Quasi-Classical Theory of Electronic Flux Density in Electronically

Article pubs.acs.org/JPCA

Quasi-Classical Theory of Electronic Flux Density in Electronically Adiabatic Molecular Processes D. J. Diestler University of NebraskaLincoln, Lincoln, Nebraska 68585, United States ABSTRACT: The standard Born−Oppenheimer (BO) description of electronically adiabatic molecular processes predicts a vanishing electronic flux density (EFD). A previously proposed “coupled-channels” theory permits the extraction of the EFD from the BO wave function for one-electron diatomic systems, but attempts at generalization to many-electron polyatomic systems are frustrated by technical barriers. An alternative “quasi-classical” approach, which eliminates the explicit quantum dynamics of the electrons within a classical framework, yet retains the quantum character of the nuclear motion, appears capable of yielding EFDs for arbitrarily complex systems. Quasi-classical formulas for the EFD in simple systems agree with corresponding coupledchannels formulas. Results of the application of the new quasi-classical formula for the EFD to a model triatomic system indicate the potential of the quasi-classical scheme to elucidate the dynamical role of electrons in electronically adiabatic processes in more complex multiparticle systems. fails spectacularly (i.e., always gives ⟨je(x,t)⟩ = 0). This is so because the electrons are constantly in a stationary state (i.e., the lowest energy eigenstate of the electronic Hamiltonian) and the EFD vanishes identically for such a state.3 A couple of ways to skirt this obstacle have been proposed. One way, pioneered by Barth et al.,4 is to describe the flow of electrons in terms of (scalar) electronic fluxes (Fe,S) through surfaces (S) that partition the system appropriately. They applied this approach to study the rate of flow of electrons relative to nuclei through planes normal to the internuclear axis in oriented H2+ vibrating in the electronic ground state. Using the same technique, Kenfack et al. examined the dependence of Fe,S on the initial state of the vibration in oriented D2+;5 they also investigated isotopic effects in oriented H2+.6 Employing a similar scheme, Andrae et al.7 monitored electronic fluxes that accompany the breaking and making of covalent bonds during the Cope rearrangement of semibullvalene. Another way around the obstacle (i.e., ⟨je(x,t)⟩ = 0) was proposed by Okuyama and Takatsuka.8 They define a complex “time-shift flux” operator, which they use to compute complex EFDs associated with intramolecular vibrations in H2, NaCl, and the formic-acid dimer. Because the BOA is the bedrock of the quantum description of electronically adiabatic molecular processes, we prefer not to abandon it forthwith, but rather to embrace it and to attempt to draw forth a useful approximation to ⟨je(x,t)⟩ from ΨBOA(r,R,t) directly. In this spirit we have developed a “coupled-channels” theory9 and tried it on the oriented hydrogen molecule ion (H2+) vibrating in the electronic ground state,10 which is the only system for which highly accurate (“exact”) maps of ⟨je(x,t)⟩ are available.4 For this simplest of all molecules the

I. INTRODUCTION The mechanism of many an electronically adiabatic elementary molecular process (i.e., a process that does not involve electronic excited states) would be illuminated by a knowledge of the flow of electrons that attends the process. We presume that the most detailed picture of that flow would comprise a time sequence of three-dimensional (vector) maps of the electronic (population) flux density (EFD) je(x,t) (i.e., the instantaneous rate of flow of electrons per unit area at points of observation x at time t). The classical expression for the EFD is je (x,t ) =

∑ δ[x − ri(t )]ri̇(t ) i

(1)

where the Dirac distributions δ[x − ri(t)] describe the electrons as point particles at positions ri(t) with velocities ṙi(t) ≡ dri/dt. In quantum mechanics the state of the system is characterized utterly by the wave function, and the observed EFD is given by the expectation value ⟨je(x,t)⟩. Many dynamic properties of electronically adiabatic processes (e.g., population densities, fluxes and yields (timeintegrated fluxes) of particles) can be adequately described by the Born−Oppenheimer approximation (BOA),1,2 which implies that the motion of the light, fast electrons responds instantaneously to that of the heavy, slow nuclei. The BOA wave function is expressed as a simple product, ΨBOA(r,R,t ) = Φ(r;R) ψ (R,t )

