Adiabatic and Diabatic Collision Processes at Low Energies

Abstract

Adiabatic and diabatic electronic states of a system of atoms are defined and their properties are described. Nonadiabatic interaction for slow quasi-classical motion of atoms is discussed within the two-state approximation. Analytical formulae for nonadiabatic transition probabilities are presented for particular models with reference to single and double-passage of coupling regions. The generalization for multiple passages is described.

1 Basic Definitions

1.1 Slow Quasi-Classical Collisions

Slow collisions of atoms or molecules (neutral or charged) are defined as collisions for which the velocity of the relative motion of colliding particles v is substantially lower than the velocity of valence electrons v_e

$$\displaystyle\frac{v}{v_{\text{e}}}\ll 1\;.$$

(52.1)

If v_e is estimated as $v_{\text{e}}\approx{\mathrm{1}}\,{\mathrm{\text{a.u.}}}\approx{\mathrm{10^{8}}}\,{\mathrm{cm/s}}$, then Eq. (52.1) is fulfilled for medium-mass nuclei (≈ 10 amu) up to several ke V.

Quasi-classical collisions are those for which the de Broglie wavelength λ_dB for relative motion is substantially smaller than the range parameter a of the interaction potential (the WKB (Wentzel–Kramers–Brillouin) condition 1 ; 2 ; 3 ; 4 )

$$\displaystyle\lambda_{\text{dB}}\ll a\;.$$

(52.2)

The two conditions in Eqs. (52.1) and (52.2) define the energy range within which collisions are slow and quasi classical. For medium-mass nuclei, this energy range covers collision energies above room temperature and below hundreds of e V. The parameter a should not be confused with another important parameter, L₀, which characterizes the extent of the interaction region. For instance, for the exchange interaction between two atoms, L₀ corresponds to the distance of closest approach of the colliding particles, while a is the range of the exponential decrease of the interaction. Typically, L₀ noticeably exceeds a.

1.2 Adiabatic and Diabatic Electronic States

Let r refer to a set of electronic coordinates in a body-fixed frame related to the nuclear framework of a colliding system and let ℛ refer to a set of nuclear coordinates determining the relative position of nuclei in this system. A configuration of electrons and nuclei in a frame fixed in space is completely determined by r, ℛ, and the set of Euler angles Ω, which relate the body-fixed frame to the space-fixed frame. With the total Hamiltonian of the system ℋ(r , ℛ , Ω), the stationary state wave function satisfies the equation

$$\displaystyle\mathcal{H}(r,\mathcal{R},\Omega)\Psi_{\text{E}}(r,\mathcal{R},\Omega)=E\Psi_{\text{E}}(r,\mathcal{R},\Omega)\;.$$

(52.3)

The electronic adiabatic Hamiltonian H(r ; ℛ) is defined to be the part of ℋ(r , ℛ , Ω) in which the kinetic energy of the nuclei and certain weak interactions are ignored. Typically, H(r ; ℛ) includes the electron–electron and electron–nuclear electrostatic interaction and, possibly, the spin–orbit interaction. The adiabatic electronic functions ψ_n(r ; ℛ) are defined as eigenfunctions of H(r ; ℛ) at a fixed nuclear configuration ℛ with set of electronic coordinates r

$$\displaystyle H({r};\mathcal{R})\psi_{n}(r;\mathcal{R})=U_{n}(\mathcal{R})\psi_{n}(r;\mathcal{R})\;.$$

(52.4)

The eigenvalues U_n(ℛ) are called adiabatic potential energy surfaces (adiabatic PES). In the case of a diatom, the set ℛ collapses into a single coordinate, the internuclear distance R, and the PES become potential energy curves PEC, U_n(R). The functions ψ_n(r ; ℛ) depend explicitly on ℛ and implicitly on the Euler angles Ω. The significance of the adiabatic PES is related to the fact that in the limit of very low velocities, a system of nuclei will move across a single PES. In this approximation, called the adiabatic approximation, the function U_n(ℛ) plays the part of the potential energy, which drives the motion of the nuclei 1 ; 2 ; 3 ; 4 .

An electronic diabatic Hamiltonian is defined formally as a part of H, i.e., H₀ = H + ΔH. The partitioning of H into H₀ and ΔH is dictated by the requirement that the eigenfunctions of H₀, called diabatic electronic functions ϕ_n, depend weakly on the configuration ℛ. The physical meaning of this weak dependence is different for different problems. A perfect diabatic basis set ϕ_n(r) is ℛ independent; for practical purposes, one can use a diabatic set that is considered ℛ independent within a certain region of the configuration space ℛ.

Two basis sets ψ_n and ϕ_n generate the matrices

$$\displaystyle\begin{aligned}\displaystyle\langle\phi_{m}|H|\phi_{n}\rangle=&\displaystyle H_{mn}\;,\\ \displaystyle\langle\phi_{m}|\mathcal{H}|\phi_{n}\rangle=&\displaystyle H_{mn}+D_{mn}\;,\\ \displaystyle\langle\psi_{m}|\mathcal{H}|\psi_{n}\rangle=&\displaystyle U_{n}(\mathcal{R})\updelta_{mn}+\mathcal{D}_{mn}\;.\end{aligned}$$

(52.5)

The eigenvalues of the matrix H_mn are U_n; D_mn and 𝒟_mn are the matrices of interactions ignored in the passage from the Hamiltonian ℋ to H. All the above matrices are, in principle, of infinite order. For low-energy collisions, the use of finite matrices of moderate dimension will usually suffice.

Diabatic PES are defined as the diagonal elements H_nn. The significance of the diabatic PES is that for velocities that are high [but still satisfy Eq. (52.1)] the system moves preferentially across diabatic PES, provided that the off-diagonal elements H_mn are small enough.

An adiabatic function ψ_n(r ; ℛ) can be constructed as a linear combination of the diabatic functions ϕ_m(r) as

$$\displaystyle\psi_{n}\left({r};\mathcal{R}\right)=\sum_{m}C_{nm}\left(\mathcal{R}\right)\phi_{m}({r})\;.$$

(52.6)

1.3 Nonadiabatic Transitions: The Massey Parameter

Deviations from the adiabatic approximation manifest themselves in transitions between different PES, which are induced by the dynamic coupling matrix 𝒟. At low energies, the transitions usually occur in localized regions of nonadiabatic coupling (NAR). In these regions, the motion of nuclei in different electronic states is coupled, and, in general, it cannot be interpreted as being driven by a single potential 3 ; 5 ; 6 ; 7 .

An important simplifying feature of slow adiabatic collisions is that, typically, the distance between different NAR is substantially larger than the extents of each NAR. This makes it possible to formulate simple models for the coupling in isolated NAR, and, subsequently, to incorporate the solution for nonadiabatic coupling into the overall dynamics of the system.

For a system of s nuclear degrees of freedom, there are the following possibilities for the behavior of PES within NAR:

(i):

If two s-dimensional PES correspond to electronic states of different symmetry, they can cross along an (s − 1)-dimensional line. For a system of two atoms, s = 1, and so two potential curves of different symmetry can cross at a point.

(ii):

If two s-dimensional PES correspond to electronic states of the same symmetry, they can cross along an (s − 2)-dimensional line. For a system of two atoms, s = 1, and so two potential curves of different symmetry cannot cross.

(iii):

If two s-dimensional PES correspond to electronic states of the same symmetry in the presence of spin–orbit coupling, they can cross along an (s − 3)-dimensional line.

Statement (ii) applied to a two-atom system is known as the Wigner–Witmer noncrossing rule. In applications to a system with s > 1, it is frequently discussed in terms of the conical intersection 8 .

