ARTICLE pubs.acs.org/JPCA
Coupled-Channels Quantum Theory of Electronic Flux Density in Electronically Adiabatic Processes: Fundamentals D. J. Diestler Institut f€ur Chemie und Biochemie, Freie Universit€at Berlin, 14195 Berlin, Germany, and University of Nebraska—Lincoln, Lincoln, Nebraska 68583, United States
bS Supporting Information ABSTRACT: The BornOppenheimer (BO) description of electronically adiabatic molecular processes predicts a vanishing electronic flux density (je), even though the electrons certainly move in response to the movement of the nuclei. This article, the first of a pair, proposes a quantum-mechanical “coupled-channels” (CC) theory that allows the approximate extraction of je from the electronically adiabatic BO wave function . The CC theory is detailed for H2+, in which case je can be resolved into components associated with two channels α (=a,b), each of which corresponds to the “collision” of an “internal” atom α (proton a or b plus electron) with the other nucleus β (proton b or a). The dynamical role of the electron, which accommodates itself instantaneously to the motion of the nuclei, is submerged in effective electronic probability (population) densities, Δα, associated with each channel (α). The Δα densities are determined by the (time-independent) BO electronic energy eigenfunction, which depends parametrically on the configuration of the nuclei, the motion of which is governed by the usual BO nuclear Schr€odinger equation. Intuitively appealing formal expressions for the electronic flux density are derived for H2+.
1. INTRODUCTION In the context of quantum mechanics a molecular process is described completely by the time-dependent state vector |Ψ(t)æ, which evolves according to Schr€ odinger’s equation governing the simultaneous coupled motions of electrons and nuclei. In the most common instance, with which we are concerned in this article, the process is taken to be electronically adiabatic (i.e., the light, fast electrons adjust instantaneously to the movements of the heavy, slow nuclei). To describe such a process, one typically invokes the BornOppenheimer approximation (BOA),13 expressing the total wave function as a simple product Æq, Q jΨðtÞæ Ψðq, Q , tÞ ¼ Φðq; Q ÞψðQ , tÞ
ð1:1Þ
where q and Q stand for collections of electronic (q) and nuclear (Q) coordinates (i.e., eigenvalues of electronic and nuclear position and spin operators, in general). In eq 1.1 Φ is an eigenfunction of the Hamiltonian describing the motion of the electrons in the field of the nuclei fixed in the configuration Q. The associated electronic eigenvalue, a function of Q, serves as the potential energy surface (PES) in the Schr€odinger equation obeyed by the nuclear wave function ψ(Q,t). The course of the process can be monitored variously through the densities of the particles (electrons or nuclei), fluxes of particles through prescribed surfaces, or flux densities of particles. The last are especially useful in that they afford a picture of the instantaneous rate of flow of particles. A nice example can be seen in the article by McCullough and Wyatt,4 who plot two-dimensional maps of the nuclear (probability) flux density computed from the nuclear wavepacket obtained by numerical solution of Schr€odinger’s equation for the symmetric collinear exchange reaction H + H2 f H2 + H on the ground-state PES. One’s understanding of the mechanism of r 2011 American Chemical Society
this fundamental reaction, as well as those of other arguably more important electronically adiabatic processes, should be significantly advanced by a knowledge of the electronic flux density (je). That the electronically adiabatic BOA wave function yields je ¼ 0
ð1:2Þ
is therefore not a little frustrating, although it is understandable. For every Q the electronic energy eigenstate [Φ(q;Q)] is stationary and the flux density of particles in a stationary state vanishes. Nevertheless, because the electrons certainly move in response to the movement of nuclei, there must be a concomitant electronic flux. Most prior theoretical studies in which nonzero je is observed deal with electronically nonadiabatic processes that couple two or more BOA states. One way to mix states is to apply an external field. For example, Steiner and Fowler5 analyze patterns of πelectronic ring currents in aromatic molecules subject to a stationary magnetic field and Barth et al.6 study ring currents induced in model Mgporphyrin by few-cycle femtosecond laser pulses. Other previous theoretical investigations abandon the BOA by coupling adiabatic states through the nuclear kineticenergy operator, which is neglected in reaching the BOA wavefuntion (eq 1.1). Nafie,7 for example, uses perturbation theory to compute nonadiabatic wave functions for vibronic states of molecules. He defines a “transition current density” that provides a visualization of the electronic motion accompanying transitions between these states. More recently Okuyama and Takatsuka8 Special Issue: Femto10: The Madrid Conference on Femtochemistry Received: August 15, 2011 Revised: October 19, 2011 Published: November 21, 2011 2728
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implement the semiclassical Ehrenfest theory in a numerical simulation of je associated with intramolecular vibrations in H2, NaCl, and the formic acid dimer . Since myriad valuable insights into molecular processes have been attained within the strictures of the BOA, and since the infrastructure of quantum chemistry rests on the BOA, one is naturally loath to abandon it merely in pursuit of the elusive je. One would rather confine oneself to the BOA and devise alternative measures to characterize the process of interest. Thus, Barth et al.9 investigate the fundamental question of the relative rate of rearrangement of electrons and nuclei by monitoring simultaneously the electronic and nuclear fluxes and yields (i.e., cumulative fluxes) through key “dividing surfaces”. They compare BOA with highly accurate numerical (to which we refer loosely henceforth as “exact”) results for H2+. They also present the first very accurate three-dimensional maps of je computed from the “exact” wave function for H2+. In a similar vein Andrae et al.10 study the flux of electrons associated with the breaking and making of chemical bonds that take place during a pericyclic reaction (in particular, the Cope rearrangement of semibullvalene). Taking a very different tack, Okuyama and Takatsuka8 introduce a complex “time-shift flux”, which has the same form as je, except that the time is displaced by the parameter Δt/2 in |Ψ(t)æ and by Δt/ 2 in ÆΨ(t)|. It has the apparent virtue of yielding nonzero values in the BOA. Okuyama and Takatsuka compute the “time-shift flux” for intramolecular vibrations in H2, NaCl and the formic acid dimer and interpret the real and imaginary parts thereof. The present piece is the first of two articles that are intended to introduce and to test a new (coupled-channels(CC)) approach to the approximate computation of je for strictly electronically adiabatic processes described by eq 1.1. The first article (I) presents the formal derivation of the CC theory. The second (hereafter referred to as II) is devoted to a numerical application of the CC theory to H2+. We choose H2+ in particular because of the availability of highly accurate numerical (“exact”) solutions of Schr€odinger’s equation that account fully for the electronicnuclear coupling.9 The outline of the rest of article I is as follows. By comparing BOA and exact treatments of the hydrogen atom, we elucidate in section 2 the BOA’s failure to furnish a nonzero je. Exploiting this insight in section 3, we propose a scheme through which je can be elicited from the BOA wave function. The results of Section 3 suggest a coupled-channels approach, which we develop for H2+ in the a frame (i.e., the observer is stationed on nucleus a) in section 4 and in the nuclear center-of-mass frame in section 5. The principal results are summarized in section 6, and their interpretation is discussed in classical terms.
