Editing Born–Oppenheimer approximation (section)

== Born–Oppenheimer approximation with correct symmetry ==
To include the correct symmetry within the Born–Oppenheimer (BO) approximation,<ref name=BornOppie/><ref>{{cite book|first1=M. |last1=Born|author-link1=Max Born| first2=K. |last2=Huang|author-link2=Huang Kun|title= [[Dynamical Theory of Crystal Lattices]]|year=1954 |publisher=Oxford University Press|location=New York|chapter=IV}}</ref> a molecular system presented in terms of (mass-dependent) nuclear coordinates <math>\mathbf{q}</math> and formed by the two lowest BO adiabatic potential energy surfaces (PES) <math>u_1(\mathbf{q})</math> and <math>u_2 (\mathbf{q})</math> is considered. To ensure the validity of the BO approximation, the energy ''E'' of the system is assumed to be low enough so that <math>u_2 (\mathbf{q})</math> becomes a closed PES in the region of interest, with the exception of sporadic infinitesimal sites surrounding degeneracy points formed by <math>u_1(\mathbf{q})</math> and <math>u_2(\mathbf{q})</math> (designated as (1, 2) degeneracy points).

The starting point is the nuclear adiabatic BO (matrix) equation written in the form<ref>{{cite book | title=Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections
| chapter=Born-Oppenheimer Approach: Diabatization and Topological Matrix | publisher=John Wiley & Sons, Inc. | location=Hoboken, NJ, USA | date=28 March 2006 | isbn=978-0-471-78008-3 | doi=10.1002/0471780081.ch2 | pages=26–57}}</ref>
: <math>-\frac{\hbar^2}{2m} (\nabla + \tau)^2 \Psi + (\mathbf{u} - E)\Psi = 0, </math>
where <math>\Psi(\mathbf{q}) </math> is a column vector containing the unknown nuclear wave functions <math>\psi_k(\mathbf{q})</math>, <math>\mathbf{u}(\mathbf{q})</math> is a diagonal matrix containing the corresponding adiabatic potential energy surfaces <math>u_k(\mathbf{q})</math>, ''m'' is the reduced mass of the nuclei, ''E'' is the total energy of the system, <math>\nabla</math> is the [[gradient]] operator with respect to the nuclear coordinates <math>\mathbf{q}</math>, and <math>\mathbf{\tau}(\mathbf{q})</math> is a matrix containing the vectorial non-adiabatic coupling terms (NACT):
: <math>\mathbf{\tau}_{jk} = \langle \zeta_j | \nabla\zeta_k \rangle.</math>

Here <math>|\zeta_n\rangle</math> are eigenfunctions of the [[electronic Hamiltonian]] assumed to form a complete [[Hilbert space]] in the given region in [[Configuration space (physics)|configuration space]].

To study the scattering process taking place on the two lowest surfaces, one extracts from the above BO equation the two corresponding equations:
: <math>-\frac{\hbar^2}{2m} \nabla^2\psi_1 + (\tilde{u}_1 - E)\psi_1 - \frac{\hbar^2}{2m} [2\mathbf{\tau}_{12}\nabla + \nabla\mathbf{\tau}_{12}]\psi_2 = 0,</math>
: <math>-\frac{\hbar^2}{2m} \nabla^2\psi_2 + (\tilde{u}_2 - E)\psi_2 + \frac{\hbar^2}{2m} [2\mathbf{\tau}_{12}\nabla + \nabla\mathbf{\tau}_{12}]\psi_1 = 0,</math>
where <math>\tilde{u}_k(\mathbf{q}) = u_k(\mathbf{q}) + (\hbar^{2}/2m)\tau_{12}^2</math> (''k'' = 1, 2), and <math>\mathbf\tau_{12} = \mathbf\tau_{12}(\mathbf{q})</math> is the (vectorial) NACT responsible for the coupling between <math>u_1(\mathbf{q})</math> and <math>u_2(\mathbf{q})</math>.

Next a new function is introduced:<ref>{{cite journal | last1=Baer | first1=Michael | last2=Englman | first2=Robert | title=A modified Born-Oppenheimer equation: application to conical intersections and other types of singularities | journal=Chemical Physics Letters | publisher=Elsevier BV | volume=265 | issue=1–2 | year=1997 | issn=0009-2614 | doi=10.1016/s0009-2614(96)01411-x | pages=105–108| bibcode=1997CPL...265..105B }}</ref>
: <math> \chi = \psi_1 + i\psi_2, </math>
and the corresponding rearrangements are made:
# Multiplying the second equation by ''i'' and combining it with the first equation yields the (complex) equation <math display="block">-\frac{\hbar^2}{2m} \nabla^{2}\chi + (\tilde{u}_1 - E)\chi + i\frac{\hbar^2}{2m}[2\mathbf{\tau}_{12}\nabla + \nabla\mathbf{\tau}_{12}]\chi + i(u_1 - u_2)\psi_2 = 0.</math>
# The last term in this equation can be deleted for the following reasons:  At those points where <math>u_2(\mathbf{q})</math> is classically closed, <math>\psi_{2}(\mathbf{q}) \sim 0</math> by definition, and at those points where <math>u_2(\mathbf{q})</math> becomes classically allowed (which happens at the vicinity of the (1, 2) degeneracy points) this implies that: <math>u_1(\mathbf{q}) \sim u_2(\mathbf{q})</math>, or <math>u_1(\mathbf{q}) - u_2(\mathbf{q}) \sim 0</math>. Consequently, the last term is, indeed, negligibly small at every point in the region of interest, and the equation simplifies to become <math display="block">-\frac{\hbar^2}{2m} \nabla^{2}\chi + (\tilde{u}_1 - E)\chi + i\frac{\hbar^2}{2m}[2\mathbf{\tau}_{12}\nabla + \nabla\mathbf{\tau}_{12}]\chi = 0.</math>