The efficiency of the nonadiabatic coupling between two adiabatic electronic states is determined according to the adiabatic principle of mechanics (both classical and quantum), by the value of the Massey (M) parameter ζ^M, which represents the product of the electronic transition frequency ω_el and the time τ_nuc that characterizes the rate of change of electronic function due to the nuclear motion. Putting ω_el ≈ ΔU(ℛ) ∕ ℏ and τ_nuc = ΔL ∕ v(ℛ), we get

$$\displaystyle\zeta^{\text{M}}(\mathcal{R})=\omega_{\text{el}}\tau_{\text{nuc}}=\Delta U(\mathcal{R})\Delta L/\hbar v(\mathcal{R})\;,$$

(52.7)

where ΔU is the spacing between any two adiabatic PES, and ΔL is a certain range that depends on the type of coupling. The nonadiabatic coupling is inefficient at those configurations ℛ where ζ^M(ℛ) ≫ 1. If ζ^M(ℛ) is less than or of the order of unity, the nonadiabatic coupling is efficient, and a change in adiabatic dynamics of nuclear motion is very substantial.

The following relations usually hold for the parameters ΔL , a , L₀ for slow collisions

$$\displaystyle\Delta L\ll a\ll L_{0}\;.$$

(52.8)

When the nonadiabatic coupling is taken into account, the total (electronic and nuclear) wave function, Ψ_E, can be represented as a series expansion in ψ_n or ϕ_n (the Euler angles Ω are suppressed for brevity)

$$\displaystyle\begin{aligned}\displaystyle\Psi_{\text{E}}({r},\mathcal{R})=&\displaystyle\sum_{n}\psi_{n}(r;\mathcal{R})\chi_{n{\text{E}}}(\mathcal{R})\\ \displaystyle=&\displaystyle\sum_{n}\phi_{n}({r})\kappa_{n{\text{E}}}(\mathcal{R})\;.\end{aligned}$$

(52.9)

Here, χ_nE(ℛ) and κ_nE(ℛ) are the functions that have to be found as solutions to the coupled equations formulated in the adiabatic or diabatic electronic basis, respectively 3 ; 5 ; 6 ; 7 ; 9 . In general, different contributions to the first sum in Eq. (52.9) can be associated with nonadiabatic transition probabilities between different electronic states.

A practical means of calculating functions χ_nE(ℛ) [or κ_nE(ℛ)] consists of expanding them over certain basis functions Ξ_nν(ℛ^′), where ℛ^′ denotes all coordinates ℛ save a single coordinate R. Writing

$$\displaystyle\chi_{n{\text{E}}}(\mathcal{R})=\sum_{\nu}\Xi_{n\nu}(\mathcal{R^{\prime}})\xi_{n\nu{\text{E}}}(R)\,\;,$$

(52.10)

one arrives at a set of coupled second-order equations for the unknown functions ξ_nνE(R) (the scattering equations). In the adiabatic approximation, the total wave function is represented by a single term in the first sum of Eq. (52.9)

$$\displaystyle\Psi_{\text{E}}(r,\mathcal{R})=\psi_{n}(r;\mathcal{R})\chi_{n{\text{E}}}(\mathcal{R})\;.$$

(52.11)

2 Two-State Approximation

2.1 Relation Between Adiabatic and Diabatic Basis Functions

In the two-state approximation, two adiabatic functions ψ_k(r ; ℛ) are assumed to be expressed as a linear combination of two diabatic functions ϕ_k(r) via a rotation angle θ

$$\displaystyle\begin{aligned}\displaystyle\psi_{1}(r;\mathcal{R})=&\displaystyle\cos\theta(\mathcal{R})\phi_{1}(r)+\sin\theta(\mathcal{R})\phi_{2}(r)\;,\\ \displaystyle\psi_{2}(r;\mathcal{R})=&\displaystyle-\sin\theta(\mathcal{R})\phi_{1}(r)+\cos\theta(\mathcal{R})\phi_{2}(r)\;.\end{aligned}$$

(52.12)

The rotation angle, θ(ℛ), is expressed via the diagonal and off-diagonal matrix elements of the adiabatic Hamiltonian H in the diabatic basis ϕ₁ , ϕ₂

$$\displaystyle\tan 2\theta(\mathcal{R})=\frac{2H_{12}(\mathcal{R})}{H_{11}(\mathcal{R})-H_{22}(\mathcal{R})}\;.$$

(52.13)

The eigenvalues of H are

$$\displaystyle\begin{aligned}\displaystyle U_{1,2}=&\displaystyle\bar{U}\pm\Delta U\;,\\ \displaystyle\bar{U}=&\displaystyle(H_{11}+H_{22})/2=\bar{H}\;,\\ \displaystyle\Delta U=&\displaystyle\frac{1}{2}\sqrt{(\Delta H)^{2}+4H_{12}^{2}}\;,\\ \displaystyle\Delta H=&\displaystyle H_{11}=H_{22}\;,\end{aligned}$$

(52.14)

implying

$$\displaystyle\begin{aligned}\displaystyle\Delta H=&\displaystyle\Delta U\cos 2\theta\;,\\ \displaystyle H_{12}=&\displaystyle(1/2)\Delta U\sin 2\theta\;.\end{aligned}$$

(52.15)

2.2 Coupled Equations and Transition Probabilities in the Common Trajectory Approximation

A two-state nonadiabatic wave function Ψ(r , ℛ) can be written as an expansion into either adiabatic or diabatic electronic wave functions

$$\displaystyle\begin{aligned}\displaystyle\Psi(r;\mathcal{R})=&\displaystyle\psi_{1}(r;\mathcal{R})\,\alpha_{1}(\mathcal{R})+\psi_{2}(r;\mathcal{R})\,\alpha_{2}(\mathcal{R})\;,\\ \displaystyle\Psi(r;\mathcal{R})=&\displaystyle\phi_{1}(r)\beta_{1}(\mathcal{R})+\phi_{2}(r)\beta_{2}(\mathcal{R})\;,\end{aligned}$$

(52.16)

in which the nuclear wave functions satisfy two coupled s-dimensional Schrödinger equations 1 ; 2 ; 3 .

In the common trajectory (CT) approximation, the motion of the nuclei is described by the classical trajectory, i.e., by a one-dimensional manifold 𝒬(t) embedded in the s-dimensional manifold ℛ. A section of PES along this one-dimensional manifold determines a set of effective PEC. In the case of atomic collisions, Q coincides with the interatomic distance R, and the effective PEC are just ordinary PEC. A definition of a CT, or an effective potential that drives it, represents a challenging task which is not discussed here (e.g., references in the collection of notes 10 ; 11 ).

A CT counterpart of Eq. (52.16) is

$$\displaystyle\begin{aligned}\displaystyle\Psi(r,t)=&\displaystyle\psi_{1}\left[r;\mathcal{Q}(t)\right]a_{1}(t)+\psi_{2}\left[r;\mathcal{Q}(t)\right]a_{2}(t)\;,\\ \displaystyle\Psi(r;t)=&\displaystyle\phi_{1}(r)b_{1}(t)+\phi_{2}(r)b_{2}(t)\;.\end{aligned}$$

(52.17)

The expansion coefficients a_k(t) satisfy the set of equations

$$\displaystyle\begin{aligned}\displaystyle\mathrm{i}\hbar{}\frac{\mathrm{d}a_{1}}{\mathrm{d}t}&\displaystyle=U_{1}(\mathcal{Q})a_{1}+\mathrm{i}\dot{\mathcal{Q}}g(\mathcal{Q})a_{2}\;,\\ \displaystyle\mathrm{i}\hbar{}\frac{\mathrm{d}a_{2}}{\mathrm{d}t}&\displaystyle=-\mathrm{i}\dot{\mathcal{Q}}g(\mathcal{Q})a_{1}+U_{2}(\mathcal{Q})a_{2}\;.\end{aligned}$$

(52.18)

Here, the dynamical coupling function (defined in the adiabatic basis)

$$\displaystyle g(\mathcal{Q})=\langle\psi_{1}|\frac{\partial}{\partial\mathcal{Q}}|\psi_{2}\rangle=-\langle\psi_{2}|\frac{\partial}{\partial\mathcal{Q}}|\psi_{1}\rangle=\frac{\mathrm{d}\theta}{\mathrm{d}\mathcal{Q}}\;,$$

(52.19)

arises from the action of the operator $\mathrm{i}\hbar\partial/\partial t$ on the adiabatic functions and is expressed in terms of the angle θ(𝒬) (defined in the diabatic basis by Eq. (52.13)).