2. EXACT TREATMENT OF THE H ATOM To demonstrate quantitatively how the BOA leads to eq 1.2, we apply it to a simple system that can otherwise be handled exactly, the H atom. We ignore the effects of relativity and intrinsic spins of particles. We wish to compute the electronic and protonic (probability) flux densities in the space-fixed laboratory coordinate frame, the classical expressions for which are je, L ðx, tÞ ¼ δ½x re ðtÞ_re ðtÞ
ð2:1aÞ
jp, L ðx, tÞ ¼ δ½x R p ðtÞR_ p ðtÞ
ð2:1bÞ
Here δ designates the Dirac distribution, the parameter x denotes the point of observation of j (i.e., the distance of the
point of observation from the laboratory origin), re and Rp refer to the positions of the electron and proton, respectively, and r_ e and R_ p are the respective velocities. The subscript “L” emphasizes that the flux densities depend on the perspective of the observer, who in the present instance is located at the origin of the laboratory (L) frame. The corresponding quantum expectation values are Æje, L ðx, tÞæ ¼
p 2ime
¼ RefÆΨðtÞjδðx re Þ_re jΨðtÞæg
Z
Z dre
dR p δðx re Þ½Ψðre , R p , tÞ∇re Ψðre , R p , tÞc:c:
ð2:2aÞ Æjp, L ðx, tÞæ ¼ RefÆΨðtÞjδðx R p ÞR_ p jΨðtÞæg ¼
Z Z p dre dR p δðxR p Þ½Ψðre , R p , tÞ∇R p Ψðre , R p , tÞc:c: 2iMp
ð2:2bÞ where the classical functions r and r_ have been replaced by their respective operators r and ip3r/m in the position representation and “c.c.” stands for the complex conjugate of the immediately preceding term. The BOA wave function can be written Ψðre , R p , tÞ = ϕðre ; R p ÞχðR p , tÞ
ð2:3Þ
where ϕ is an energy eigenfunction of the H atom and χ is a wavepacket describing the motion of the proton (see Appendix I of the Supporting Information). In the usual case, where one is dealing with a polyatomic system, the energy eigenvalue E corresponding to the electronic state ϕ depends on the nuclear configuration and serves as the effective potential energy governing the nuclear motion. However, in the present instance there is only a single nucleus, so E is independent of the position of the proton. The proton moves through the laboratory as a free particle with the electron adjusting instantaneously to its motion. Substituting eq 2.3 into eq 2.2a, we obtain Æje, L ðx, tÞæBOA Z p ¼ dR p jχðR p , tÞj2 ½ϕðre ; R p Þ∇re ϕðre ; R p Þ c:c:re ¼ x 2ime ¼0 ð2:4aÞ
The second line follows because ϕ is real. We take ϕ to be a nondegenerate state (in particular, the 1s ground state). In a similar way we get from eq 2.2b Æjp, L ðx, tÞæBOA Z p ¼ dre jϕðre ; R p Þj2 ½χðR p , tÞ∇R p χðR p , tÞ c:c:R p ¼ x 2iMp p ¼ ½χ ðR p , tÞ∇R p χðR p , tÞ c:c:R p ¼ x ð2:4bÞ 2iMp The second equality in eq 2.4b results from the assumed normalization of ϕ. Thus, although the BOA gives zero electronic flux density, it yields a nonzero nuclear flux density, given by an expression equivalent to those usually derived by analogy with hydrodynamics for single-particle systems.1114 2729
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Since the electron is dragged along by the proton, one intuitively expects to see an electronic flux parallel with the protonic flux. To discern how the BOA misses it, we compute je from the exact wave function of H, whose derivation is summarized in Appendix II of the Supporting Information. We introduce the two-body Jacobi coordinates R ¼ ðMp R p þ me re Þ=M r ¼ re R p
ð2:5Þ
