t ) ⎜ ⎣ =⎜ ⎜ ⎡ ⎜⎜ 9⎢Ψ*(x ,y ,z ,X ,Y ,Z ,t ) ⎝ ⎢⎣

(37)

∫0

(39)

jtotal (x ,y ,z ,X ,Y ,Z ,t )

where r̂ = r, R̂ = R, and ψ̂ e(0,0,r̂;X̂ ,Ŷ,Ẑ ) now denotes the electronic wave function that is evaluated originally in the molecular frame and then transformed into the electron frame. Upon substitution R̂ → R, Θ̂ → Θ, Φ̂ → Φ, we obtain

=

dϕ ̃ |ψẽ (r = ̃ r ,θ ̃,ϕ;̃ R̃ =R )|2

where the time derivative of the nuclear density ∂/∂tρn(R,t) is given by eq 34. 2.4. Flux Densities. Using the “total” wave function Ψ(x,y,z,X,Y,Z,t), eq 15, the six-dimensional (6D) flux density of all particles (the electron and the nuclei) in ncom/laboratory frame is defined as36,37

dϕ ̃ |ψẽ (r = ̃ r ,θ ̃,ϕ;̃ R̃ =R )|2

2π

2π

dR R2ρe (r ;R ) ρn (R ,t )

∫0

∂ ρ (r ,t ) = ∂t e

ρe (0,0,r ̂,t ) ∞

∫0

analogous to eq 33. Its time derivative is

Equation 36 already allows us to conclude that the electronic density is isotropic, because its right-hand side does not depend on the orientation angles θ, ϕ of the electronic vector r in the ncom/laboratory frame. To support this conclusion, let us also evaluate the electron density in the electron frame. Again, using the isotropy of the nuclear density, we obtain

∫0

dθ ̃ sin θ ̃

denotes the isotropic time-dependent electron density. It is normalized according to

(36)

=

π

(40)

ρe (r ,θ ,ϕ ,t ) ∞

∫0

ρe (r ;R ) = ρẽ (r = ̃ r ;R̃ =R )

dΦ |ψẽ (r ̃,θ ̃,ϕ;̃ R ,0,0)|2

where (x, y, z; X, Y, Z) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ; R sin Θ cos Φ, R sin Θ sin Φ, R cos Θ) and ψe(x,y,z;X,Y,Z) denotes the electronic wave function, which is evaluated first in the molecular frame and then transformed into the ncom/laboratory frame, and r̃ = r, and θ̃ = θ̃(θ,ϕ;Θ,Φ) and ϕ̃ = ϕ̃ (θ,ϕ;Θ,Φ), in accord with the transformation of coordinates, eq 10. Now if the nuclear orientation angles are varied over the whole sphere [0 ≤ Θ ≤ π, 0 ≤ Φ ≤ 2π], then the electronic orientations are also varied over the whole sphere, [0 ≤ θ̃ ≤ π,0 ≤ ϕ̃ ≤ 2π]. Furthermore, one can rewrite the angular part of the Jacobean for integration over the nuclear variables in terms of the equivalent expression for the electron, sin Θ dΘ dΦ = sin θ̃ dθ̃ dϕ̃ . Physically, this means that integration over all orientations of the internuclear axis (for fixed electronic position in the ncom/lab frame) has been transformed into the equivalent integration over all orientations of the electron (with respect to the internuclear axis in the molecular frame). The result is

∫0

∞

dR R2ρn (R ,t )

where

(35)

=

∞

dΦ |ψe(0,0,r ;X ,Y ,Z)|2

∫−∞ ×

(38) 8416

+∞

dX

∫−∞

+∞

dY

∫−∞

⎤ Ψ(x ,y ,z ,X ,Y ,Z ,t )⎥ me ⎦

⎡ dZ 9⎢Ψ*(x ,y ,z ,X ,Y ,Z ,t ) ⎣

pr

(44)



∫−∞

⎡ dz 9⎢Ψ*(x ,y ,z ,X ,Y ,Z ,t ) ⎢⎣

⎤ PR Ψ(x ,y ,z ,X ,Y ,Z ,t )⎥ ⎥⎦ μn

(45)

+∞

jn (X ,Y ,Z ,t ) =

∫−∞ ×

Article

+∞

dx

∫−∞

+∞

dy

2.5. Continuity Equations. The density and flux density of all particles (electron and nuclei) are related to each other by the continuity equation, in the ncom/laboratory frame,36,37 ∂ ρ(x ,y ,z ,X ,Y ,Z ,t ) + (∇r ,∇R ) ·jtotal (x ,y ,z ,X ,Y ,Z ,t ) = 0 ∂t (49)

analogous to the reductions (30) and (31) of the “total” density to the electronic and nuclear densities. Using the BOA wave function (15) with normalization (17) of the electronic wave function, the nuclear flux density is

The reduction of the “total” density and flux density to electronic and nuclear densities and flux densities, by corresponding integrations over the complementary electronic and nuclear coordinates, respectively, in the ncom/laboratory frames, eqs 30, 31 and eqs 44, 45, suggest the analogous reduction of the “total” continuity equation (49) to electronic and nuclear ones. Accordingly,