For low energies, function g(𝒬) is localized near the NAR center, and Eqs. (52.18) decouple away from the NAR center. This property of coupled equations may be lost if they are formulated in the diabatic basis. The diabatic expansion coefficients b_k(t) satisfy the set of equations

$$\displaystyle\begin{aligned}\displaystyle\mathrm{i}\hbar\frac{\mathrm{d}b_{1}}{\mathrm{d}t}=&\displaystyle H_{11}(\mathcal{Q})b_{1}+H_{12}(\mathcal{Q})b_{2}\;,\\ \displaystyle\mathrm{i}\hbar\frac{\mathrm{d}b_{2}}{\mathrm{d}t}=&\displaystyle H_{21}(\mathcal{Q})b_{1}+H_{22}(\mathcal{Q})b_{2}\;.\end{aligned}$$

(52.20)

Clearly, for a system of two atoms, 𝒬 ≡ R.

Solutions to Eqs. (52.18) and (52.20) are equivalent, provided that the initial conditions are matched, and the transition probability is properly defined with account taken for the coupling of the diabatic states away from the NAR center. It is customary to identify the center of the NAR with a value of 𝒬 = 𝒬_p, which corresponds to the real part of the complex-valued coordinate 𝒬_c at which two adiabatic PES cross and which possesses the smallest imaginary part 1 . The crossing conditions in the adiabatic and diabatic representations are

$$\displaystyle\begin{aligned}\displaystyle\Delta U(\mathcal{Q}_{\text{c}})=&\displaystyle 0\;,\\ \displaystyle\big[\Delta H(\mathcal{Q}_{\text{c}})\big]^{2}+4H^{2}_{12}(\mathcal{Q}_{\text{c}})=&\displaystyle 0\;.\end{aligned}$$

(52.21)

Since 𝒬 represents a one-dimensional manifold, the crossing condition Eq. (52.21) is satisfied for complex values of 𝒬 = 𝒬_c, unless H₁₂ = 0. Then, by definition, the location of the NAR center is identified with 𝒬_p = ℜ𝒬_c. A characteristic width of an isolated NAR is determined by the width Δ𝒬_p of a peaked (at Δ𝒬 = 𝒬_p) function g(𝒬). Normally, Δ𝒬_p is about ℑ𝒬_c 3 . A more general discussion of the crossing in the complex coordinate plane is given in the context of the hidden crossing 10 .

For low energies, the Eq. (52.20) along a particular CT are partitioned into sets of adiabatic evolution and the sets of nonadiabatic transformation of the amplitudes. The spatial (or temporal) extension of the former is normally much larger compared to the latter, such that nonadiabatic transformation of the amplitudes can be described by Eq. (52.18) applied to an isolated NAR. Since the characteristic width of a NAR is small on the scale of the whole CT, the part of CT that is essential in describing the nonadiabatic transformation of a_k(t) can be simplified in order to make Eq. (52.18) more easily handled. This simplification (to be called the local common trajectory, LCT) is a crux of the analytical solution for model cases of nonadiabatic coupling (Sect. 52.3) within an isolated NAR.

With the given functions U₁(𝒬), U₂(𝒬), g(𝒬) (or H₁₁(𝒬), H₂₂(𝒬), and H₁₂(𝒬)), the Eq. (52.18) are completely defined by setting the LCT with 𝒬 = 𝒬_LCT(t). Equation (52.18) decouple on both sides of the NAR, say, at t < t⁽⁻⁾ and t > t⁽⁺⁾. At these values of t, the two-state function Ψ(r , t) evolves adiabatically, and this behavior can be singled out by transformation ($t^{(-)}<t_{\text{p}}<t^{(+)}$)

$$\displaystyle a_{k}(t)=\bar{a}_{k}(t)\exp\left[-\frac{\mathrm{i}}{\hbar}\int_{t_{\text{p}}}^{t}U_{k}[\mathcal{Q}_{\text{LCT}}(t)]\mathrm{d}t\right]\;.$$

(52.22)

The amplitudes $\bar{a}_{k}(t)$, which become time-independent outside NAR, satisfy equations

$$\displaystyle\begin{aligned}\displaystyle\frac{\mathrm{d}\bar{a}_{1}}{\mathrm{d}t}=&\displaystyle\bar{a}_{2}\dot{\mathcal{Q}}_{\text{LCT}}g(\mathcal{Q}_{\text{LCT}})\exp\left[\frac{\mathrm{i}}{\hbar}\int_{t_{\text{p}}}^{t}\Delta U[\mathcal{Q}_{\text{LCT}}(t)]\mathrm{d}t\right]\;,\\ \displaystyle\frac{\mathrm{d}\bar{a}_{2}}{\mathrm{d}t}=&\displaystyle-\bar{a}_{1}\dot{\mathcal{Q}}_{\text{LCT}}g(\mathcal{Q}_{\text{LCT}})\\ \displaystyle&\displaystyle\times\exp\left[-\frac{\mathrm{i}}{\hbar}\int_{t_{\text{p}}}^{t}\Delta U[\mathcal{Q}_{\text{LCT}}(t)]\mathrm{d}t\right]\;.\end{aligned}{}$$

(52.23)

Thus, the amplitudes $\bar{a}_{k}(t)$ are determined by two functions, ΔU(𝒬) and g(𝒬), that evolve along a chosen LCT. These two functions are related to the two diabatic basis functions H₁₂(𝒬) and ΔH(𝒬) by Eq. (52.14). It follows from Eq. (52.23) that $\bar{a}_{1}$ and $\bar{a}_{2}$ become time independent outside NAR. This property of the amplitudes in the adiabatic basis is lost for their counterparts in the diabatic basis.

A solution of the equations of the nonadiabatic coupling across an isolated maximum of g(𝒬) within the time interval $t^{(-)}\leq t\leq t^{(+)}$ (LCT interval $\mathcal{Q}_{\text{LCT}}^{(-)}\leq\mathcal{Q}_{\text{LCT}}\leq\mathcal{Q}_{\text{LCT}}^{(+)}$, accordingly) gives the single-passage (or one-way) matrix of the nonadiabatic evolution, $\mathcal{N}_{n,m}^{+,-}$, which connects the amplitudes $\{a_{1}^{(-)},a_{2}^{(-)}\rightarrow a_{1}^{(+)},a_{2}^{(+)}\}$ on both sides of NAR. In particular, the probability P₁₂ of the nonadiabatic transition 1 → 2 and the probability 1 − P₁₂ of the nonadiabatic survival 1 → 1 are

$$\displaystyle P_{12}=\left|\mathcal{N}_{2,1}^{+,-}\right|^{2}\quad\mathrm{and}\quad 1-P_{12}=\left|\mathcal{N}_{1,1}^{+,-}\right|^{2}\;.$$

(52.24)

The efficiency of nonadiabatic coupling in crossing an isolated NAR can be estimated by the Landau formula for transition probability in the near-adiabatic limit when P₁₂ ≪ 1

$$\displaystyle P_{12}=\exp\left[-\frac{2}{\hbar}\left|\Im\int_{t_{\text{r}}}^{t_{\text{c}}}\Delta U\left[\mathcal{Q}_{\text{LCT}}(t)\right]\mathrm{d}t\right|\right]\;,$$

(52.25)

where t_c is a root of the equation 𝒬_LCT(t_c) = 𝒬_c, and t_r is any real-valued time. The single-passage transition probabilities are discussed in Sect. 52.3.