where R is the center of mass (c.o.m.) and M = Mp + me is the total mass of the system. Then we can rewrite eq 2.2a as Z Z p dr dRδ½x ðR þ Mp r=MÞ Æje, L ðx, tÞæ ¼ 2ime 2 jχ~ðR, tÞj ½ϕ~ðrÞ∇r ϕ~ðrÞ c:c: Z Z p dr dRδ½x ðR þ Mp r=MÞ þ 2iM 2 jϕ~ðrÞj ½χ~ðR, tÞ∇R χ~ðR, tÞ c:c: ð2:6Þ We have also used eqs II.1b, II.4, and II.6a in reaching eq 2.6. Tildes on symbols are intended to distinguish exact functions from the corresponding BOA ones. We restrict our consideration again to nondegenerate states. Hence ϕ ~ is real and the first term on the right side of eq 2.6 disappears. Performing the integration on r, we get from the second term Æje, L ðx, tÞæ ¼
Z p 2 ~ ðR, tÞ∇R χ~ðR, tÞc:c: ðM=Mp Þ3 dRjϕ~½MðxRÞ=Mp j ½χ 2iM
ð2:7aÞ Following a similar procedure, we obtain from eqs 2.2b, (II.1b), (II.4), and (II.6b) Æjp, L ðx, tÞæ Z p 2 drjϕ~ðrÞj ½χ~ðR, tÞ∇R χ~ðR, tÞ c:c:R ¼ xþme r=M ¼ 2iM ð2:7bÞ The phenomena represented by the exact expressions in eq 2.7 can be illuminated by the introduction of the small parameter me ¼ 5:44 104 ð2:8Þ ε M
¼
Æjp, L ðx, tÞæ pð1 εÞ ¼ 2iMp
drjϕ~ ðrÞj2 ½χ~ðR, tÞ∇R χ~ðR, tÞ c:c:R ¼ x
p ½χ~ ðR, tÞ∇R χ~ðR, tÞ c:c:R ¼ x 2iMp
Æje, L ðx, tÞæ =
p 2iMp
ð2:10Þ
Z
2 ~ ðR, tÞ∇R χ~ðR, tÞ c:c: dRjϕ~ðx RÞj ½χ
ð2:11Þ From eqs 2.10 and 2.11 we infer Z 2 dRjϕ~ðx RÞj Æjp, L ðR, tÞæ Æje, L ðx, tÞæ ¼
ð2:12Þ
This last formula is especially appealing in that it accords with one’s intuition. At any instant t, the electronic flux density is just equal to the summation (integration) of the protonic flux density at the position of the proton weighted by the probability density of the electron at the point of observation x with respect to the proton. The electron is dragged instantaneously and coherently by the proton; its spatial distribution relative to the proton does not change with time. The motion of the proton determines je entirely. From a mathematical viewpoint the flux is generated by the c.o.m. motion (i.e., the second term on the right side of eq 2.6). In other words, the contribution of the c.o.m. component of the electron’s velocity (ip3R/M) (see eq II.6a in Appendix II of the Supporting Information) is nonvanishing, whereas that of the relative component (ip3r/me) vanishes for the same reason that of the “bare” electronic component (ip3re /me) vanishes in the BOA treatment. The electron is in a stationary (internal) state with respect to the proton in both cases.
3. EXTRACTION OF THE ELECTRONIC FLUX DENSITY FROM THE BOA WAVEFUNCTION FOR THE HYDROGEN ATOM The analysis of section 2 suggests that we try to draw forth je approximately from the BOA wave function by transforming from laboratory to Jacobi coordinates defined in eq 2.5. Thus, from eqs 2.2a and I.4 (see Appendix I of the Supporting Information) we have Æje, L ðx, tÞæBOA ¼
p 2ime
Z
Z dre
dR p δðx re Þ½ϕ ðre R p ÞχðR p , tÞ
∇re ϕðre R p ÞχðR p , tÞ c:c:
ð2:9aÞ ¼
Z
Z
where the second line depends on the normalization of ϕ~. Hence, noting that Rp f R and χ f χ~ in the limit ε f 0, we conclude that the protonic flux density given by eq 2.4b in the BOA is equal to the “exact” c.o.m. flux density, given by eq 2.10. Likewise, in the same limit we can write eq 2.9a as
in terms of which we can recast eq 2.7 as Z p 1 2 Æje, L ðx, tÞæ ¼ ð1 εÞ2 dRjϕ~½ð1 εÞ ðx RÞj 2iMp ½χ~ðR, tÞ∇R χ~ðR, tÞc:c:
p 2iMp
Æjp, L ðx, tÞæ =
p 2ime
Z
Z dRδfx ½R þ ð1 εÞrg
dr
fϕðrÞχðRεr, tÞ∇r ½ϕðrÞχðRεr, tÞc:c:g
2 drjϕ~ðrÞj ½χ~ðR, tÞ∇R χ~ðR, tÞ c:c:R ¼ xþεr
þ
ð2:9bÞ Now exploiting the puniness of ε compared to 1, we set ε = 0 in eq 2.9b to obtain
pð1 εÞ 2iMp
Z
Z dr
dRδfx ½R þ ð1 εÞrgjϕðrÞj2
½χðR εr, tÞ∇R χðR εr, tÞ c:c: 2730
ð3:1Þ
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The second equality follows by eqs 2.8, II.1a, and II.6a (see Appendix II of the Supporting Information). Mindful of the smallness of ε, we expand the BOA protonic wavepacket in the integrand of eq 3.1 formally as χðR εr, tÞ = χðR, tÞ εr 3 ð∇R χÞε ¼ 0 þ Oðε2 Þ