⎡ ⎛ iℏ ⎞ jn (X ,Y ,Z ,t ) = 9⎢Ψ*nuc,JM = 00(X ,Y ,Z ,t )⎜⎜ − ∇R ⎟⎟ ⎢⎣ ⎝ μn ⎠ ⎤ × Ψnuc,JM = 00(X ,Y ,Z ,t )⎥ ⎥⎦

∂ ρ (x ,y ,z ,t ) + ∇r ·je (x ,y ,z ,t ) = 0 ∂t e

(46)

for the electron and

The isotropy of the nuclear wave function (JM = 00) suggests to re-express the operator ∇R in terms of spherical coordinates,38 1 ∂ 1 ∂ ∂ ∇R = eR + eΘ + eΦ R ∂Θ R sin Θ ∂Φ ∂R

∂ ρ (X ,Y ,Z ,t ) + ∇R ·jn (X ,Y ,Z ,t ) = 0 ∂t n

∂ ρ (r ,t ) + ∂t e 1 + r sin θ 1 + r sin θ

jn (R ,t ) = jnR (R ,t )eR ⎡ ⎤ ⎛ iℏ ∂ ⎞ ⎟⎟Ψnuc,JM = 00(R ,t )⎥eR = 9⎢Ψ*nuc,JM = 00(R ,t )⎜⎜ − ⎢⎣ ⎥⎦ ⎝ μn ∂R ⎠ ⎡ ⎤ P 1 9⎢Ψ (R ,t )* R Ψn(R ,t )⎥eR 2 ⎢ n ⎥⎦ μn 4πR ⎣

(51)

for the nuclei. The isotropy of the electronic and nuclear densities and the radial symmetry of the nuclear flux density suggest to re-express the divergences in eqs 50 and 51 in spherical coordinates,38

(47)

with unit vectors eR, eΘ, eΦ in radial (R) and angular (Θ, Φ) directions along and perpendicular to the internuclear vector R, respectively. Because the nuclear wave function (JM = 00) depends on R but not on Θ and Φ, eq 19, it follows that the angular components of the nuclear flux density are equal to zero; i.e., jn has radial symmmetry. Using the expression (19) for the nuclear wave function, its radial (R) component is

=

(50)

1 ∂ 2 r j (r ,θ ,ϕ ,t ) r 2 ∂r er ∂ sin θjeθ (r ,θ ,ϕ ,t ) ∂θ ∂ j (r ,θ ,ϕ ,t ) = 0 ∂ϕ eϕ

(52)

and 1 ∂ 2 ∂ R j (R ,Θ,Φ,t ) ρn (R ,t ) + 2 ∂t R ∂R nR 1 ∂ sin ΘjnΘ (R ,Θ,Φ,t ) + R sin Θ ∂Θ 1 ∂ j (R ,Θ,Φ,t ) = 0 + R sin Θ ∂Φ nΦ

(48)

with radial nuclear momentum operator PR = −iℏ∂/∂R. Note that the TDNSE (23) yields complex-valued nuclear wave functions Ψn(R,t), thus fulfilling a necessary condition for nonzero flux densities.36 It is a temptation to try to evaluate the electronic flux density by an analogous way, i.e., insertion of the BOA ansatz (15) into eq 44, use of the isotropy of the nuclear wave function (JM = 00), decomposition of the electronic momentum operator into radial and angular components analogous to eq 47, etc. But it is wellknown that this approach must fail, due to a fundamental failure of the BOA (ii), which yields real valued electronic wave functions ψe(x,y,z;X,Y,Z) representing the electronic ground state. The BOA violates, therefore, the necessary condition for nonzero components of the flux density,36 which means nonzero flux components call for complex-valued wave functions. This necessary (but not automatically sufficient!) condition would be satisfied, for example, by complex-valued electronic wave functions representating degenerate electronic excited states,15−17 such as excited 2Π, 2Δ, etc. states of H2+. We shall evaluate the electronic flux density of the vibrating H2+( 2Σg+,JM=00), therefore, by the alternative approach pioneered in refs 2−4, i.e., by invoking the electronic continuity equation.

(53)

respectively. It is instructive to consider first the nuclear continuity equation (53). Because we already know that its angular components are equal to zero, and the radial component depends just on R, cf. eq 48, it simplifies to ∂ 1 ∂ 2 ρn (R ,t ) + 2 R j (R ,t ) = 0 ∂t R ∂R nR

(54)

It is gratifying that the expressions 32 and 48 for the nuclear density and flux density satisfy this continuity equation automatically. Explicitly, inserting eqs 32 and 48 into 54 yields ⎤ P ∂ ∂ ⎡ * |Ψn(R ,t )|2 + 9⎢Ψ n (R ,t ) R Ψn(R ,t )⎥ = 0 ⎥⎦ ∂t ∂R ⎢⎣ μn

(55)