(2)

where r and R stand collectively for the electronic and nuclear coordinates, respectively, Φ(r;R) is the ground-state eigenfunction of the Hamiltonian describing the motion of the electrons in the field of the nuclei f ixed in the configuration R, and ψ(R,t) is a wave packet governing the motion of the nuclei which satisfies a nuclear Schrödinger equation in which the ground-state eigenvalue of the electronic Hamiltonian serves as the potential energy. Despite the utility of the BOA, a straightforward attempt to compute ⟨je(x,t)⟩ from ΨBOA(r,R,t) © XXXX American Chemical Society

Special Issue: A: Jorn Manz Festschrift Received: May 16, 2012 Revised: July 5, 2012

A

dx.doi.org/10.1021/jp304747a | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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coupled-channels theory gives an EFD strictly parallel with the internuclear axis that agrees very well with the parallel component of the “exact” EFD. The coupled-channels scheme can be arduously extended to two-electron systems (e.g., H2), but attempts to generalize the theory to polyatomic systems are thwarted by crippling technical problems. To circumvent these, we put forth here an alternative approach to the extraction of ⟨je(x,t)⟩ directly from ΨBOA(r,R,t) for arbitrarily complex systems.

∫ dx ρa(1)(x;R) = N(a)

to Mulliken’s “gross atomic population” N(a) on atom a. We now replace eq 1 with the electronic analogue of eq 3: je (x,t ) =

∑ δ[x − Ra(t )]Ṙ a(t ) a

∑ niϕi 2(x;R) i

III. QUASI-CLASSICAL FORMULAS FOR THE ELECTRON FLUX DENSITY IN ONE-ELECTRON SYSTEMS A trivial example is the H atom translating freely in the electronic ground state (1s). Because there is only a single nucleus, eq 5 reduces to

(3)

ϕ1(x;R) = χ (1) (x;R) = φ1s(|x − R|)

∑ χa(i) (x;R)

ρ(1)(x;R) = φ1s 2(|x − R|)

je (x,t ) = ρ(1)(x;R(t )) Ṙ (t ) = φ1s 2(|x − R(t )|)Ṙ (t )

(4)

⟨je (x,t )⟩ = Re[⟨ψ (t )|φ1s 2(|x − R|) Ṙ |ψ (t )⟩] =

∑ cl(i)(R)φl (x − Ra) a

a

(10)

which has an intuitively appealing interpretation. Because the proton is strictly localized, jn vanishes everywhere except at x = R(t) (see eq 3). In contrast, the electron is smeared spherically about the proton. Therefore, je is proportional to the proton’s velocity by the EPD at x − R(t). If the quantum behavior of the proton is significant, one must regard je in eq 10 as an operator and compute the expectation value using the protonic wave packet ψ(R,t). Thus, we have

where χa(i) (x;R) =

(9b)

where R now signifies the position of the proton. From eq 8 we get the quasi-classical result

(5a)

a

(9a)

and eq 6 to

where ni is the number of electrons that occupy the ith MO. The MOs are expanded as linear combinations of (normalized) atomic orbitals (AOs) φla centered on the nuclei, ϕi =

(8)

Although the expression for jn in eq 3 is strictly classical, the formula for je in eq 8 is “quasi-classical” in that the quantum nature of the electrons is implicit in ρ(l) a . The dynamics of the electrons that is manifest in eq 1 is submerged in eq 8; only the explicit dynamics of the nuclei remains. Hence, in the quasiclassical scheme the expectation value ⟨je(x,t)⟩ is computed from ψ(R,t) alone.

where Ra(t) and Ṙ a(t) stand for the position and velocity of nucleus a in the laboratory coordinate frame. The BOA yields an accurate (nonzero) map of ⟨jn(x,t)⟩. The reason is that the nuclear motion is well characterized by the generally nonstationary nuclear wave packet ψ(R,t). This suggests that if the electrons could be represented by distributions ρ(1) a (x;R) associated with the individual nuclei a (analogous to the Dirac distributions in eq 3), then a nonvanishing ⟨je(x,t)⟩ could be computed from ψ(R,t). However, rather than being strictly localized, as are the nuclei themselves, the ρ(1) a are expected to be diffuse. To find ρ(1) a , we recall that for a given (fixed) nuclear configuration R the electrons are distributed over the entire nuclear framework according to the electronic population density (EPD)ρ(1)(x;R). Making the LCAO-MO ansatz11 (i.e., assuming that Φ can be expressed as an antisymmetrized product of molecular orbitals (MOs) ϕi), we obtain ρ(1)(x;R) =