p 2ime
Z
p 2iMp
Z
M ¼ Ma þ Mb þ m e Mαe ¼ Mα þ me
dRδfx ½R þ ð1 εÞrg
(The subscript “EBOA” (extended BOA) emphasizes the additional approximation ε f 0, which goes beyond the usual BOA.) Again, because ϕ is taken to be real, the relative contribution (first term on the right side of eq 3.3) vanishes. Finally, in the limit ε f 0 we perform the integration on r in the second term, obtaining
ð4:3Þ
(For the sake of generality we regard the nuclei as distinguishable for the time being.) R stands for the position of the c.o. m. of the whole system, r0 α for the distance of the electron from nucleus α, and R0 α for the distance to the c.o.m. of “atom” α (i.e., nucleus α plus electron) from the other nucleus (β). In the a frame the classical expressions for the flux densities of the electron and nucleus b are, respectively, je, a ðx, tÞ ¼ δfx ½re ðtÞ R a ðtÞg½_re ðtÞ R_ a ðtÞ
ð4:4bÞ
ð3:4Þ
Now x means the distance of the point of observation from nucleus a. Clearly ja,a(x,t) = 0. The corresponding quantum expectation values can be written Z p dR L δ½x ðre R a Þ Æje, a ðx, tÞæ ¼ 2i fΨðR L , tÞ½∇re =me ∇R a =Ma ΨðR L , tÞ c:c:g ð4:5aÞ
which is the analogue of eq 2.12. We conclude that our trick (i.e., transforming to the two-body Jacobi coordinates and then dropping terms proportional to ε in the argument of χ) yields an accurate approximation to the “coherent” electronic flux density that “should” be observed for the state of the H atom characterized by the BOA.
4. COUPLED-CHANNELS TREATMENT OF THE HYDROGENMOLECULE ION In the analysis of the H atom in section 3 we adopt an “external” perspective, namely, that of an observer stationed at the origin of the L frame. We now extend the treatment to H2+, in order to observe how the electron flows with respect to the nuclei during the process of interest. For this purpose, an “internal” viewpoint (i.e., the observer stands on a particular nucleus or at the c.o.m. of the nuclei) is appropriate. From the perspective of nucleus a (i.e., the a frame) we can view this system as the “collision” of “atom” b with nucleus a, which corresponds to “channel b” in the language of scattering theory.15 We describe the motion using channel-b Jacobi coordinates R ¼ ðMa R a þ Mb R b þ me re Þ=M r0b ¼ re R b R 0b ¼ ðMb R b þ me re Þ=Mbe R a
ð4:1Þ
Alternatively, from the viewpoint of nucleus b we describe the motion in channel a in terms of the channel-a Jacobi
ð4:4aÞ
jb, a ðx, tÞ ¼ δfx ½R b ðtÞ R a ðtÞg½R_ b ðtÞ R_ a ðtÞ
dRjϕðx RÞj2 ½χðR, tÞ∇R χðR, tÞ c:c:
SinceR f Rp, χ f χ~, and ϕ f ϕ~ as ε f 0, the formula in eq 3.4 agrees with that in eq 2.11 in the same limit. The combination of eqs 3.4 and 2.4b gives Z Æje, L ðx, tÞæEBOA ¼ dR p jϕðx R p Þj2 Æjp, L ðR p , tÞæBOA ð3:5Þ
ð4:2Þ
In eqs 4.1 and 4.2 Rα denotes the position of nucleus α (=a,b) in the L frame; M α is the mass of nucleus α and
Z
dr
jχðR, tÞj2 ½ϕðrÞ∇r ϕðrÞ c:c: Z Z pð1 εÞ dr dRδfx ½R þ ð1 εÞrgjϕðrÞj2 þ 2iMp ½χðR, tÞ∇R χðR, tÞ c:c: ð3:3Þ
Æje, L ðx, tÞæEBOA ¼
R ¼ ðMa R a þ Mb R b þ me re Þ=M r0a ¼ re R a R 0a ¼ ðMa R a þ me re Þ=Mae R b
ð3:2Þ
Then dropping diminutive terms of order ε and higher on the right side of eq 3.2, we rewrite eq 3.1 as Æje, L ðx, tÞæEBOA ¼
coordinates
Z p dR L δ½x ðR b R a Þ 2i fΨðR L , tÞ½∇R b =Mb ∇R a =Ma ΨðR L , tÞ c:c:g
Æjb, a ðx, tÞæ ¼
ð4:5bÞ where RL = (Ra,Rb,re) compactly represents the configuration in terms of laboratory coordinates. Appendix III of the Supporting Information summarizes the derivation of the BOA wave function for H2+. We assume that the (real) BOA electronic energy eigenfunction Φ can be adequately represented on the basis of atomic orbitals ψlα centered on the nuclei: Φðr; RÞ