Once passage matrices $\mathcal{N}_{n,m}^{+,-}$ are known, they can be incorporated into the general scheme of nonadiabatic dynamics. In particular, the transition probabilities for the double passage of the same NAR are discussed in Sect. 52.4, and the nonadiabatic dynamics with multiple transitions is described in Sect. 52.5.

2.3 Selection Rules for Nonadiabatic Coupling

In the general case, the coupling between adiabatic states or diabatic states is controlled by certain selection rules. The most detailed selection rules exist for a system of two colliding atoms, since this system possesses a high symmetry (𝒞_∞v or, for identical atoms, D_∞h point symmetry in the adiabatic approximation). In the adiabatic representation, the coupling is due to the elements of the matrix 𝒟. They fall into two different categories: those proportional to the radial nuclear velocity (coupling by radial motion or radial coupling) and those proportional to the angular velocity of rotation of the molecular axis (coupling by rotational motion or Coriolis coupling). Besides, 𝒟 includes the spin–orbit and other weak interactions if they are not included in the adiabatic Hamiltonian.

In a diabatic representation, the coupling is due to the parts of the interaction potential neglected in the definition of the diabatic Hamiltonian H₀. In typical cases, these parts (besides the nuclear motion) are: the electrostatic interaction between different electronic states constructed as certain electronic configurations (H₀ corresponds to a self-consistent field Hamiltonian); spin–orbit interaction (H₀ corresponds to a nonrelativistic Hamiltonian); hyperfine interaction (H₀ ignores the magnetic interaction of electronic and nuclear spins); as well as the electrostatic interaction between electrons and quadrupole moments of nuclei. Different definitions of the adiabatic electronic states (i.e., different Hund coupling cases 1 ) of a system of two atoms are discussed in 11 ; the respective selection rules are summarized in 12 . The selection rules for the above interactions in this case are listed in Table 52.1 for two conventional nomenclatures for molecular states: Hund's case (a), ^2S+1Λ ^(σ)_w , and Hund's case (c), Ω ^(σ)_w .

Tab. 52.1 Selection rules for the coupling between diabatic and adiabatic states of a diatomic quasi molecule (w = g , u; $\sigma=+,-$)

For molecular systems with more than two nuclei, the selection rules cannot be put in a detailed form since, in general, the symmetry of the system is quite low. For the important case of three atoms, a general configuration is planar (𝒞_s symmetry); particular configurations correspond to an isosceles triangle if two atoms are identical (𝒞_2vsymmetry), to an equilateral triangle for three identical atoms (D_3h symmetry), or to a linear configuration. For the last case, the selection rules are the same as for a system of two atoms. The selection rules for the dynamic coupling between adiabatic states classified according to the irreducible representations of the 𝒞_s and 𝒞_2v groups are listed in Table 52.2. In Table 52.2, z and y refer to two modes of the relative nuclear motion in the system plane, R_z and R_y refer to two rotations about principal axes of inertia lying in the system plane, and R_x refers to a rotation about the principal axis of inertia perpendicular to the system plane. In-plane motion of nuclei couples the state of the same reflection symmetry; rotation of the system plane couples the state of the different reflection symmetry. The spin–orbit interaction, if included in matrices D_mn and 𝒟_mn, couples the states of different reflection symmetry. For a more detailed discussion see 12 .

Tab. 52.2 Selection rules for dynamic coupling between adiabatic states of a system of three atoms

3 Single-Passage Transition Probabilities in Common Trajectory Approximation

3.1 Transitions Between Noncrossing Adiabatic Potential Energy Curves

Single-passage transition probabilities between noncrossing adiabatic potential curves for different models can be classified by either diabatic or adiabatic Hamiltonians using the common trajectory with the NAR region. The following discussion refers to the cases of one-dimensional motion, when the transition probability can be expressed analytically. Here, a quite general model corresponds to exponential diabatic potentials and coupling (Nikitin (N) model 13 ), with residual dynamical coupling neglected. It is formulated as

$$\displaystyle\begin{aligned}\displaystyle\Delta H^{\text{N}}(\mathcal{Q})&\displaystyle=-\Delta E\left[1-\cos 2\vartheta\,\mathrm{e}^{-\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}})}\right]\;,\\ \displaystyle H_{12}^{\text{N}}(\mathcal{Q})&\displaystyle=(\Delta E/2)\sin 2\vartheta\,\mathrm{e}^{-\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}})}\;.\end{aligned}$$

(52.26)

Here, ϑ is a parameter that defines the interplay between the off-diagonal element H ^N₁₂ (𝒬) and the difference in the diagonal elements ΔH^N(𝒬). In addition, Eq. (52.26) includes asymptotic spacing ΔE of diabatic PEC (for α(𝒬 − 𝒬_p) ≫ 1) and characteristic scale 1 ∕ α of the exponential interaction. The meaning of the characteristic coordinate 𝒬_p is revealed, when one passes to the adiabatic basis with the spacing ΔU(𝒬) and dynamical coupling g(𝒬)

$$\displaystyle\begin{aligned}\displaystyle\Delta U(\mathcal{Q})&\displaystyle=\Delta E\sqrt{1-2\cos 2\vartheta\,\mathrm{e}^{-\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}}})+\,\mathrm{e}^{-2\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}})}}\;,\\ \displaystyle g(\mathcal{Q})&\displaystyle=\frac{\alpha}{2}\frac{\sin 2\vartheta\,\mathrm{e}^{-\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}}})}{1-2\cos 2\vartheta\,\mathrm{e}^{-\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}}})+\,\mathrm{e}^{-2\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}})}}\;.\end{aligned}$$

(52.27)

As ϑ changes from very small values to π ∕ 2, the spacing of adiabatic PEC changes from the narrow avoided crossing to the wide one and ultimately displays strong divergence. The adiabatic PEC cross at

$$\displaystyle\mathcal{Q}_{\text{c}}=\mathcal{Q}_{\text{p}}\pm 2\mathrm{i}\vartheta/\alpha\;.$$

(52.28)

This expression defines 𝒬_p as the center of NAR, 𝒬_p = ℜ𝒬_c, while the characteristic range of NAR is $\Delta\mathcal{Q}_{\text{p}}=\Im\mathcal{Q}_{\text{c}}=2\vartheta/\alpha$. The adiabatic states become uncoupled outside the NAR, while diabatic states remain coupled on the asymptotic side of NAR.

For this model, adiabatic wave functions coincide with diabatic functions before entering the coupling region (in the limit α(𝒬 − 𝒬_p) ≫ 1), but after exiting the coupling region (in the limit $\alpha(\mathcal{Q}-{\mathcal{Q}_{\text{p}}})\ll-1$) they are ϑ-dependent linear combinations of the diabatic functions.

The definition of the exponential model within the LCT concept is completed once one specifies the LCT that crosses the NAR. It is taken to be a segment of a rectilinear trajectory,

$$\displaystyle\mathcal{Q}(t)-\mathcal{Q}_{\text{p}}=v_{\text{p}}(t-t_{\text{p}})\;,$$

(52.29)

where $v_{\text{p}}=\dot{\mathcal{Q}}(t)|_{t=t_{\text{p}}}$ is additional parameter of the model, and t_p is the time corresponding to the center of NAR.

The solution of coupled equations yields the transition probability between adiabatic states

$$\displaystyle P^{\text{N}}_{12}=\frac{\sinh(\zeta\cos^{2}\vartheta)}{\sinh\zeta}\,\mathrm{e}^{-\zeta\sin^{2}\vartheta}\;,$$

(52.30)

where ζ = πΔE ∕ (ℏαv_p) is the asymptotic Massey parameter. Its value at the center of NAR is ζ_p = ζsin²ϑ.