Equation 55 holds as a consequence of the radial TDNSE (23). By analogy with the nuclear continuity eq 54, the angular components of the electronic flux density must also be equal to zero. This conjecture may be rationalized in two ways that support each other. First, if one assumes that the vibrating H2+(2Σg+,JM=00) has any nonzero angular electronic flux 8417



Article

• W2: Use ψ̃ e(r̃,θ̃,ϕ̃ ;R̃ = R) to obtain the isotropic electron density ρe(r;R), eq 40. • W3: Use V(R) to solve the TINSE (26) for the nuclear eigenfunctions ξν(R) and eigenenergies Eν. The eigenfunctions ξν(R) are expanded in a set of B-spline functions whose derivatives are calculated analytically and whose integrals are performed by using Gauss−Legendre quadrature, which gives exact results for polynomials.34,35 • W4: Specify the initial (t = 0) nuclear wave function Ψn0(R). This will be done, exemplarily, in the beginning of section 3. • W5: Calculate the coefficients cν for the expansion of Ψn0(R) in terms of the nuclear eigenfunctions ξν(R), eq 25. • W6: Use the results for the cν, ξν(R), and Eν to obtain the time dependent nuclear wave function Ψn(R,t) as a solution of the TDNSE (23) in the form (28). • W7: Using Ψn(R,t), obtain the isotropic nuclear density ρn(R,t) and its time derivative, by means of eqs 32 and 34, respectively. • W8: Again using Ψn(R,t), obtain the nuclear flux density jn(r,t) with radial symmetry, by means of eq 48. • W9: Using the isotropic time independent electron density ρe(r;R) (see W2) and the nuclear density ρn(R,t) (see W7), obtain the time-dependent isotropic electron density ρe(r,t), by means of eq 39. • W10: Likewise, using ρe(r;R) and the time derivative of the nuclear density ∂ρn(R,t)/∂t (see W7), obtain the time derivative of the electron density ∂ρe(r,t)/∂t, by means of eq 42. • W11: Finally, using ∂ρe(r,t)/∂t, obtain the electronic flux density je(r,t) with radial symmetry, by means of eq 59. In step W1, the time-independent electronic wave function in the molecular frame is centered at the nuclei a and b. By extrapolation, one may anticipate that the resulting time dependent isotropic electron density will also be centered at the isotropic densities of the nuclei a and b. For comparison of the nuclear and electronic densities and flux densities, it is, therefore, illuminating to consider the corresponding nuclear wave functions in terms of the coordinates Xa, Ya, Za or Ra = R/2, Θa = Θ, Φa = Φ of the nuclear vector Ra (=Rb), instead of the internuclear vector R with its coordinates (3). The corresponding “total” wave functions are related to each other,

components in the ncom/laboratory frame and integrates them over a half plane that extends from the z-axis to infinity, then the resulting angular flux must be zero because otherwise it would imply nonzero electronic orbital angular momentum,39 in conflict with zero orbital angular momentum for the present electronic and rotational ground state, 2Σ+g , JM = 00. The zero value of the angular flux integral would imply, however, that angular flux densities flowing in one angular direction must be compensated by opposite ones, somewhere else in the same halfplane. This type of angular flux pattern would represent at least two counter-rotating partial waves, which can be ruled out for the electronic and rotational ground state of the molecular ion. Second, nonzero angular flux density would break the symmetry of the system in the nuclear center of mass frame; this is not allowed. Consider, for example, the hypothetical case that the angular electronic flux density jeϕ(r,θ,ϕ,t) is nonzero at some time t for some value of the coordinates (r, θ, ϕ) in the ncom/ laboratory frame, defined by the set of unit vectors ex, ey, ez. One could then rotate this set by an angle ϕ′ about the z-axis, thus generating alternative unit vectors ex′ , ey′, ez′ = ez, with corresponding coordinates (r′ = r, θ′ = θ, ϕ′ + ϕ) where jeϕ(r′,θ′,ϕ′+ϕ,t) = jeϕ(r,θ,ϕ,t), again in the ncom/laboratory frame. The substitutions ex′, ey′, ez′ = ez → ex, ey, ez and r′, θ′, ϕ′ + ϕ → r, θ, ϕ with arbitrary angle ϕ′ then reveal that the hypothetical nonzero jeϕ(r,θ,ϕ,t) represents a unidirectional angular (ϕ) component of the flux density. Because the time dependent Schrödinger equation is invariant with respect to the inversion ϕ → −ϕ, it follows that jeϕ(r,θ,ϕ,t) = −jeϕ(r,θ,ϕ,t) = 0. The same type of argument applies to the other angular component of the flux density, thus jeθ(r,θ,ϕ,t) = 0. As a consequence, the electronic continuity equation is simplified to the radial continuity equation which can be written as ∂ 1 ∂ 1 ∂ ρ (r ,t ) = − 2 r 2jer (r ,θ ,ϕ ,t ) = − 2 r 2jer (r ,t ) ∂t e r ∂r r ∂r

(56)

analogous to the nuclear continuity eq 54. Because the left-hand side of eq 56 depends only on r, the right-hand side cannot depend on θ and ϕ. Hence the electronic flux density must have radial symmetry; it consists of just a single, i.e., the radial, component, which depends exclusively on r, je (r ,t ) = jer (r ,t )er

(57)

Ψa(r ,θ ,ϕ ,R a ,Θa=Θ,Φa =Φ,t ) =

Radial integration of eq 56 with the boundary condition lim r 2jer (r ,t ) = 0

(60)

The factor √2 ensures proper normalization, compensating the Jacobean (=1/2) of the transformation (4). A similar relation is valid for the nuclear wave function in BOA, e.g.