∑ ∑ cl ðRÞψl ðr, RÞ ¼ ∑ Φα ðr; RÞ ¼ ∑ Φα ðr0α , R 0α Þ α ¼ a, b α ¼ a, b ¼
α ¼ a, b lα
α
α
ð4:6Þ
where the second line defines implicitly the components Φα of Φ that are associated with the channels (α) characterized above in terms of Jacobi coordinates (eqs 4.1 and 4.2). Note that Φα(r0 α,R0 α) is to be interpreted as the function in channel coordinates (r0 α,R0 α) that results from the transformation of Φα(r;R) = ∑lαclα(R)ψlα(r,R) in the Jacobi coordinates (r,R) employed in Appendix III. We stress that within the context of the analysis of H2+, we use the symbol R(=Rb Ra) for the internuclear separation, in order to conform to convention. Using eqs 4.6, III.6, and III.8, we can express the total BOA 2731
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wave function as a summation over channels Ψ¼
∑
α ¼ a, b
Ψα
ð4:7Þ
coupled-channels Ansatz, which supplements the ε f 0 limit procedure of the “EBOA” (extended BOA) . Appendix V also derives the following formula for the flux density of nucleus b in the a frame
where the component in channel α is given by Ψα
Æjb, a ðx, tÞæBOA ¼
¼ ξðR , tÞΦα ðr; RÞχðR, tÞ ¼ ξðR , tÞΦα ðr0α , R 0α Þχðr0α , R 0α , tÞ
ð4:8Þ
Henceforward in this article we focus on the electronic ground state of H2+, assuming the minimal basis (i.e., a single 1s orbital on each nucleus). In this case Φðr; RÞ
¼ NðRÞ½ψ1sa ðjr þ R=2jÞ þ ψ1sb ðjr R=2jÞ ¼
∑
α ¼ a, b
Nðr0α , R 0α Þψ1sα ðr 0α Þ ¼
∑
α ¼ a, b
Φα ðr0α , R 0α Þ ð4:9Þ
where NðRÞ ¼ f2½1 þ SðRÞg1=2 is the normalization constant and Z SðRÞ ¼ drψ1sa ðjr þ R=2jÞψ1sb ðjr R=2jÞ
ð4:10aÞ
ð4:10bÞ
is the overlap between the orbitals. We emphasize that we employ the minimal basis for purely didactic purposes, in order to express the formulas in specific familiar terms. The derivations to follow hold as well for the general case of the extended basis. Using eq 4.7, we can recast the expression for the electronic flux density in eq 4.5a as Æje, a ðx, tÞæBOA ¼
∑ Æje, a ðx, tÞæα α ¼ a, b
ð4:11Þ
ð4:15Þ Equation 4.15 is the analogue of eq 2.10 that pertains to the H atom in the L frame. (Incidentally, the formula in eq 4.15 can be derived directly without the CC procedure in the Jacobi coordinate system employed in Appendix III of the Supporting Information. That is the reason for the subscript “BOA” instead of “CC”.) Combining eqs 4.13 and 4.15, we get Z dRΔb ðx; RÞÆjb, a ðR, tÞæBOA ð4:16Þ Æje, a ðx, tÞæCC ¼ Equation 4.16 for the H2+ molecule corresponds to eq 3.5 for the H atom, if we make the following correlations: a T L; b T p; R T Rp; Δb T |ϕ|2. Thus, Δb is interpreted as the probability (number) density of the electron that is associated with nucleus b in H2+. The analogy can be rendered more vivid by imagining the mass of nucleus a of H2+ to approach infinity while the charge on nucleus a approaches zero. Thus, as the nucleus a becomes virtual, the H2+ molecule metamorphoses into the H atom and Φ(r;R) f ϕ(re Rp), Δb f [ϕ]2, Æjb,a(R,t)æBOA f Æjp,L(Rp,t)æBOA, and Æje,a(x,t)æCC f Æje,L(x,t)æEBOA. By interchanging the labels on the nuclei, we can write je in the b frame as Z dR 0 Δa ðx 0 ; R 0 ÞÆja, b ðR 0 , tÞæBOA ð4:17Þ Æje, b ðx0 , tÞæCC ¼ where Δa ðx 0 ; R 0 Þ Φðx 0 ; R 0 ÞΦa ðx 0 ; R 0 Þ ¼ Φðx 0 ; R 0 ÞNðR 0 Þψ1sa ðjx 0 R 0 jÞ ¼ NðR 0 Þ½ψ1sa ðjx 0 R 0 jÞ þ ψ1sb ðx 0 ÞNðR 0 Þψ1sa ðjx 0 R 0 jÞ
where the component from channel α is given by Æje, a ðx, tÞæα Z p ¼ dR L δ½x ðre R a ÞfΨ½∇re =me ∇R a =Ma Ψα c:c:g 2i
ð4:18aÞ Æja, b ðR 0 , tÞæBOA ¼