Special cases of the exponential model correspond to the linearly crossing diabatic potentials with the constant coupling (Landau–Zener (LZ) model 14 ; 15 ; 16 ), parallel diabatic potentials with the exponential coupling (Demkov (D) model 17 ), and the asymptotically degenerate diabatic potentials (resonance (R) model 13 ).

The LZ model 14 ; 15 ; 16 is obtained from the exponential model by retaining Eq. (52.27) terms linear in ϑ and α(𝒬 − 𝒬_p) (i.e., for ϑ ≪ 1, ζ ≫ 1)

$$\displaystyle\begin{aligned}\displaystyle\Delta H^{\text{LZ}}(\mathcal{Q})&\displaystyle=\Delta E\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}})\equiv\Delta F\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}})\;,\\ \displaystyle H_{12}^{\text{LZ}}&\displaystyle=\Delta E\vartheta\equiv V\;.\end{aligned}{}$$

(52.31)

Here, the right-hand-side corresponds to the standard parameterization of the LZ model, where ΔF = ΔEα denotes the difference in slopes of the diabatic crossing PEC at 𝒬_p, and V is a constant. The pattern of the adiabatic PEC, with ΔU(𝒬_p) calculated from Eq. (52.31) as

$$\displaystyle\Delta U(\mathcal{Q})=\left|\sqrt{\Delta F^{2}(\mathcal{Q}-\mathcal{Q}_{\text{p}})^{2}+4V^{2}}\right|,$$

(52.32)

corresponds to an avoided crossing (or pseudo crossing). For this model, adiabatic wave functions coincide with diabatic functions on one side of the NAR, but on the other side, the former interchange with the latter accompanied by a sign reversal.

When ϑ ≪ 1 and ζ ≫ 1, Eq. (52.30) yields the LZ transition probability

$$\displaystyle P^{\text{LZ}}_{12}=\exp(-2\pi\zeta\vartheta^{2})=\exp(-2\pi V^{2}/\Delta Fv_{\text{p}}).$$

(52.33)

Remarkable properties of the Landau–Zener model are:

(i):

Expression Eq. (52.33) for the probability P ^LZ₁₂ is valid for arbitrary values of the exponent and not only for the large ones. In the latter case, the probability is very low, so that the system moves preferentially along a noncrossing adiabatic PEC.

(ii):

In the weak-coupling case (small value of the exponent in Eq. (52.33) termed narrow avoided crossing), the system moves preferentially along a crossing diabatic PES. The survival probability for a motion along the adiabatic PEC is then very low, $1-P^{\text{LZ}}_{12}=2\pi V^{2}/(\Delta Fv_{\text{p}})$.

The D model 17 is obtained from Eq. (52.27) by setting ϑ = π ∕ 4

$$\displaystyle\begin{aligned}\displaystyle\Delta H^{\text{D}}(\mathcal{Q})&\displaystyle=\Delta E\;,\\ \displaystyle H_{12}^{\text{D}}(\mathcal{Q})&\displaystyle=(\Delta E/2)\exp\left[-\alpha(\mathcal{Q}-\mathcal{Q}_{\text{p}})\right]\;.\end{aligned}$$

(52.34)

The transition probability in this case is

$$\displaystyle P^{\text{D}}_{12}=\frac{\exp(-\zeta)}{1+\exp(-\zeta)}\;.$$

(52.35)

For this model, the adiabatic wave functions coincide with the diabatic functions on the one side of NAR, but on the other side, the former are expressed through the linear combinations of the latter with equal weights.

The R model is defined by the condition ΔE = 0, i.e., ζ = 0. For arbitrary ϑ, this case corresponds to the so-called accidental resonance (AR), and for ϑ = π ∕ 4, it corresponds to the symmetric resonance (SR). In these two cases, the transition probabilities read

$$\displaystyle\begin{aligned}\displaystyle P^{\text{AR}}_{12}=&\displaystyle\left.P^{\text{N}}_{12}\right|_{\zeta=0}=\cos^{2}\vartheta\;,\\ \displaystyle P^{\text{SR}}_{12}=&\displaystyle\left.P^{\text{N}}_{12}\right|_{\zeta=0,\vartheta=\pi/4}=1/2\;.\end{aligned}$$

(52.36)

3.2 Transitions Between Crossing Adiabatic Potential Curves

The nomenclature of the adiabatic PEC within and outside a NAR differs for noncrossing and crossing adiabatic PES. For the former case, the assignment of the asymptotic states can be done according to their energies (e.g., U₁(𝒬) > U₂(𝒬)), while for the latter case, the inequality sign is reversed as a system passes the NAR. When applying the model for the noncrossing adiabatic PEC to the crossing PEC, one should take into account this reversal of the nomenclature. Then the transition probability P₁₂ between the crossing adiabatic PEC can be recovered from the survival probability for the noncrossing adiabatic PEC with the proper modification of parameters.

A particular case of the linear crossing (LC) adiabatic potentials and the constant dynamical coupling corresponds to the Hamiltonian

$$\displaystyle\begin{aligned}\displaystyle\Delta U^{\text{LC}}(\mathcal{Q})=&\displaystyle\Delta F\alpha(\mathcal{Q}-\mathcal{Q}_{\text{c}})\;,\\ \displaystyle\mathcal{D}_{12}^{\text{LC}}=&\displaystyle D=\text{const}\;.\end{aligned}$$

(52.37)

This Hamiltonian can be mapped onto the LZ Hamiltonian Eq. (52.31) by replacing the velocity v_p at the NAR center with the velocity v_c at the crossing point 𝒬_c. Taking into account the reversal of the LZ nomenclature for the adiabatic and diabatic functions in crossing the NAR, the nonadiabatic transition probability for the LC P ^LC₁₂ is related to P ^LZ₁₂ as

$$\displaystyle\begin{aligned}\displaystyle P^{\text{LC}}_{12}=&\displaystyle 1-\left.P^{\text{LZ}}_{12}(V,v_{\text{p}})\right|_{V\rightarrow D,v_{\text{p}}\rightarrow v_{\text{c}}}\\ \displaystyle=&\displaystyle 1-\exp\left[-2\pi D^{2}/(\hbar\Delta Fv_{\text{c}})\right]\;.\end{aligned}$$

(52.38)

The matrix element D is often related to the Coriolis coupling between the adiabatic electronic states of different spatial symmetry. With D being proportional to the angular velocity of rotation of the molecular frame at the crossing point 𝒬_c, this coupling is normally weak, and in this limit Eq. (52.38) reads

$$\displaystyle P^{\text{LC}}_{12}\approx 2\pi D^{2}/(\hbar\Delta Fv_{\text{c}})\;.$$

(52.39)

With P ^LC₁₂  ≪ 1, the system moves preferentially along the crossing PEC. However, sometimes the adiabatic PES are defined with the spin–orbit interaction neglected. In this case, the adiabatic electronic states are associated with different values of the electronic spin, and then the spin–orbit interaction is included into the coupling matrix element D. Then the approximation in Eq. (52.39) may not be valid, and one should use the general relation in Eq. (52.38) with the quantity D² in the exponent accounting for the contributions due to the dynamical (Coriolis) and static (spin–orbit) couplings.