(58)

r→0

yields the electronic flux density je (r ,t ) = −

1 r2

∫0

r

dr′ r′2

∂ ρ (r′,t )er ∂t e

2 Ψn(r ,θ ,ϕ ,R =2R a ,Θ,Φ,t )

Ψa,JM = 00(R a ,t ) =

(59)

2.6. Working Equations. The quantum theory of subsections 2.1−2.5 results in the following workplan for the evaluations of the four concerted observables ρn, ρe, jn, and je of the vibrating H2+(2Σ+g ,JM=00) in the ncom/laboratory frame, with steps W1, W2, etc. as listed below. • W1: Use ab initio methods of quantum chemistry to solve the TIESE 16 in the molecular frame, on a grid of internuclear distances R. This yields the time-independent electronic wave ̃ ̃ ;R̃ =R) with normalization (17), together function ψ̃ e(r̃,θ,ϕ with the electronic energy, which serves as potential energy curve V(R). In the present application, we use an equidistant space grid R ⊂ [0.7a0, 10.0a0] with ΔR = 0.1a0.

Ψa(R a ,t ) = 4π R a

8 Ψn(R =2R a ,t ) (R =2R a) 4π

(61)

Using Ψa,JM=00(Ra,t) instead of Ψn,JM=00(R,t), one obtains the expressions for the isotropic density ρa (R a ,t ) =

1 |Ψa(R a ,t )|2 = 8ρn (R =2R a ,t ) 4πR a 2

(62)

and the flux density ja (R a ,t ) =

⎡ ⎤ PR 1 9⎢Ψ*a (R a ,t ) a Ψa(R a ,t )⎥e R a 2 mp ⎥⎦ 4πR a ⎢⎣

= 4jn (R =2R a ,t )eR 8418

(63)



Article

the H2(1Σ+g ) molecule. For this purpose, we performed a CASSCF(2,10) calculation combined with a subsequent MRCI calculation using the MOLPRO program32 with the aug-ccpVTZ basis set.33 The resulting potential curve VH2(1Σ+g )(R) is shown in Figure 2. It is evaluated on the same radial grid that has been specified before; see item W1 of the working plan. Second, the initial nuclear wave function ξH2(1Σ+g ,v=0),n(R) is obtained as a solution of the radial TINSE (26), again using B-spline functions.34,35 Third, we mimic the above-mentioned vertical FC type process by putting the vibrational wave function ΨH2(1Σ+g ,v=0),n(R) from the potential curve VH2(1Σ+g )(R) for H2(1Σ+g ) to V(R) for the molecular ion H2+(2Σ+g ), with some modification. Specifically, we define the initial wave function in the form

of the nucleus a (idential to those for nucleus b), analogous to eqs 32 and 48, respectively.