∑ ρa(1)(x;R(t ))Ṙ a(t ) a

II. QUASI-CLASSICAL ROUTE TO THE ELECTRONIC FLUX DENSITY We start with the analogue of eq 1 for the nuclear flux density jn: jn (x,t ) =

(7) 12

ℏ 2iM p

∫ dR φ1s2(|x − R|)[ψ *(R,t )∇R ψ (R,t )

− ψ (R,t )∇R ψ *(R,t )] (5b)

(11)

The coefficients are determined through the variational principle (i.e., the c(i) la are varied so as to minimize the expectation value of the electronic Hamiltonian at the fixed nuclear configuration R). Combining eqs 4 and 5a, we can write ρ(1)(x;R) as

In the second line of eq 11 we employ the coordinate representation and introduce the classical-quantum correspondences R → R and Ṙ → −iℏ∇R/Mp, where Mp stands for the mass of the proton. By the same procedure we obtain the quantum flux density of the proton

la

c(i) la

ρ(1)(x ; R) =

a

≡

⟨jp (x,t )⟩ = Re[⟨ψ (t )|δ(x − R)Ṙ |ψ (t )⟩]

∑ [∑ ni χa(i) (x;R) ϕi(x;R)] i

∑ ρa(1)(x;R) a

=

ℏ [ψ *(R,t )∇R ψ (R,t ) − ψ (R,t )∇R ψ *(R,t )]R = x 2iM p (12)

(6)

Combining eqs 11 and 12, we get

The second line of eq 6 identifies the electronic population distributions (EPDs) associated with the nuclei. Incidentally, this definition of ρ(1) a (x;R) is related by

⟨je (x,t )⟩ = B

∫ dR φ1s2(|x − R|)⟨jp(R,t )⟩

(13)



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x′ = x − 9

Because the proton is no longer precisely localized, the EPD must be weighted by the protonic flux density at R. We note that the expression in eq 13 reached via the quasi-classical pathway is identical with the coupled-channels expression (see eq 3.5 of ref 9). In the case of molecules we are primarily interested in intramolecular flow of electrons. The simplest example is afforded by the H2+ molecule vibrating in the electronic ground state. To isolate the intramolecular component of je, we transform to Jacobi coordinates 9 = (R a + R b)/2

(14a)

R = R b − Ra

(14b)

He sees the nuclei at positions R′a = R a − 9 = −R/2

(21a)

R′b = R b − 9 = R/2

(21b)

Substitution of eqs 21 into eq 8 yields je,NCM (x′,t ) = ρa(1)(x′;R(t ))[Ṙ ′a(t ) + 9̇ (t )] + ρ (1)(x′;R(t ))[Ṙ ′b(t ) + 9̇ (t )] b

=

where a and b label the protons. Equation 8 can then be recast as je (x,t ) = ρ(1)(x;9,R)9̇ +

where the subscript stresses that the point of observation x′ is referred to the NCM. The second line of eq 22 follows from eq 21 and the condition 9̇ (t ) = 0 that obtains in the perspective of the NCM coordinate frame. The formulas in eqs 16 and 22 agree, except for the argument of ρ(1) α . In the laboratory frame the EFD depends on both NCM and relative coordinates, whereas in the NCM frame it depends only on the relative coordinate. If 9 = 9̇ = 0, then x coincides with x′, according (1) to eq 20, and ρ(1) α (x;0,R) = ρα (x′;R).