4 Double-Passage Transition Probabilities

4.1 Transition Probabilities in the Classically-Allowed and Classically-Forbidden WKB Regimes: Interference and Tunneling

If a system traverses several NAR, its dynamics can be characterized by successive matching of the different 𝒩 matrices by the intermediate matrices of the adiabatic evolution. The latter have a simple expression when the system motion satisfies the standard WKB condition. In the case of an atomic collision, the set ℛ shrinks into a single coordinate, R, and $v_{\text{p}}=\sqrt{2(E-U_{\text{p}})/\mu}$ is expressed through the radial energy, E, of the relative motion of the atoms having the reduced mass, μ, and the potential energy, $U_{\text{p}}=\bar{U}(R)|_{R=R_{\text{p}}}$. If there is only one NAR over the whole range of R, the colliding system traverses it twice, as the atoms approach and then recede, and there are two different paths between the center of the NAR, R_p, and the turning points, R_t1 and R_t2, on the adiabatic potential curves, U₁(R) and U₂(R). The double-passage nonadiabatic event 1 → 2 consists of: (i) the single-passage transition 1 → 2, the adiabatic evolution R_p → R_t2 → R_p in the state | 2⟩, the single-passage survival 2 → 2; (ii) the single-passage survival 1 → 1, the adiabatic evolution R_p → R_t1 → R_p in state | 1⟩, and the single-passage transition 1 → 2. Accordingly, the double-passage nonadiabatic amplitude 𝒜 ^WKB₁₂ for the transition 1 → 2 reads

$$\displaystyle\begin{aligned}\displaystyle\mathcal{A}_{12}^{\text{WKB}}&\displaystyle=\sqrt{P_{12}}\exp(2\mathrm{i}\phi_{2}+2\mathrm{i}\Delta\phi_{12})\sqrt{P_{22}}\\ \displaystyle&\displaystyle\quad\,+\sqrt{P_{11}}\exp(2\mathrm{i}\phi_{1}+2\mathrm{i}\Delta\phi_{21})\sqrt{P_{12}}\;,\end{aligned}$$

(52.40)

where $P_{22}=P_{11}=1-P_{12}$ are the survival probabilities, 2ϕ₁ and 2ϕ₂ are the WKB phases accumulated during the adiabatic motion of a diatom from the center of NAR to the turning points and back, and 2Δϕ₁₂ and 2Δϕ₂₁ are the so-called dynamical phases that originate from nonadiabatic dynamics in crossing the NAR. Then the double-passage transition probability, $\mathcal{P}_{12}^{\text{WKB}}=\left|\mathcal{A}_{12}^{\text{WKB}}\right|^{2}$, assumes the form

$$\displaystyle\begin{aligned}\displaystyle&\displaystyle\left.\mathcal{P}_{12}^{\text{WKB}}(E)\right|_{E> U_{\text{p}}}=\\ \displaystyle&\displaystyle 4P_{12}(1-P_{12})\sin^{2}(\Phi_{12}^{\text{WKB}}+\Delta\Phi_{12})\;.\end{aligned}$$

(52.41)

Here,

$$\displaystyle\begin{aligned}\displaystyle\hbar\Phi_{12}^{\text{WKB}}(E)&\displaystyle={}\hbar(\phi_{1}-\phi_{2})\\ \displaystyle&\displaystyle=\int^{R_{\text{t1}}}_{R_{\text{p}}}p_{1}(R)\,\mathrm{d}R-\int^{R_{\text{t2}}}_{R_{\text{p}}}p_{2}(R)\mathrm{d}R\;,\\ \displaystyle\Delta\Phi_{12}&\displaystyle={}\Delta\phi_{12}-\Delta\phi_{21}\;,\end{aligned}{}$$

(52.42)

where p_k are the classical moments for motion across adiabatic potentials U_k(R). Equation (52.41) was first derived by Stückelberg 18 for an avoided crossing situation under assumption ΔΦ₁₂ = 0.

If the single-passage transition probability is calculated in the LCT approximation, Eq. (52.41) represents a combination of LCT and WKB approximations. A further simplification of this expression corresponds to the replacement of the WKB phase, Φ ^WKB₁₂ , with its CT counterpart,

$$\displaystyle\Phi_{12}^{\text{CT}}=\frac{1}{\hbar}\int^{t_{\text{t}}}_{t_{\text{p}}}\Delta U\big(R(t)\big)\mathrm{d}t\;,$$

(52.43)

where ΔU(R) = U₁(R) − U₂(R), R(t) is a CT, and, t_p and t_t are the time moments corresponding to the NAR center and the turning points for the motion in the field of the CT potential, U^CT(R), respectively. An ambiguity in the definition of U^CT(R) shows up in the approximation for 𝒫₁₂ resulting from the passage Φ ^WKB₁₂  → Φ ^LC₁₂ .

When P₁₂ in Eq. (52.41) is considered within the LZ or D models, the respective expressions are called Landau–Zener–Stückelberg (LZS) probability and Rosen–Zener–Demkov probability (although the interference phase was calculated in Eq. 19 for a special case of time-dependent coupling). Analytical expressions for the dynamical phases for the models discussed in Sect. 52.3 are also available 3 ; 6 .

A particular example of the LZS probability corresponds to the narrow avoided crossing between the adiabatic potentials (or to the weak coupling between the crossing diabatic curves). Here, one replaces the adiabatic potentials in Eq. (52.42) by their diabatic counterparts that define the phase, Φ ^WKB-D₁₂ , with ΔΦ₁₂ = π ∕ 4 3 . Then Eq. (52.41), in the weak-coupling limit and with the LZS parameterization assumes the form derived by Landau (L) 14

$$\displaystyle\begin{aligned}\displaystyle\left.\mathcal{P}_{12}^{\text{L}}(E)\right|_{E> U_{\text{p}}}&\displaystyle=\frac{8\pi V^{2}}{\hbar\Delta F}\left[\frac{\mu}{2(E-U_{\text{p}})}\right]^{1/2}\\ \displaystyle&\displaystyle\quad\,\times\sin^{2}\left(\Phi_{12}^{\text{WKB-D}}(E)+\pi/4\right)\;,\end{aligned}$$

(52.44)

which is valid for $\left.\mathcal{P}_{12}^{\text{L}}(E)\right|_{E> U_{\text{p}}}\ll 1$. The unphysical divergence in Eq. (52.44) at E → U_p is due to the use of the WKB approximation, when turning points are close to the crossing point. For a correct expression in that case, see Sect. 52.4.2.

In many applications, one can use the mean transition probability ⟨𝒫₁₂⟩, which is obtained from 𝒫₁₂, by averaging over several oscillations,

$$\displaystyle\langle\mathcal{P}_{12}\rangle=2P_{12}(1-P_{12})\;,$$

(52.45)

such that the double-passage transition probability ⟨𝒫₁₂⟩ is expressed only in terms of the single-passage transition probability, P₁₂. Equation (52.45) is a simple example of the surface-hopping approximation 9 , when one calculates the total transition probability as a results of two independent hop/survival events at the first and the second crossings of NAR: hop 1 → 2, (survival 2 → 2) + (survival 1 → 1), hop 2 → 1 (Sect. 52.5.2).

The main condition of applicability of Eqs. (52.40)–(52.45) is Φ ^WKB₁₂  ≫ 1, i.e., the energy, E, is well above the mean adiabatic potential at the NAR center, U_p. In this case, the NAR region is classically accessible, and the single-passage probabilities depend on the local kinetic energy E − U_p at the NAR center. With decreasing collision energy, the condition Φ ^WKB₁₂  ≫ 1 for the WKB interference eventually breaks down, implying also the inapplicability of the LCT approximation for the single-passage transition probability. When the energy, E, drops well below U_p, one can again use the WKB approximation for the description of tunneling nonadiabatic transitions 1 ; 3 . Then the system reaches NAR by motion in the classically forbidden range of R, and this motion is characterized by the imaginary-value quantity $\left.\Phi_{12}^{\text{WKB}}\right|_{E<U_{\text{p}}}$, which is obtained from the real-valued quantity, $\left.\Phi_{12}^{\text{WKB}}\right|_{E> U_{\text{p}}}$, by the analytical continuation into the complex energy plane from the classically accessible to the classically forbidden WKB motion. Under the condition $\left|\Im\Phi_{12}^{\text{WKB}}(E)\right|_{E<U_{\text{p}}}\gg 1$, the transition probability, $\left.\mathcal{P}_{12}^{\text{WKB}}(E)\right|_{E<U_{\text{p}}}$, which is a classically forbidden counterpart of $\left.\mathcal{P}_{12}^{\text{WKB}}(E)\right|_{E> U_{\text{p}}}$ in Eq. (52.41), is proportional to a small exponential factor

$$\displaystyle\left.\mathcal{P}_{12}^{\text{WKB}}(E)\right|_{E<U_{\text{p}}}\propto\exp\left[-2\left|\Im\Phi_{12}^{\text{WKB}}(E)\right|\right]\;.$$

