Probing the nonlocal approximation to resonant collisions ofelectrons with diatomic molecules Page: 4 of 15
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XtJk). We can easily see that the state IXvi#q) is an
within the Born-Oppenheimer approximation which will
be defined explicitly later (see Eq. (48) below).
By employing the two-potential formula for the scat-
tering T-matrix (see , p. 202) we get
TKE,=f (Xvf9fVlKXvi J) + (XvfoV2K +) (28)
where T+(R, r) is given by Eq. (8), or equivalently, apart
from the Born-Oppenheimer approximation used for the
perturbed initial state, by
The last equation shows that we can take Xw (R)0 (r; R)
as the initial state of the system to determine the wave
function T+(R, r) which we will use later to derive the
effective equation for the nuclear dynamics.
The second term of Eq. (28) corresponds to the reso-
nant part of the T-matrix as defined in  and is fully
determined by the resonant part Q41+ of the full wave
function defined below in Eq. (40). Using Eqs. (16),
(21), (26) and the orthogonality (Okq-d) =0 we obtain
TyL = (Xv 7 H01- PHeliPq+)
(Xvfokf-PHeIQ4+) . (30)
This expression can be further simplified if we define
+(R) = (0d p )T where (...)T means an integration
over the electronic coordinate r only. In terms of the res-
onant nuclear wavefunction I'd, for which we will derive
the effective Schrddinger equation in the following sub-
section, the resonant part of the T-matrix can be written
T v* f= (XerfV K*d .
Note that this expression differs slightly from the re-
sult of Domcke (ref. , Eq. (4.14)) where the matrix
Vk without a superscript, which corresponds to the ma-
trix element Vk defined by Eq. (21), was, in our opin-
ion, used incorrectly. This small difference becomes im-
portant when the background terms defined below are
added to the resonant T matrix (which was not usually
the case in previous studies of resonant electron-molecule
collisions), since the coupling matrix elements Vi are in
general complex even when the the discrete state is real.
The reason why we cannot use V + instead of V is that
in general, in spite of the fact that p7 belongs to P space,
(0kf) # 3(kf/2 k2/2) (32)
(- IPHelQIpd) # (03 3He)lIa).
Instead, if we consider a special case of the real discrete
state and if we realize that for the radial case with a real
p -(r) = (4 (r))*,
we can simplify the matrix element between electronic
wave functions in (30) as
(0-IPHIQd) = (Kf-HejlId) -- (OdHeJllf,) = Vtf
where we assumed that Hei is a Hermitian operator.
Note that in this special case we can use the matrix ele-
ment V+ but without complex conjugation. In the three-
dimensional case, Eq. (34) must be modified to
k (# )* (36)
and thus V* in Eq. (4.14) of  should be replaced by
V - under the assumption that Od is real, otherwise
V must be used.
We now return to Eq. (28). Its first term is generally
called the background scattering T-matrix and reads
Tbgof (Xvf 4kf PH.1P He + ntXvi kiJ)
(Xvf- jVnt PH.iQIXviJ )
KXvfQQk IV nt IXvi Ji)
(Xvf XkfJt, x).
J (R) = dro*(r; R)Ji (r).
is an overlap of the unperturbed incoming wave with the
discrete state. These background terms are non-zero even
for inelastic vibrational excitation but generally small
when compared to the resonant part of the T-matrix.
For an example where these terms are not negligible, see
the results for the F2-like model in Section VI below.
C. Nuclear wave equation
To derive the basic equation of the nonlocal model
which determines the effective nuclear dynamics, we be-
gin by defining the outgoing, scattered wave part of the
full wave function
T+(R, r)= 4+(R, r) - Xvi (R)pt(r).
Since P + Q = 1 we can next write
(R, r) = Q<(R, r) + P< (R, r)
= d4(R)Od(r; R) + J k (R)p 0(r; R)k dk
where we have used Eqs. (15) and (19) and defined
4'k(R) _ (0f 4P), the P space counterpart of TdZ(R).
We next write Eq. (29) in differential form
(E - H) 141) = (He1 - PHeiP) Xzi 0) . (41)
I4'+) = IXvi #f) + V2X1f).
E -H + ir/z X ik )
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Houfek, Karel; Rescigno, Thomas N. & McCurdy, C. William. Probing the nonlocal approximation to resonant collisions ofelectrons with diatomic molecules, article, September 7, 2007; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc902556/m1/4/: accessed March 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.