(15)

1 (1) [ρ (x;0,R(t )) − ρa(1)(x;0,R(t ))]Ṙ (t ) 2 b

IV. QUASI-CLASSICAL TREATMENT OF THE TRIATOMIC SYSTEM To apply the quasi-classical technique to the general triatomic system (abc), we adopt the perspective of an observer riding on nucleus b. The relation between points of observation in the laboratory (x) and b (x″) coordinate frames is

(16)

By symmetry the EPDs satisfy the relation ρb(1)(x;0,R) = ρa(1)( −x;0,R)

(17)

Therefore, an observer facing b measures a value of je at x equal to the electronic distribution on b times the velocity of b relative to himself at the origin, which is one-half the relative velocity Ṙ of the protons. On the other hand, facing a, he measures a value of je at −x, which is exactly opposite. The corresponding quantum expression is 1 ⟨je (x,t )⟩ = 2

∫ dR

[ρb(1)(x;0,R)

−

x″ = x − R b

(23)

and the positions of the nuclei in the b frame are

ρa(1)(x;0,R)]⟨jb ,a (R,t )⟩ (18)

where ⟨jb ,a (x,t )⟩ =

1 (1) [ρ (x′;R(t )) − ρa(1)(x′;R(t ))]Ṙ (t ) 2 b (22)

1 (1) [ρ (x;9,R) − ρa(1)(x;9,R)]Ṙ 2 b

where we suppress the argument t on the right-hand side (RHS). The first term on the RHS of eq 15 is analogous to the RHS of eq 10 and has a similar intuitively satisfying interpretation. The second term represents the intramolecular contribution to je, which we interpret as follows. We suppose that the nuclear center of mass (NCM) 9 is f ixed at the origin (i.e., 9 = 9̇ = 0). Then eq 15 reduces to je (x,t ) =

(20)

ℏ [ψ *(R,t )∇R ψ (R,t ) − ψ (R,t )∇R ψ *(R,t )]R = x 2iμab (19)

R″a = R a − R b ≡ R1

(24a)

R″b = R b − R b = 0

(24b)

R″c = R c − R b ≡ R 2

(24c)

From eqs 8 and 24 we have je,b (x″,t ) = ρa(1)(x″;R1(t ),R 2(t ))[Ṙ ″a(t ) + Ṙ b(t )] + ρb(1)(x″;R1(t ),R 2(t ))[Ṙ ″b(t ) + Ṙ b(t )] + ρ(1)(x″;R1(t ),R 2(t ))[Ṙ ″c (t ) + Ṙ b(t )]

is the flux density of nucleus b relative to nucleus a and μab = Mp/2 is the reduced mass associated with the motion of b relative to a. (Again, we note that the expression in eq 18 reached by the quasi-classical approach agrees with the coupledchannels formula in eq 5.7 of ref 9.) Thus, by analogy with the H atom, the intramolecular EPD (i.e., the combination (1) + [ρ(1) b (x;0,R) − ρa (x;0,R]) in H2 must be weighted by the relative quantum nuclear flux density, if quantum behavior of the nuclei is to be taken into account.10 The expression in eq 16 for the intramolecular contribution to the EFD can be reached by an alternate pathway. Instead of fixing the NCM at the origin of the laboratory coordinate frame we assume that the observer rides on the NCM, where his point of observation (x′) is related to that (x) of an observer in the laboratory frame by

c

=

ρa(1)(x″;R1(t ),R 2(t ))Ṙ 1(t ) + ρc(1)(x″;R1(t ),R 2(t ))Ṙ 2(t )

(25)

The second equality in eq 25 is due to eqs 24. Note that the EFD generally depends on six degrees of nuclear freedom (i.e., the rotational and vibrational modes). These are described classically by a trajectory in the 12-dimensional nuclear phase space. Quantum effects can be accounted for by calculating the expectation value of the operator corresponding to je,b(x″,t), which we can show to be expressible as C


The Journal of Physical Chemistry A ⟨je,b (x″,t )⟩ =

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electron occupies ϕ2. Therefore, from eqs 4−6 we deduce the formula

∫ dR1 ∫ dR 2 [ρa(1)(x″;R1,R 2) + Ma ρc(1)(x″;R1,R 2)/Mab]j1(R1,R 2,t ) +

∫ dR1 ∫

ρα(1) = 2ϕ1 χα(1) + ϕ2 χα(2)

dR 2 [ρc(1)(x″;R1,R 2)