(52.46)

The pre-exponential factor for Eq. (52.46) can be calculated either by specifying the integration contour for calculating the WKB transition amplitude or by the analytical continuation of Eq. (52.44) from the classically accessible to the classically forbidden WKB regions. In particular, the tunneling weak-coupling probability, $\left.\mathcal{P}_{12}^{\text{L}}(E)\right|_{E<U_{\text{p}}}$, which is a counterpart of $\left.\mathcal{P}_{12}^{\text{L}}(E)\right|_{E> U_{\text{p}}}$ in Eq. (52.44), reads

$$\displaystyle\begin{aligned}\displaystyle\left.\mathcal{P}_{12}^{\text{L}}(E)\right|_{E<U_{\text{p}}}&\displaystyle=\frac{2\pi V^{2}}{\hbar\Delta F}\left[\frac{2\mu}{E-U_{\text{p}}}\right]^{1/2}\\ \displaystyle&\displaystyle\quad\,\times\exp\left[-2\left|\Im\Phi_{12}^{\text{WKB-D}}(E)\right|\right]\;.\end{aligned}$$

(52.47)

Another practically important example of the nonadiabatic transition that occurs in the deep tunneling region corresponds to the Landau–Teller 20 model of the vibrational relaxation of a diatomic molecule in a collision with atoms. The probability, P^LT, of the transition with the energy release, ΔE (which is induced by the relative motion in a field of a repulsive exponential potential of the range 1 ∕ α) is proportional to a small exponential factor with the exponent that is different in the CT and WKB approximations

$$\displaystyle\begin{aligned}\displaystyle P^{\text{LT-CT}}&\displaystyle\propto{}\exp\left[-2\pi\Delta E\sqrt{\mu/2}/(\hbar\alpha\sqrt{E})\right]\;,\\ \displaystyle P^{\text{LT-WKB}}&\displaystyle\propto{}\exp\left[-2\pi\frac{\sqrt{2\mu}\left(\sqrt{E+\Delta E}-\sqrt{E}\right)}{\hbar\alpha}\right]\;,\end{aligned}$$

(52.48)

where E is the initial collision energy. Since the LT-CT exponent in Eq. (52.48) is the first term of the expansion of the LT-WKB exponent in powers of the ratio ΔE ∕ E, the former provides a good approximation to the latter for ΔE ∕ E ≪ 1.

4.2 Nonadiabatic Transitions near Turning Points

The effect of the turning points onto the double-passage probability shows itself in Eq. (52.41) through the phase Φ ^WKB₁₂ , assumed to be large enough. With E approaching U_p, the turning points become closer to the NAR center. In this situation, one can use a model of linearly crossing (LC) diabatic PEC.

In the CT approach, one keeps the LZ Hamiltonian Eq. (52.31) but introduces a CT that explicitly describes the incoming and outgoing motion within the NAR. For the case when the slopes, −F₁ and −F₂, of the diabatic potentials at R_c are of the same sign, the CT can be defined as the decelerated and accelerated motion in the field of the effective force $\bar{F}$

$$\displaystyle R(t)-R_{\text{p}}=\bar{F}t^{2}/(2\mu)-(E-U_{\text{p}})/\bar{F}\;,$$

(52.49)

where $\bar{F}$ is assumed to be a certain mean of F₁ and F₂. For E > U_p, the trajectory R(t) reaches the center of NAR twice at $t_{\text{p}}=\pm\sqrt{2\mu(E-U_{\text{p}})}/\bar{F}$. For E < U_p, t_p is imaginary, and the center of NAR is classically inaccessible; then the trajectory R(t) reaches the NAR center through the classically forbidden range of potentials. It turns out that when $\bar{F}=\sqrt{F_{1}F_{2}}$, the semiclassical problem defined by the LZ Hamiltonian Eq. (52.31) and the CT Eq. (52.49) becomes equivalent to the quantum problem with the Hamiltonian Eq. (52.31) supplemented by the mean potential $\bar{U}(R)=U_{\text{p}}-(F_{1}+F_{2})(R-R_{\text{p}})$ 3 .

In the weak-coupling limit, the expression for the LC transition probability, 𝒫 ^LC₁₂  ≪ 1, reads

$$\displaystyle\begin{aligned}\displaystyle\mathcal{P}_{12}^{\text{LC}}(E)&\displaystyle=4\pi^{2}V^{2}\left(\frac{2\mu}{\hbar^{2}\bar{F}\Delta F}\right)^{2/3}\\ \displaystyle&\displaystyle\quad\,\times\mathrm{Ai}^{2}\left[(U_{\text{p}}-E)\left(\frac{2\mu\Delta F^{2}}{\hbar^{2}\bar{F}^{4}}\right)^{1/3}\right]\;,\end{aligned}$$

(52.50)

where Ai is the Airy function, and the difference U_p − E can be of either sign.

The large absolute values of the argument of the Airy function correspond to the cases when the turning points are far from the NAR center; then Eq. (52.50) passes into the WKB expressions Eqs. (52.44) and (52.47), where the phases refer to the linear potentials

$$\displaystyle\mathcal{P}_{12}^{\text{LC}}(E)=\begin{cases}\left.\mathcal{P}_{12}^{\text{LC}}(E)\right|_{E> U_{\text{p}}},&\Phi_{12}^{\text{WKB-D}}\rightarrow\Phi_{12}^{\text{WKB-DL}}\gg 1\;,\\ \left.\mathcal{P}_{12}^{\text{LC}}(E)\right|_{E<U_{\text{p}}},&\left|\Phi_{12}^{\text{WKB-D}}\right|\rightarrow\left|\Phi_{12}^{\text{WKB-DL}}\right|\gg 1\;.\end{cases}$$

(52.51)

Here, the phase Φ ^WKB-DL₁₂ , which is real for E > U_p and imaginary for E < U_p, is

$$\displaystyle\Phi_{12}^{\text{WKB-DL}}(E)=\frac{2}{3}(E-U_{\text{p}})^{3/2}\sqrt{\frac{2\mu}{\hbar^{2}}}\frac{\Delta F}{\bar{F}^{2}}\;.$$

(52.52)

In the intermediate cases, when the Airy function cannot be approximated by its asymptotic expressions, Eq. (52.51) describes the situation when the center of NAR is close to the turning points. A further discussion on the transitions near turning points can be found in 3 . A review of different two-state models for single and double-passage nonadiabatic transitions is presented in 21 .