ð4:12Þ (For economy of notation, we drop the subscript “BOA” on the channel components.) The evaluation of the contributions from each channel is detailed in Appendix V of the Supporting Information. The resulting total is Æje, a ðx, tÞæCC
p ¼ 2iμab
Z
dRΔb ðx; RÞ½χðR, tÞ∇R χðR, tÞ c:c:
ð4:13Þ where Δb ðx; RÞ ¼ Φðx; RÞΦb ðx; RÞ ¼ Φðx; RÞNðRÞψ1sb ðjx RjÞ ¼ NðRÞ½ψ1sa ðxÞ þ ψ1sb ðjx RjÞNðRÞψ1sb ðjx RjÞ
ð4:14Þ and μab = MaMb/(Ma + Mb)is the reduced mass of the nuclei. Note that eq 4.13 pertaining to H2+ is the analogue of eq 3.4 for H. The new subscript “CC” stresses the role of the
p ½χ ðR, tÞ∇R χðR, tÞ c:c:R ¼ x 2iμab
p 0 ½χ ðR , tÞ∇R 0 χðR 0 , tÞ c:c: 2iμab ð4:18bÞ
Now R0 ¼ R
ð4:19Þ 0
is the distance from nucleus b to nucleus a and x is the distance of the point of observation from b. Since a and b are identical for H2+, Δb(x; R) at observation point x in the a frame is equal to Δa(x0 ; R0 ) at observation point x0 (=x) in the b frame. Also, since the Schr€odinger equation (eq III.10 of Appendix III of the Supporting Information) that governs the relative (internal) nuclear motion in the b frame is identical with that in the a frame, χ(R0 ,t) = χ(R,t). We conclude that Æje,a(x,t)æCC = Æje,b(x0 , t)æCC. That is Æje,a(x,t)æCC observed at distance x from nucleus a is equal to Æje,b(x0 ,t)æCC observed at distance x0 from nucleus b. The points of observation x and x0 in the a and b frames, respectively, are related by x0 ¼ x R 2732
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Plugging eq 4.20 into eq 4.18a and invoking eq 4.19, we obtain
The general result is
Δa ðx; RÞ Φðx; RÞΦa ðx; RÞ ¼ NðRÞ½ψ1sa ðxÞ þ ψ1sb ðjx RjÞNðRÞψ1sa ðxÞ ð4:21Þ
Æje, NCM ðx 0 , tÞæCC ¼
ð4:22Þ
Æje, NCM ðx0 , tÞæCC ¼
ð5:1Þ where the c.o.m. of the nuclei is given by R NCM ¼ ðMa R a þ Mb R b Þ=ðMa þ Mb Þ
ð5:2Þ
We continue, for the time being, to treat the nuclei as distinguishable. Now x0 signifies the distance of the point of observation from the nuclear c.o.m. The prime is intended to distinguish the perspective in the NCM frame from that in the a frame. The quantum expectation value corresponding to the classical electronic flux density is Z p dR L δfx0 ½re ðMa R a þ Mb R b Þ=Mab Æje, NCM ðx0 , tÞæ ¼ 2i fΨðR L , tÞ½∇re =me ð∇R a þ ∇R b Þ=Mab ΨðR L , tÞ c:c:g
where Æje, NCM ðx 0 , tÞæα ¼
p 2i
Z
dR½Δb ðx 0 ; RÞ Δa ðx 0 ; RÞÆjb, a ðR, tÞæBOA
Δa ðx 0 ; RÞ Φðx 0 ; RÞΦa ðx 0 ; RÞ ¼ Φðx 0 ; RÞNðRÞψ1sa ðjx 0 þ R=2jÞ ¼ NðRÞ½ψ1sa ðjx0 þ R=2jÞ þ ψ1sb ðjx 0 R=2jÞNðRÞψ1sa ðjx 0 þ R=2jÞ
ð5:8aÞ
Δb ðx 0 ; FÞ Φðx 0 ; RÞΦb ðx 0 ; RÞ ¼ Φðx 0 ; RÞNðRÞψ1sb ðjx 0 R=2jÞ ¼ NðRÞ½ψ1sa ðjx 0 þ R=2jÞ þ ψ1sb ðjx 0 R=2jÞNðRÞψ1sb ðjx 0 R=2jÞ
ð5:8bÞ
We note in passing that, as demonstrated in Appendix VI, eq 5.6 can be formally recast as Z 0 dσΔa ðx 0 ; σÞÆja, NCM ð σ, tÞæBOA Æje, NCM ðx , tÞæCC ¼ Z þ
dσΔb ðx0 ; σÞÆjb, NCM ðσ, tÞæBOA
ð5:9Þ where σ = R/2. Though this expression has an appealing form, as a sum of separate channel components involving the corresponding nuclear flux densities in the same frame, its numerical implementation is rather less convenient than that of the alternate formula in eq 5.7 (see article II). If the nuclei are constrained to a line, say the z axis, then from eq 5.7 we obtain Æje, NCM ðx 0 , tÞæCC ¼
1 2
Z
dR½Δb ðx0 ; Rez Þ Δa ðx0 ; Rez ÞÆjb, a ðR, tÞæBOA ez
ð5:10Þ where now Δa ðx0 ; Rez Þ Φðx 0 ; Rez ÞΦa ðx 0 ; Rez Þ ¼ Φðx 0 ; Rez ÞNðRÞψ1sa ðjx0 þ Rez =2jÞ
Following the development of section 4, we express the flux density as a summation over channels as Æje, NCM ðx0 , tÞæα ∑ α ¼ a, b
Z
where
ð5:3Þ
Æje, NCM ðx 0 , tÞæCC ¼
1 2
ð5:7Þ
Hence, Δα may as well be called the “gross atomic density” on atom α.