+ Mc ρa(1)(x″;R1,R 2)/Mbc]j2 (R1,R 2,t )

(26)

ℏ [χ *(R1,R 2,t )∇R1 χ (R1,R 2,t ) 2iμab − χ (R1,R 2,t )∇R1 χ *(R1,R 2,t )]

j2 (R1,R 2,t ) ≡

(30)

We treat the nuclear motion classically, assuming that the nuclei a and c move in concert along the reaction path (RP, i.e., the path of steepest descent from the saddle point on the PES). On the RP the phase (R1, R2, Ṙ 1, Ṙ 2) of the nuclear motion is constrained by the relation R2 = R2(R1) and by energy conservation

where for the sake of notational economy, we define the auxiliary nuclear flux densities j1(R1,R 2,t ) ≡

α = a, c

M(R1̇ 2 + Ṙ 2 2)/2 + V (R1 ,R 2) = E

(31)

where M = Ma = Mc, V(R1,R2) is the ground-state PES, and E is the (fixed) total energy. We model the RP as depicted in Figure 1. In the asymptotic region (ab + c), where atom c does not

(27a)

ℏ [χ *(R1,R 2,t )∇R 2 χ (R1,R 2,t ) 2iμbc − χ (R1,R 2,t )∇R 2 χ *(R1,R 2,t )]

(27b)

In eq 27 the six-dimensional normalized wavepacket describing rotational and vibrational motions obeys the nuclear Schrödinger equation ℏ2 2 [∇R1 /μab + ∇2R2 /μbc − 2∇R1 ·∇R2 /M ]+ V (R1,R 2)}χ (R1,R 2,t ) 2 ∂χ (R1,R 2,t ) = iℏ (28) ∂t

{−

where μαβ ≡ MαMβ/Mαβ, Mαβ ≡ Mα + Mβ and M ≡ Ma + Mb + Mc, and V(R1,R2) is the potential energy surface (PES) of the ground electronic state. We note that the general formula in eq 26 concurs with the expression for the one-electron triatomic system that can be derived by the coupled-channels approach.9 We now illustrate the application of the quasi-classical formula in eq 25 through the model symmetric, collinear exchange reaction L + MaL → LMa + L, where the L's stand for identical “light” end atoms and Ma for a “massive” central atom.13 We assume that the nuclei are constrained to the x-axis and so arranged that Ra ≤ Rb ≤ Rc. For this highly idealized system the b frame coincides with the NCM frame, whose origin we take to be coincident with the origin of the laboratory frame. Then eq 25 reduces to je (x,t ) = ρa(1)(x;R1 ,R 2)R1̇ + ρc(1)(x;R1 ,R 2)Ṙ 2

Figure 1. Schematic of reaction path (RP) for model collinear, symmetric exchange reaction L + MaL → LMa + L. Parameters are M = 2 × 103, V† =10−2, E = 2 × 10−2, and R01 = 2 in atomic units.

interact with diatom ab, R1 = −R01 (where R01 is the equilibrium internuclear separation) and 2R01 ≤ R2 < ∞; in the other asymptotic region (a + bc), where a does not interact with bc, R2 = R01 and −∞ < R1 ≤ −2R01. Along the asymptotic portions of the RP the PES is constant, which we set to zero for convenience (i.e., V = 0 on the asymptotic RP). In the region of configuration space where all three atoms interact (−2R01 ≤ R1 ≤ 0; 0 ≤ R2 ≤ 2R01) the RP is approximated by the arc of a circle of radius R01 centered at (−2R01, 2R01), along which R1 = R01(cos θ − 2) and R2 = R01(sin θ + 2), where θ is defined in Figure 1. We approximate the PES along the RP in the interaction region by the form