3. RESULTS AND DISCUSSIONS This section is divided into two parts: In subsection 3.1, we specify the initial nuclear wave function Ψn0(R) of the vibrating H2+(2Σ+g ,JM=00). In BOA (eq 15), this automatically also specifies the total wave function for the electron and the nuclei. In subsection 3.2, we propagate the wave functions in time, to evaluate and discuss the four concerted observables ρn, ρe, jn, and je. For this purpose, we shall apply the workplan (subsection 2.6), which is based on the quantum theory derived in subsections 2.1−2.5. 3.1. Choice of the Initial Nuclear Wave Function. The quantum theory for the four observables ρn, ρe, jn, and je for the vibrating H2+(2Σ+g ,JM=00) derived in section 2 is applicable to arbitrary initial nuclear wave function Ψn0(R). Various scenarios lead to entirely different initial wave functions. For example, one can employ an ultrashort ultraviolet pump laser pulse combined with a properly delayed low frequency control pulse to induce large amplitude vibrations of H2+(2Σ+g ).40 Alternatively, one can design two ultrashort pump and dump laser pulses to kick the wave function, which represents the vibrational ground state of H2+(2Σ+g ) such that the kick induces the vibration, with control of the initial momentum.25 Here we adapt still another scenario, that is, the familiar, ultrafast near resonant single photon photoionization of the hydrogen molecule H2 in its electronic and vibrotational ground state 1Σ+g , νJM = 000. Essentially, this will excite the nuclear wave function ΨH2(1Σ+g ,v=0),n(R) representing the vibrational ground state of H2(1Σ+g ,vJM=000) in a vertical Franck−Condon (FC) type way from the potential energy curve VH2(1Σ+g )(R) of the precursor H2(1Σ+g ) to the potential V(R) of H2+(2Σ+g ). This scenario is illustrated in Figure 2. This process has already been investigated in great detail; see, e.g., ref 41 and references therein. It is not the purpose of this paper to carry out additional quantum simulations of the initiation process. Instead, we focus on the evaluation of the concerted time dependent observables ρn, ρe, jn, and je after the preparation of the initial state. For this reason, suffice it here to specify the initial wave function Ψn0(R) within the frame of some reasonable model assumptions and approximations, as follows: First, let us consider the precursor molecule H2(1Σ+g ,vJM=000) in the laboratory frame. The preparation of almost perfectly pure para-H2 molecules in the rovibrational ground state in gas at low temperature ( 1.6a0, they still point away from the ncom, in accord with the bond stretch documented at 5 fs, whereas for R and r < 1.6a0, they already point to the ncom, anticipating the bond compression which is documented at t = 15 fs. Around the border with equal nuclear and electronic radii Ra = Ra0 = 1.6a0 and r = r0 = 1.6a0 between the two “inner” and “outer” domains, the electronic and nuclear flux densities vanish, jaRa(Ra=Ra0,t=9fs) = je(r=r0,t=9fs) = 0. This may be interpreted as a consequence of the delocalizations of the nuclear and electronic wave functions, as discussed above: In an ultrashort time interval (compare with refs 1 and 3−6) near t = 9 fs the “heads” of the wavepackets turn direction; i.e., they already run toward small values of r and R, before the “tails”, which still run away from the ncom. These effects in the ncom/ laboratory frame correspond to those that have been discovered in the molecular frame; cf. refs 4 and 5. Extrapolation of the results for the snapshots at t = 5, 9, and 15 fs may lead astray to a (wrong!) hypothesis; i.e., the radially symmetric topologies (that means the signs) of the nuclear and electronic flux densities are always the same. This is falsified, however, by the event documented in the snapshot at t = 18 fs. Here, the nuclear flux density changes sign four times, in contrast with a single sign change for the electronic flux density. A blow up of this phenomenon is documented in Figure 5, revealing the different topologies of jaRa(Ra,t) and jer(r,t) in an even more obvious way than in Figure 4. The multiple changes of the directions of the nuclear flux density remind us of the “quantum accordion effect”, which has been discussed in ref 29 for the nuclear flux densities of vibrating D2+(2Σ+g ) and Na2(21Πg), derived from pump−probe spectroscopy with ultrahigh spatiotemporal resolutions (see refs 30 and 31, respectively). The phenomenon is again a consequence of the interferences of different parts (“head” and “tail”) of the wave function. Because wavepacket dispersion increases with time, the resulting interferences in the nuclear flux densities at t = 18 fs are richer than those at t = 9 fs. This conjecture is supported by wavepacket interferometry, which may induce analogous interferences not only in the nuclear densities but also in the flux densities.27 For comparison, Figure 5 also documents the related interferences in

Figure 5. Nuclear and electronic densities ρa and ρe and the radial nuclear and electronic flux densities jaRa and jer versus the distance Ra and r of the nucleus a and the electron from the nuclear center of mass of the vibrating H2+(2Σ+g ,JM=00). Note the triple layer structure of ρa and the quadruple changes of the directions of jaRa during the turn from quantum bubble contraction to expansion at t = 18 fs. Compare with Figure 4 bottom panel.

the nuclear density ρaRa(Ra,t=18fs). At the same time, Figure 5 also shows the electronic density ρe(r,t=18fs). Apparently, most of the fine structures in the nuclear density ρaRa(Ra,t=18fs) based on nuclear wavepacket interferences are “washed out” in the electronic densities, similar to the “washing out” effect in the electronic flux density jer(r,t=18fs), compared to the nuclear jaRa(Ra,t).

4. CONCLUSIONS The snapshots of the concerted nuclear and electronic densities and flux densities of the vibrating H2+(2Σ+g ,JM=00) in the nuclear center of mass/laboratory frame shown in Figure 4 remind us of radially symmetric pulsation of an isotropic bubble that consists of two different components: a rather narrow, sticky spherical layer for the nuclei that is overcast by a broad, soft gel for the electrons. Hence the vibrating H2+(2Σ+g ,JM=00) appears as a pulsating bubble in the laboratory frame, with a very short (approximately 18 fs) period of pulsation. At the turns from expansion to contraction, or vice versa, the nuclear layer exhibits prominent interference patterns that appear as isotropic spherical multilayers. At some instances, the individual nuclear sublayers may exhibit typically two, sometimes even three isotropic sub-sublayers that run in opposite directions, with corresponding changes of signs in the radial nuclear flux densities. To some extent, the electronic gel sticks to the nuclear layer(s), thus mimicking part of the nuclear interferences, but all the fine structures tend to be washed out. It still remains a challenge to discover an instant (possibly for a different scenario) with more than one changes of sign in the radial electronic flux density. In any case, the origin of the more or less prominent patterns in the nuclear layer and its electronic overcast are quantum interferences of the “heads” and “tails” of the underlying wavepacket that run in opposite directions when turning from expansion to contraction, or vice versa. We thus conclude that the vibrating H2+(2Σ+g ,JM=00) appears as a pulsating quantum bubble in the laboratory frame. In contrast, for example, classical soap-bubbles do not exhibit any multilayers with alternating expansions and contractions at any instant. The present quantum dynamics simulations are based on the quantum theory derived in section 2. It is presented step-by-step and with all details so that it can serve as a reference for various extensions including investigations of concerted nuclear and electronic densities and flux densities of homo- and heteronuclear 8422