je, NCM ðx 0 , tÞ ¼ δfx 0 ½re ðtÞ R NCM ðtÞg½_re ðtÞ R_ NCM ðtÞ
Ma Mb Δb ðx 0 ; RÞ Δa ðx 0 ; RÞ Æjb, a ðR, tÞæBOA Mab Mab
For H2+ in particular, Ma = Mb, and we obtain from eq 5.6
which says that the total electronic probability (number) density is equal to the sum of the densities assigned to the nuclei. Therefore, the interpretation of Δα as the electronic probability density associated with nucleus α is consistent. We remark in passing on the connection between the electron number density Δα assigned to nucleus α and the “gross atomic population” N(α) on atom α defined by Mulliken.16 That is Z dxΔα ðx; RÞ ¼ NðαÞ ¼ 1=2, α ¼ a, b ð4:23Þ
5. COUPLED-CHANNELS FORMULA FOR ELECTRONIC FLUX DENSITY FOR H2+ IN THE NUCLEAR CENTER-OFMASS FRAME For the observer stationed on the nuclear c.o.m. (NCM) of H2+, the electronic flux density is given classically by
dR
ð5:6Þ
which is just the electronic probability density assigned to nucleus a, as expressed in the a frame. Addition of eqs 4.14 and 4.21 yields Δa ðx; RÞ þ Δb ðx; RÞ ¼ ½Φðx; RÞ2
Z
Δb ðx0 ; Rez Þ ¼ Φðx0 ; Rez ÞΦb ðx 0 ; Rez Þ ¼ Φðx0 ; Rez ÞNðRÞψ1sb ðjx0 Rez =2jÞ
ð5:4Þ
ð5:11Þ Φðx0 ; RÞ ¼ NðRÞ½ψ1sa ðjx 0 þ Rez =2jÞ þ ψ1sb ðjx 0 Rez =2jÞ
dR L δfx 0 ½re ðMa R a þ Mb R b Þ=Mab
fΨ½∇re =me ð∇R a þ ∇R b Þ=Mab Ψα c:c:g
Since ψ1sα ðj x0 þ Rez =2jÞ ¼ ψ1sβ ðjx0 Rez =2jÞ
ð5:5Þ
Employing the minimal basis for Φ, we evaluate the channel contributions in Appendix VI of the Supporting Information.
ð5:12Þ
α, β ¼ a, b, α 6¼ β 2733
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we have from eq 5.11 Φð x0 ; Rez Þ ¼ Φðx0 ; Rez Þ
ð5:13aÞ
and Δα ½ x 0 ; Rez ¼ Δβ ½x0 ; Rez
ð5:13bÞ
α, β ¼ a, b, α 6¼ β and consequently from eq 5.10 Æje, NCM ð x 0 , tÞæCC ¼ Æje, NCM ðx 0 , tÞæCC
ð5:14Þ
In other words, je is antisymmetric under inversion of the observation point through the nuclear c.o.m. By similar reasoning it can be shown that je is antisymmetric under reflection of the observation point in the plane passing through the nuclear c.o.m. perpendicularly to the internuclear separation.
6. SUMMARY AND DISCUSSION The straightforward approach to the calculation of the electronic flux density (je) that accompanies an electronically adiabatic process described in the BOA leads to je = 0. This result can be rationalized by noting that the electrons are in a stationary state [Φ(q; Q)] for every nuclear configuration Q and that je = 0 for stationary states. Therefore, although the vanishing of je is comprehensible, it is nevertheless contrary to intuition, since the electrons of course must move as the adiabatic process takes place. Rather than forsake the BOA, we resolve to abide by its restrictions and to find an approximate scheme through which je can be extricated from the BOA wave function. To understand how the BOA fails to provide a nonzero je, we compute je exactly for the H atom. We notice that the small parameter ε = me/(Mp + me) appears in analytic expressions for the flux densities (see eq 2.9) and that taking the limit ε f 0 yields an “exact” formula for je (eq 2.12) that is especially appealing intuitively. This ε f 0 limit procedure is then adapted in modified form within the framework of the BOA. To evaluate the quantum expression for je,L in the context of the BOA, we first transform from the original laboratory coordinates to Jacobi coordinates. We then formally expand the BOA nuclear wavepacket (χ) in a power series in the small parameter ε = me/(Mp + me). Taking the limit ε f 0 before carrying out the required integration over configuration space yields (see eq 3.5)
time. Thus, at any point x in the laboratory the observer sees je,L rise as the H atom approaches and then fall as it recedes. By the way, we point out an interesting connection between this classical picture of the free H atom and the time-dependent description of asymptotic states in atomic collisions in terms of “traveling atomic orbitals” (TAOs).17,18 That is, the electronic flux density associated with a TAO has exactly the same form as that given by eq 6.3 with constant velocity. In a first step toward extending the procedure to the calculation of electronic flux densities within molecules, we focus on H2+. We define flux densities from two internal perspectives. The observer is stationed either on one of the two protons (i.e., the α (=a,b)frame) or on the nuclear center of mass (i.e., the NCM frame). The (internal) motion of the system can now be described as a “collision” of one internal H “atom” α (i.e., proton α (a, b) plus the electron) with the other proton β (b, a). The flux densities in either of the two reference frames can be decomposed into contributions from two coupled channels, each of which correlates with the collision of one H “atom” with the other proton. The procedure for eliciting nonzero je from the BOA wave function is adapted from that employed for the H atom, as described above. The contribution from each channel is evaluated by first transforming to Jacobi coordinates appropriate to that channel and then invoking the ε f 0 limiting technique analogous to that described above for the H atom itself. In the a frame, for example, we obtain (see eq 4.16) Z dRΔb ðx, RÞÆjb, a ðR, tÞæBOA ð6:4Þ Æje, a ðx, tÞæCC ¼ This expression is the analogue of that in eq 6.1 for the H atom. We make the following correlations: a T L; R T Rp; Δb T |ϕ|2. In the classical limit of the relative nuclear motion, we have the analogue of eq 6.2 _ Æjb, a ðR, tÞæBOA f δ½R RðtÞRðtÞ Substitution of eq 6.5 into eq 6.4 gives _ Æje, a ðx, tÞæCC ¼ Δb ðx, RðtÞÞRðtÞ