5 Multiple-Passage Transition Probabilities

5.1 Multiple Passage in Atomic Collisions

In the case of atomic collisions, there is only one nuclear coordinate R. If there exist several NAR on the R-axis, those that are classically accessible (for given total energy, E, and total angular momentum, J) can be traversed several times. In the semiclassical approximation 22 , the multiple-passage transition amplitude, A_if, for a given transition between initial state, i, and final state, f, can be calculated as a sum of transition amplitudes A ^ℒ_if , over all possible classical paths, ℒ, which connect these states and which run along a one-dimensional manifold R

$$\displaystyle A_{\mathrm{if}}=\sum_{\mathcal{L}}A^{\mathcal{L}}_{\mathrm{if}}\;,$$

(52.53)

where each A ^ℒ_if can be expressed through the probability, 𝒫 ^ℒ_if , and the phase, Φ ^ℒ_if , by 23

$$\displaystyle A^{\mathcal{L}}_{\mathrm{if}}=\big[\mathcal{P}^{\mathcal{L}}_{\mathrm{if}}\big]^{1/2}\exp\big(\mathrm{i}\Phi^{\mathcal{L}}_{\mathrm{if}}\big)\;.$$

(52.54)

The net transition probability is then

$$\displaystyle\begin{aligned}\displaystyle\mathcal{P}_{\mathrm{if}}&\displaystyle=|A_{\mathrm{if}}|^{2}\\ \displaystyle&\displaystyle=\sum_{\mathcal{L}}\mathcal{P}^{\mathcal{L}}_{\mathrm{if}}{+}{\sum_{\mathcal{L,L^{\prime}}}}^{\prime}\big[\mathcal{P}^{\mathcal{L}}_{\mathrm{if}}\mathcal{P}^{\mathcal{L^{\prime}}}_{\mathrm{if}}\big]^{1/2}\cos\big(\Phi^{\mathcal{L}}_{\mathrm{if}}-\Phi^{\mathcal{L^{\prime}}}_{\mathrm{if}}\big)\;.\end{aligned}$$

(52.55)

The first sum runs over all different paths and the second (primed) over all different pairs of paths. The primed sum usually yields a contribution to the transition probability, which oscillates rapidly with a change of the parameters entering into 𝒫_if (i.e., E and J) and represents a multiple-passage counterpart to the Stückelberg oscillations.

If the Stückelberg oscillations are neglected, 𝒫_if is equivalent to a mean transition probability ⟨𝒫_if⟩

$$\displaystyle\langle\mathcal{P}_{\mathrm{if}}\rangle=\sum_{\mathcal{L}}{\mathcal{P}}^{\mathcal{L}}_{\mathrm{if}}\;.$$

(52.56)

For one NAR, there are two equivalent paths, and ${\mathcal{P}}^{(1)}_{\mathrm{if}}=\mathcal{P}^{(2)}_{\mathrm{if}}=P_{\mathrm{if}}(1-P_{\mathrm{if}})$. Then Eq. (52.56) yields Eq. (52.45).

5.2 Surface Hopping

For molecular collisions, Eqs. (52.53) and (52.54) apply as well. However, the manifold of ℛ to which a trajectory 𝒬(t) belongs now comprises 3N − 5 (for a linear arrangement of nuclei) or 3N − 6 (for a nonlinear arrangement) degrees of freedom, where N is the number of atoms in the system. The approximation Eq. (52.56) is called, in the context of inelastic molecular collisions, the surface-hopping approximation 24 ; 5 . Each time a trajectory reaches a NAR, it bifurcates, and the system makes a hop from one PES to another with a certain probability. Keeping track of all the bifurcations and associated probabilities, one calculates 𝒫 ^ℒ_if along a path ℒ made up of different portions of trajectories running across different PES. Because of the complicated sequence of nonadiabatic events leading from the initial state to the final state, each 𝒫 ^ℒ_if is a complicated function of different single-passage transition probabilities, P_nm, and survival probabilities, 1 − P_nm. Even if all P_nm are known in analytical form, the calculation of ⟨𝒫_if⟩ requires numerical computations to keep track of individual nonadiabatic events 24 .

In order to formulate the approach implementing the surface-hopping approximation, often referred to as molecular dynamics with electronic transitions, it is convenient to rewrite Eqs. (52.18) for the time evolution of the wave function expansion coefficients in the more general density matrix notation. The diagonal density matrix elements in the adiabatic representation, ρ₁₁ = a₁a ^*₁ and ρ₂₂ = a₂a ^*₂ , determining the electronic state populations then satisfy the following equations

$$\displaystyle\frac{\mathrm{d}\rho_{22}}{\mathrm{d}t}=-\frac{\mathrm{d}\rho_{11}}{\mathrm{d}t}=\dot{\mathcal{Q}}g(\mathcal{Q})(\rho_{12}+\rho_{21})\;,$$

(52.57)

and the coherence defined by ρ₁₂ = a₁a ^*₂ satisfies

$$\displaystyle\frac{\mathrm{d}\rho_{12}}{\mathrm{d}t}=\mathrm{i}\hbar^{-1}\Delta U(\mathcal{Q})\rho_{12}+\dot{\mathcal{Q}}g(\mathcal{Q})(\rho_{11}-\rho_{22})\;.$$

(52.58)

Equation (52.57) shows that it is the dynamical coupling function g(𝒬) that promotes the changes in the electronic state populations, i.e., the electronic transitions. Due to these transitions the trajectory bifurcates, and the surface-hopping approach can be considered a procedure for selection of the proper branch for the system to evolve. The most popular version of that approach, fewest-switches surface hopping (FSSH) 25 , proposed by J. C. Tully is considered here. Its key feature is the algorithm maintaining the statistical distribution of electronic state populations in agreement with those obtained by numerically integrating the system of differential equations (52.57) and (52.58). Let the system be at time t on a trajectory 𝒬₁ corresponding to the state 1 with population ρ₁₁(t). According to Eq. (52.57) the nonadiabatic coupling g(𝒬) induces the electronic transitions in a small time interval δt, leading to the change of the state 1 population, $\delta\rho_{1}=\rho_{11}(t+\delta t)-\rho_{11}(t)$. If δρ₁ ≥ 0, i.e., there is a gain in the state 1 population, then the probability of the switch is set to zero, and the system remains on the trajectory 𝒬₁(t). If, in contrast, δρ₁ < 0, there exists a nonzero probability P₁₂ for the trajectory to switch to the trajectory branch 𝒬₂ corresponding to the state 2

$$\displaystyle P_{12}(t,\delta t)=-\frac{\delta\rho_{1}}{\rho_{11}(t)}\approx\frac{\lambda(t)\delta t}{\rho_{11}(t)}\;,$$

(52.59)

where λ(t) denotes the right-hand side of Eq. (52.57). The decision to switch from state 1 to state 2 is made when the switch probability P₁₂ is larger than a random number between 0 and 1 generated from the uniform distribution.

The FSSH algorithm automatically incorporates a number of important properties: (i) as P₁₂ is proportional to dynamic coupling g(𝒬), the transitions between electronic states can occur at any location and time, not only in the NAR specified by some predefined conditions like was done in Sect. Sect. 52.2.2 or in the earlier version of surface hopping, which makes the method applicable to any number of passages through NAR; (ii) for the large number of trajectories, their fraction located on the state i branch approaches ρ_ii(t) at any t, thus reproducing the electronic state populations; (iii) the electronic coherences are propagated properly with Eq. (52.58) enabling the reproducing of quantum interference effects; (iv) the generalization for any number of coupled electronic states is straightforward, and FSSH was recently implemented to study the nonadiabatic dynamics of an open-shell molecule at a metal surface 26 .

At the same time, surface hopping is not free from some ad hoc adjustments, which can limit its validity. For example, when a switch between state 1 and state 2 PES occurs, the velocities of classical degrees of freedom have to be rescaled to preserve the total energy, which leads to discontinuities in the classical trajectory, 𝒬(t), defined on the many-dimensional manifold, ℛ. Moreover, the amount of classical kinetic energy is not always sufficient to fill the gap between electronic states (classically forbidden transitions). In that case, the switch is rejected (a so-called “frustrated hop”), which introduces artificial distortions in the state populations.

The manifold ℛ can be reduced in size if one treats other degrees of freedom, besides electronic ones, on the same footing. In this way one introduces adiabatic vibronic (vibrational + electronic) states and adiabatic vibronic PES, and considers nonadiabatic transitions between them 27 . In the vibronic representation, the formal theory remains the same; however its implementation is more difficult since there are many more possibilities for trajectory branching. Finally, under certain conditions, one can use a fully adiabatic description of all degrees of freedom, save one – the intermolecular distance R. This approach provides a basis for the statistical adiabatic channel model (SACM) of unimolecular reactions 28 where the receding fragments are scattered adiabatically in the exit channels after leaving the region of a statistical complex.

For a recent review of the theory of molecular nonadiabatic dynamics, see 7 and papers in 9 , as well as the recent book 29 .