Article

automatically imply that the electronic flux density has the same radial symmetry? These are fascinating and demanding problems that are stimulated by the present derivation. The present results also suggest a way to the (first!) experimental measurements of intramolecular electronic flux densities, as follows: Recently, our experimental colleagues have demonstrated the first observations of moving electronic densities, with time resolutions from a few femtoseconds to subfemtoseconds;50 see also the theoretical concepts for alternative measurements of the electronic densities.51−55 We suggest that these techniques should be applied to monitor the isotropic time-dependent electron density in vibrating H2+(2Σ+g ,JM=00), in the laboratory frame. One could then employ the radial electronic continuity equation (56) together with the boundary condition (58), to determine the “experimental” electronic flux density (44) with radial symmetry, i.e., with exclusive radial component jer(r,t); see eq 59. This approach to the electronic jer(r,t) via the experimental ρe(r,t) is entirely analogous to the first determination of experimental radial nuclear flux densities jnR(R,t)29 based on experimental pump− probe spectroscopy30,31 of the nuclear reduced radial densities ρR(R,t), e.g., for vibrating D2+ or Na2.

diatomic molecules or ions with one, two, or even many electrons; work along these lines is in progress. It is illuminating to compare the present quantum theory for the scenario of the vibrating/pulsating H2+(2Σ+g ,JM=00) in the ncom/laboratory frame with the previous quantum dynamics simulations of the four concerted quantum objects, ρn, ρe, jn, and je for vibrating aligned H2+(2Σ+g ) in the laboratory frame, equivalent to vibrating H2+(2Σ+g ) in the molecular frame;4,5 see also refs 1−3. Those rather familiar scenarios imply cylindrical symmetry, instead of the present spherical symmetry. As a consequence, aligned H2+(2Σ+g ) has nuclear flux density parallel to the aligned internuclear axis, whereas the electronic flux density consists of two components, the “major” one that travels with (i.e., parallel to) the nuclei and the “minor” component that moves opposite to the nuclear direction of motion. It turned out to be rather straightforward to evaluate the “major” component of the electronic flux density, in the frame of the Born− Oppenheimer approximation,2,3 but it required the development of a demanding hierarchical approach to calculate the minor one, based on its major counterpart.4 In contrast, in the present scenario of the vibrating H2+(2Σg+,JM=00) in the ncom/ laboratory frame, there is just one nonzero component of the electronic flux density, i.e., the radial one. In practice, its evaluation is as “easy” as the evaluation of the major component for oriented vibrating H2+(2Σ+g ), but the present scenario has the huge advantage that it does not call for applications of the next steps of the hierarchical approach to calculate any “minor” components; compare with ref 4. We conclude that the present scenario of the vibrating H2+(2Σ+g ,JM=00) as a pulsating quantum bubble in the laboratory frame is much more convenient than the familar scenario of aligned H2+(2Σ+g ). The detailed approach in section 2 shows that this is a consequence of the considerations of the vibrating H2+(2Σ+g ,JM=00) with exclusive orbital angular quantum numbers JM = 00. The interplay of any partial waves with other nuclear orbital quatum numbers would prohibit important steps of the derivation, e.g., from eqs 30 and 32 to eqs 35 and 36, causing nonisotropic electronic densities and corresponding electric flux densities with rather complex structures, possibly with a “major” radial component but including also “minor” perpendicular (angular) ones. The present scenario of quasi-instantaneous FC-type preparations of the H2+(2Σg+,JM=00) ions by photoionization of the H2(1Σ+g ,vJM=000) molecules may be extended to quantum dynamics simulations of the effect of the laser pulse, from short to long pulse durations, cf. ref 50. In the continuous wave limit, this should lead to the preparation of H2+(2Σ+g ,vJM=v00) in selective isotropic rovibrational eigenstates with nuclear rovibrational quantum numbers vJM = v00, including the ground state vJM = 000. This would correspond to stationary, isotropic “quantum bubbles” in the ncom/laboratory frame, with zero nuclear and electronic fluxes. Certainly, applications of short near resonant laser pulses induce much richer quantum dynamics of the quantum bubble. It remains a challenge to extend the present derivation to a more general theory that does not make use of the Born− Oppenheimer approximation (BOA). For example, if the full (non-BOA) molecular wave function and the related density is isotropic with respect to the nuclear degrees of freedom (dofs), in the ncom/laboratory frame, does this automatically imply that the electron distribution must also be isotropic? In other words, is an anisotropic electronic density compatible with an isotropic nuclear density, or would that lead to a contradiction? Likewise, if the nuclear and the electronic angular momenta are equal to zero and if the nuclear flux density has radial symmetry, does this