Æje, L ðx, tÞæEBOA ¼
dR p jϕðx R p Þj Æjp, L ðR p , tÞæBOA
ð6:1Þ
The justification for this procedure is that the formula agrees with the “exact” expression for je,L in the same limit. The interpretation of the formula in eq 6.1 can be facilitated by considering the classical limit of the proton’s motion, in which Æjp, L ðR p , tÞæBOA f δ½R p R p ðtÞR_ p ðtÞ
ð6:2Þ
where Rp(t) represents the classical trajectory of the proton. Plugging eq 6.2 into eq 6.1, we get Æje, L ðx, tÞæEBOA ¼ jϕðx R p Þj2 R_ p ðtÞ
ð6:3Þ
In the absence of external forces (i.e., the free H atom) the proton executes rectilinear motion, dragging the electron coherently along a straight line at constant velocity R_ p(0). The distribution of electronic density relative to the proton does not change with
ð6:6Þ
where, in the case of the minimal basis (see eq 4.9) Δb ðx; FðtÞÞ
¼ Φðx; RÞΦb ðx; RÞ ¼ NðRÞ½ψ1sa ðxÞ þ ψ1sb ðjx RðtÞjÞNðRÞψ1sb ðjx RðtÞjÞ
ð6:7Þ
Z
2
ð6:5Þ
is the electronic density associated with proton (channel) b. Unlike the distribution |ϕ|2 traveling with the proton in the H atom, Δb depends on the distance R(t) between the protons of H2+. Moreover, on account of the effective potential energy of interaction between the nuclei, for sufficiently low energies the trajectory R(t) generally corresponds to vibration and rotation, in which case Æjb,a(R,t)æBOA, and consequently Æje,a(R,t)æCC, are expected to be periodic. Although these qualitative considerations in the classical limit provide an intuitively satisfying picture of the flux densities associated with simple processes, they neglect interesting and important effects due to the quantum-mechanical nature of the relative nuclear motion. These are explored in article II, which compares the results of CC and very accurate numerical simulations of H2+. Finally, we mention that the CC approach furnishes a quite simple formula (for example, the one in eq 6.4) for the electronic 2734
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flux density—a composition of two factors, Δb(x; R) and Æjb,a(R, t)æBOA. The former can be computed by standard methods of quantum chemistry and the latter by standard methods of quantum dynamics. The simple formula also permits a lucid interpretation of the electronic flux density, as demonstrated in article II.
’ ASSOCIATED CONTENT
bS
Supporting Information. Appendix I, BornOppenheimer approximation applied to the H atom; Appendix II, exact solution of Schr€odinger’s equation for the H atom; Appendix III, BOA wave function for H2+; Appendix IV, transformation of gradient operators; Appendix V, evaluation of channel contributions to flux densities for H2+ in the a frame; Appendix VI, evaluation of channel contributions to flux densities for H2+ in the NCM frame. This information is available free of charge via the Internet at http://pubs.acs.org.
’ ACKNOWLEDGMENT The author is especially indebted to Professor J€orn Manz for numerous stimulating discussions on the subject matter of this article, for many improvements in the original version of the manuscript, and for gracious hospitality. He is also thankful to the Freie Universit€at Berlin for financial support for several visits during which much of the work reported here was done. ’ REFERENCES (1) Born, M.; Huang, K. Dynamical Theory of Crystal Lattices; Oxford University Press: Oxford, 1985. (2) Born, M.; Oppenheimer, J. R. Ann. Phys. 1927, 84, 457–484. (3) Slater, J. C. Proc. Natl. Acad. Sci. U.S.A. 1927, 13, 423–430. (4) McCullough, E. A.; Wyatt, R. E. J. Chem. Phys. 1971, 54, 3578–3591. (5) Steiner, E.; Fowler, P. W. J. Phys. Chem. A 2001, 105, 9553–9562. (6) Barth, I.; Manz, J.; Shigeta, Y.; Yagi, K. J. Am. Chem. Soc. 2006, 128, 7043–7049. (7) Nafie, L. A. J. Phys. Chem. 1997, 101, 7826–7833. (8) Okuyama, M.; Takatsuka, K. Chem. Phys. Lett. 2009, 476, 109–115. (9) Barth, I.; Hege, H.-C.; Ikeda, H.; Kenfack, A.; Koppitz, M; Manz, J.; Marquardt, F.; Paramonov, G. K. Chem. Phys. Lett. 2009, 481, 118–123. (10) Andrae, D.; Barth, I.; Bredtmann, T.; Hege, H.-C.; Manz, J.; Marquardt, F.; Paulus, B. J. Phys. Chem. B 2011, 115, 5476–5483. (11) Schr€odinger, E. Ann. Phys. 1926, 81, 109–139. (12) Van Vleck, J. H. The Theory of Electric and Magnetic Susceptibilities; Oxford University Press: London, 1932. (13) Schiff, L. I. Quantum Mechanics; McGraw-Hill Book Company, Inc.: New York, 1968. (14) Cohen-Tannoudji, C.; Diu, B.; Lalo€e, F. Quantum Mechanics; John Wiley & Sons: New York, 1977. (15) Taylor, J. R. Scattering Theory; John Wiley & Sons: New York, 1972. (16) Mulliken, R. S. J. Chem. Phys. 1955, 23, 1833–1840. (17) Delos, J. B. Rev. Mod. Phys. 1981, 53, 287–357. € (18) Deumens, E.; Diz, A.; Ohrn, Y. Rev Mod. Phys. 1994, 66, 917–983.
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