(29)

where R1 and R2 are now scalars. Note that je is parallel with the x-axis because of the assumed collinearity of the collision. Because the present purpose is not to construct accurate maps of je, but rather to demonstrate semiquantitatively how the new formula in eq 8 can yield an EFD that would reasonably be expected to characterize the process, rough ̇ approximations to the EPDs (ρ(1) α ) and nuclear velocities (Ri) are sufficient. Hence, we regard the electronic Hamiltonian as a sum of effective one-electron Hamiltonians and invoke Hückel’s method.11 For simplicity, we account for only the valence electrons, of which we assume there are three (e.g., L = H and Ma is an alkali metal) and determine the MOs in the minimum basis (i.e., a 1s AO on each nucleus; φ1s(r) = π−1/2 exp(−r) in atomic units). Knowing the nuclear configuration (R1,R2), we can compute the energies and wave functions of the three lowest-energy MOs ϕ1, ϕ2, and ϕ3, which correspond respectively to bonding, nonbonding and antibonding MOs. In the electronic ground state two electrons occupy ϕ1 and one

(32) V = V† sin 2 2θ † where V is the height of the classical barrier (activation energy) at the saddle point. We now describe the flow of electrons as the nuclear configuration (R1, R2) traverses the RP from reactants ab + c to products a + bc. For this purpose we set the model parameters to the values given in the caption of Figure 1. In particular, we fix the total energy sufficiently high that the barrier is surmounted. We assume that initially c is far from ab (i.e., R2(0) ≫ 2R01) and ab has zero vibrational energy (i.e., the internuclear distance R1 is fixed at R01). Because Ṙ 1 = 0 and V = D



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the incipient interaction between ab and c results in mixing of the AO on c with those on ab, so that χ(1) c ≠ 0. Therefore, a non-negligible overlap between φ1sb in ϕ1 and χc(1) ∝ φ1sc engenders the slight skewing of the map of je(x,t) toward b that is apparent in Figure 2A. Parts B−D of Figure 2 exhibit a succession of maps of je(x,t) as the nuclear configuration moves halfway along the RP. As θ decreases from 0 to −π/4 and V increases from 0 to a maximum of V†, the magnitude of Ṙ 1 increases while that of Ṙ 2 decreases. Correspondingly, je(x,t) in the vicinity of a waxes while je(x,t) around c wanes, until they become equal at θ = −π/4. When θ = −π/4, R1 = R2 and the (1) EPDs are related by ρ(1) c (x;R1,R2) = ρa (−x;R1,R2). Thus the map of je(x,t) becomes exactly symmetric about b at θ = −π/4 (see Figure 2D). Because V is symmetric about θ = −π/4, maps of je(x,t) on the second half (−π/4 ≥ θ ≥ −π/2) of the RP are just reflections (about the y-axis) of those at corresponding points on the first half (0 ≥ θ ≥ −π/4). After the nuclear configuration exits the interaction region at (−2R01, R01) and enters the asymptotic region (a + bc), it proceeds along the line R2 = R01 at constant velocity Ṙ 1 = −(2E/ M)1/2. Since R2 is fixed, Ṙ 2 = 0 and from eq 29 we obtain je(x,t) ̇ = ρ(1) a (x;R1,R2(t))R1. Because the reaction is symmetric, as R1 becomes very large ϕ1 reduces to the bonding orbital of the isolated bc and ϕ2 to the nonbonding AO on a. Hence, ρ(1) a → ϕ22 = [φ1sa(|x − R1(t)ex|)]2. As a consequence, the map of je(x,t) stays spherically symmetric about a as it moves away from bc at constant velocity without distortion. Note that again because Ṙ 1 < 0, je(x,t) < 0 everywhere.

0, we deduce from eq 31 that c translates toward ab at constant velocity Ṙ 2(0) = −(2E/M)1/2. Likewise, from eq 29 we get je (x,t ) = ρc(1)(x;R1 ,R 2(t ))Ṙ 2(0) = [2ϕ1 χc(1) + ϕ2 χc(2) ]Ṙ 2(0)

(33)

where the second line follows from eq 30. When R2 is large, however, ϕ1 degenerates to the bonding MO for the isolated ab (i.e., χ(1) → 0) and ϕ2 to the (nonbonding) 1s AO on the c 2 2 isolated c. Thus, ρ(1) c → ϕ2 = φ1sc (|x − R2(t)ex|). Hence, far into the asymptotic region (ab + c) je (x,t ) = φ1s 2(|x − R 2(t )ex|)Ṙ 2(0) c