■

APPENDIX The purpose of this Appendix is to show that the present scenario of the preparation of the initial state; i.e., ultrafast single photon photoionization of the precursor molecule H2(1Σ+g ,vJM=000) prepares the nascent H2+(2Σ+g ,JM=00) with equal probabilities for electron spin up (α spin) or spin down (β spin). For this purpose, we use a simple, ubiquitious model, which means we assume that, in the frame of the BOA, the initial electronic wavefunction may be written by means of normalized Slater determinants ΨH (1Σ+),e(1,2) 2

g

= ψH (1Σ+),e(x1,y1,z1,σ1,x 2 ,y2 ,z 2 ,σ2 ;R ,Θ,Φ) 2

g

1 = [|χ (1) α(1) χ2 (2) β(2)| − |χ1 (1) β(1) χ2 (2) α(2)|] 2 1 1 = [χ1 (1) χ2 (2) + χ1 (2) χ2 (1)] × [α(1) β(2) − β(1) α(2)] 2 (68)

Before application of the photoionizing laser pulse, the orbitals χ1 and χ2 are the same; i.e., basically, they are σg orbitals.48 The meanings of the shorthand notations are χ1 (1) = χ1 (x1 ,y1 ,z1;R ,Θ,Φ) α(1) = α(σ1)

(69)

with spatial electronic and nuclear coordinates x1, y1, z1 and R, Θ, Φ, and with electronic spin variable σ1 for the first electron, and with analogous notations for the other occupied orbitals χ2(2), χ1(2), and χ2(1) and spin functions β(2), α(2), and β(1). Note that this Appendix is the only place of this paper where the electronic spin functions and spin variables are written explicitly; elsewhere, they are taken into account just implicitly. Immediately after ultrafast Franck−Condon type photoionization, we assume that on one hand, the nuclei did not yet have time enough to move; i.e., the values of the nuclear coordinates are still the same as before. On the other hand, either electron 1 or 2 is photoionized, which means they occupy two different orbitals. In the picture of the so-called single active electron (SAE; see, e.g., ref 56 and the references quoted 8423



■

therein), the overall expresssion for the electronic wave function (68) remains unchanged, but one of the orbitals, e.g., χ1, adapts to the electronic wave function (molecular orbital) of the ion H2+(2Σ+g ,JM=00), whereas the other one, χ2, runs so far away from the nuclei that it no longer overlaps with χ1, ∞

∞

*Y. Yang: e-mail, [email protected]; phone, +86 (0)351 701 1045; fax, +86 (0)351 701 8927. Notes

The authors declare no competing financial interest.

(70)

The electronic density corresponding to the wave function (68) of the precursor molecule H2(1Σ+g ,vJM=000) is g

= ρH (1Σ+),e (x1,y1,z1,σ1,x 2 ,y2 ,z 2 ,σ2 ;R ,Θ,Φ) 2

g

= |ψH (1Σ+),e(x1,y1,z1,σ1,x 2 ,y2 ,z 2 ,σ2 ;R ,Θ,Φ)|2 2

g

1 = [χ1 (1) χ2 (2) + χ1 (2) χ2 (1)]2 × [α(1) β(2) − β(1) α(2)]2 4 1 2 = [χ1 (1) χ2 2 (2) + 2χ1 (1) χ2 (2) χ1 (2) χ2 (1) + χ12 (2) χ2 2 (1)] 4 × [α 2(1) β 2(2) − 2α(1) β(2) α(2) β(1) + α 2(2) β 2(1)] (71)

with χ1 = χ2. After photoionization, the electronic density (71) represents the total ionized system, which consists of the H2+(2Σ+g ) ion with electronic wave function/orbital χ1 plus the electron that has run away, sitting in the nonoverlapping orbital χ2. This density of the two electrons may be reduced to the electron density of the H2+(2Σ+g ) ion by integrating over the spatial and spin variables of the electron that has run away. The antisymmetrization of the wave function implies two possibilities that we label as a and b; i.e., either electron 1 or electron 2 runs away, which means one occupies the orbital χ2 while the other one sits in orbital χ1. Accordingly, the electron density of the nascent H2+(2Σ+g ) ion is g

= ρH +(2Σ+)(x ,y ,z ,σ )a + ρH +(2Σ+)(x ,y ,z ,σ )b 2

g

2

(72)

g

where ρH +(2Σ+)(x ,y ,z ,σ )a = 2

g

∫ d2 ρH ( Σ ),e (1,2)|x ,y ,z ,σ =x ,y ,z ,σ 2

1 + g

1 1 1 1

(73)

for the case where electron 1 sits in orbital χ1 and ρH +(2Σ+)(x ,y ,z ,σ )b = 2

g

∫ d1 ρH ( Σ ),e (1,2)|x ,y ,z ,σ =x ,y ,z ,σ 2

1 + g

2 2

2

2

(74)