In order for this equation to yield a solution with the correct symmetry, it is suggested to apply a perturbation approach based on an elastic potential <math>u_0(\mathbf{q})</math>, which coincides with <math>u_1(\mathbf{q})</math> at the asymptotic region.

The equation with an elastic potential can be solved, in a straightforward manner, by substitution. Thus, if <math>\chi_0</math> is the solution of this equation, it is presented as
: <math>\chi_0(\mathbf{q}|\Gamma) = \xi_{0}(\mathbf{q}) \exp\left[-i \int_\Gamma d\mathbf{q}' \cdot \mathbf{\tau}(\mathbf{q}'|\Gamma)\right],</math>
where <math>\Gamma</math> is an arbitrary contour, and the exponential function contains the relevant symmetry as created while moving along <math>\Gamma</math>.

The function <math>\xi_0(\mathbf{q})</math> can be shown to be a solution of the (unperturbed/elastic) equation
: <math>-\frac{\hbar^2}{2m} \nabla^{2}\xi_0 + (u_0 - E) \xi_0 = 0.</math>

Having <math>\chi_0(\mathbf{q}|\Gamma)</math>, the full solution of the above decoupled equation takes the form
: <math>\chi(\mathbf{q}|\Gamma) = \chi_0(\mathbf{q}|\Gamma) + \eta(\mathbf{q}|\Gamma),</math>
where <math>\eta(\mathbf{q}|\Gamma)</math> satisfies the resulting inhomogeneous equation:
: <math>-\frac{\hbar^2}{2m} \nabla^{2}\eta + (\tilde{u}_1 - E)\eta + i\frac{\hbar^2}{2m}[2\mathbf{\tau}_{12}\nabla + \nabla\mathbf{\tau}_{12}]\eta = (u_1 - u_0)\chi_0.</math>

In this equation the inhomogeneity ensures the symmetry for the perturbed part of the solution along any contour and therefore for the solution in the required region in configuration space.

The relevance of the present approach was demonstrated while studying a two-arrangement-channel model (containing one inelastic channel and one reactive channel) for which the two adiabatic states were coupled by a [[Jahn–Teller effect|Jahn–Teller]] [[conical intersection]].<ref>{{cite journal | last1=Baer | first1=Roi | last2=Charutz | first2=David M. | last3=Kosloff | first3=Ronnie | last4=Baer | first4=Michael | title=A study of conical intersection effects on scattering processes: The validity of adiabatic single-surface approximations within a quasi-Jahn–Teller model | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=105 | issue=20 | date=22 November 1996 | issn=0021-9606 | doi=10.1063/1.472748 | pages=9141–9152| bibcode=1996JChPh.105.9141B }}</ref><ref>{{cite journal | last1=Adhikari | first1=Satrajit | last2=Billing | first2=Gert D. | title=The conical intersection effects and adiabatic single-surface approximations on scattering processes: A time-dependent wave packet approach | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=111 | issue=1 | year=1999 | issn=0021-9606 | doi=10.1063/1.479360 | pages=40–47| bibcode=1999JChPh.111...40A }}</ref><ref>{{cite journal | last1=Charutz | first1=David M. | last2=Baer | first2=Roi | last3=Baer | first3=Michael | title=A study of degenerate vibronic coupling effects on scattering processes: are resonances affected by degenerate vibronic coupling? | journal=Chemical Physics Letters | publisher=Elsevier BV | volume=265 | issue=6 | year=1997 | issn=0009-2614 | doi=10.1016/s0009-2614(96)01494-7 | pages=629–637| bibcode=1997CPL...265..629C }}</ref>  A nice fit between the symmetry-preserved single-state treatment and the corresponding two-state treatment was obtained. This applies in particular to the reactive state-to-state probabilities (see Table III in Ref. 5a and Table III in Ref. 5b), for which the ordinary BO approximation led to erroneous results, whereas the symmetry-preserving BO approximation produced the accurate results, as they followed from solving the two coupled equations.