(34)

is spherically symmetric about c and translates toward ab at constant velocity without distortion. Note that because Ṙ 2 < 0, je(x,t) < 0 everywhere. We remark on the analogy between the expression in eq 34 and that in eq 10, which pertains to the H atom moving at constant velocity in the laboratory frame. Figure 2 displays contour plots of je(x,t) for a selection of points on the RP in the interaction region. As ρ(1) is

V. CONCLUSION Our ultimate aim, toward which the present work modestly progresses, is to provide a useful semiquantitative picture of the flow in real time of electrons that accompanies an electronically adiabatic molecular process that is otherwise adequately described within the framework of the Born−Oppenheimer approximation. A previously proposed “coupled-channels” theory9 yields EFDs in accord with the those for simple systems for which analytic (e.g., H atom9) or numerically accurate (“exact”) (e.g., H2+ 10) treatments are available. However, technical problems frustrate the attempt to generalize the coupled-channels technique to polyatomic systems. The alternative quasi-classical approach proposed here yields a simple general formula for the EFD possessing the apparent virtue of applicability to arbitrarily complex systems. Input to the formula is available from standard procedures of quantum chemistry (BO electronic ground-state energy eigenfunction) and quantum dynamics (nuclear wavepacket). The quasiclassical formula yields formal expressions identical with those obtained previously for the simple systems mentioned above. Moreover, the results of its application to the model triatomic exchange reaction indicate a potential for handling more complex systems. Although the present results provide ground for optimism about the viability of the quasi-classical approach, we should elaborate here on a limitation to which we allude in the Introduction. According to the formula in eq 8, the component of the EFD contributed by an atom (nucleus) is parallel with its instantaneous velocity. In a diatomic molecule the intramolecular component of the EFD is therefore proportional to the relative velocity of the nuclei (i.e., the EFD is parallel with the internuclear axis). However, the “exact” non-BOA wave function for H2+ (oriented so that the nuclei lie on the z-axis)

Figure 2. Contour maps of je(x,t) for selected points (R1, R2) on the reaction path in the interaction region of the collinear, symmetric exchange reaction L + MaL → LMa + L(Figure 1). (A) R1 = −2.00, R2 = 4.00, Ṙ 1 = 0.00, Ṙ 2 = −4.48 × 10−3, θ = 0. (B) R1 = −2.04, R2 = 3.61, Ṙ 1 = −0.84 × 10−3, Ṙ 2 = −4.23 × 10−3, θ = −π/16. (C) R1 = −2.15, R2 = 3.24, Ṙ 1 = −1.48 × 10−3, Ṙ 2 = −3.58 × 10−3, θ = −π/8. (D) R1 = −2.59, R2 = 2.59, Ṙ 1 = −3.16 × 10−3, Ṙ 2 = −3.16 × 10−3, θ = −π/4.

cylindrically symmetric about the line of nuclear centers (xaxis), only the maps for the x−y plane are shown. Figure 2A corresponds to the point (R01, R2(t) = 2R01) where the nuclear configuration is on the threshold of the interaction region. The velocity of a relative to c (i.e., Ṙ 1) remains zero. Now, however, E



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yields both parallel (z) and transverse (r) components.4 Even though the quasi-classical EFD agrees well with the “exact” zcomponent, it has no r-component. To correct this deficiency in the quasi-classical scheme for collinear systems, one could solve the continuity equation for the tranverse component, taking the quasi-classical EFD to be the true parallel component as input.14 For more complex systems involving three or more (noncollinear) nuclei, the EFD is not in general parallel with the internuclear separations. We presume that transverse contributions analogous to those seen in oriented H2+ are present, yet unaccounted for. Our hope is that the quasi-classical approach can nevertheless furnish useful information on the dynamical role electrons play in electronically adiabatic processes in complex systems.

■ ■ ■

AUTHOR INFORMATION

Notes

The author declares no competing financial interest.

ACKNOWLEDGMENTS The author thanks Mr. Zhanping Xu for preparing Figure 2. REFERENCES

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Quasi-Classical Theory of Electronic Flux Density in Electronically

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