for the alternative case where electron 2 sits in orbital χ1. The sum (72) of the two terms (73) and (74) may be evaluated, using expression (71) for the nonoverlapping normalized orbitals (70) and also using the orthogonality of α and β spin functions, ρH +(2Σ+)(x ,y ,z ,σ ) = χ12 (x ,y ,z ;R ,Θ,Φ) × 2

g

ACKNOWLEDGMENTS

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REFERENCES

(1) Barth, I.; Hege, H.-C.; Ikeda, H.; Kenfack, A.; Koppitz, M.; Manz, J.; Marquardt, F.; Paramonov, G. K. Concerted Quantum Effecis of Electronic and Nuclear Fluxes in Molecules. Chem. Phys. Lett. 2009, 481, 118−123. (2) Diestler, D. J. Coupled-channel Quantum Theory of Electronic Flux Density in Electronically Adiabatic Processes: Fundamentals. J. Phys. Chem. A 2012, 116, 2728−2735. (3) Diestler, D. J.; Kenfack, A.; Manz, J.; Paulus, B. Coupled-Channels Quantum Theory of Electronic Flux Density in Electronically Adiabatic Processes: Application to the Hydrogen Molecule Ion. J. Phys. Chem. A 2012, 116, 2736−2742. (4) Diestler, D. J.; Kenfack, A.; Manz, J.; Paulus, B.; Pérez-Torres, J. F.; Pohl, V. Computation of ihe Elecironic Flux Density in the BornOppenheimer Approximation. J. Phys. Chem. A 2013, 117, 8519−8527. (5) Pérez-Torres, J. F. Electronic Flux Densities in Vibrating H2+ in Terms of Vibronic Eigenstates. Phys. Rev. A 2013, 87 (062512), 1−7. (6) Kenfack, A.; Barth, I.; Marquardt, F.; Paulus, B. Molecular Isotopic Effects on Coupled Electronic and Nuclear Fluxes. Phys. Rev. A 2010, 82 (062502), 1−7. (7) Okuyama, M.; Takatsuka, K. Electron Flux in Molecules Induced by Nuclear Motion. Chem. Phys. Lett. 2009, 476, 109−115. (8) Yonehara, T.; Takatsuka, K. Characterization of Electron-deficient Chemical Bonding of Diborane with Attesecond Electron Wavepacket Dynamics and Laser Response. Chem. Phys. 2009, 366, 115−128. (9) Nagashima, K.; Takatsuka, K. Electron-Wavepacket Reaction Dynamics in Proton Transfer of Formamide. J. Phys. Chem. A 2009, 113, 15240−15249. (10) Takatsuka, K.; Yonehara, T. Nonadiabatic Chemical Dynamics in Intermediate and Intense Laser Fields. Adv. Chem. Phys. 2010, 144, 93− 156. (11) Takatsuka, K.; Yonehara, T. Exploring Dynamical Electron Theory beyond the Born-Oppenheimer Framework: from Chemical Reactivity to non-Adiabatically Coupled Electronic and Nuclear Wavepackets on-the-fly under Laser Field. Phys. Chem. Chem. Phys. 2011, 13, 4987−5016. (12) Okuyama, M.; Takatsuka, K. Dynamical Electron Mechanism of Double Proton Transfer in Formic Dimer. Bull. Chem. Soc. Jpn. 2012, 85, 217−227. (13) Nagashima, K.; Takatsuka, K. Early-Stage Dynamics in Coupled Proton-Electron Transfer from the π − π* State of Phenol to Solvent

ρH +(2Σ+)(x ,y ,z ,σ ) 2

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We are grateful to Professor D. J. Diestler (Lincoln) for illuminating discusssions. J.M. also expresses his gratitude to Professors D. T. Anderson (Laramie), W. Demtrö d er (Kaiserslautern), J. Küppers (Hamburg), S. D. Peyerimhoff (Bonn), and M. Quack (Zürich) for stimulating discussions and advice on the present scenario of the preparation of the initial state; see subsection 3.1 and the Appendix. Cordial thanks also go to Professor A. Welford Castleman, Jr., for continuous engagement for femtosecond chemistry and for our scientific community.57 This work profits from financial support, in part from the Deutsche Forschungsgemeinschaft, the 973 Program of China under Grant No. 2012CB921603, the National Natural Science Foundation of China under Grant No. 11004125, and the Major Program of the Natonal Natural Science Foundation of China under Grant No. 10934004.

ρH (1Σ+),e (1,2) 2

AUTHOR INFORMATION

Corresponding Author

∞

∫−∞ dx ∫−∞ dy ∫−∞ dz χ1(x ,y ,z) χ2 (x ,y ,z) = 0

Article

1 2 [α (σ ) + β 2(σ )] 2 (75)

The result (75) shows that after photoionization, the nascent H2+(2Σ+g ) ion is prepared with equal (=0.5) probabilities for its electron with spin up (α) or spin down (β). Integrating over the spin variable yields the spatial distribution with the factor 1/2·(1 + 1) = 1 for the sum of both possibilities. For the special case of the nuclear orbital angular momentum quantum numbers JM = 00, we obtain the isotropic distribution ρe(r,R), eq 39. 8424



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