# Probing the nonlocal approximation to resonant collisions ofelectrons with diatomic molecules Page: 4 of 15

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4

XtJk). We can easily see that the state IXvi#q) is an

eigenfunction ofHo+V1 =TR+Vo+PHe1P

(27)

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 [16], 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(29)

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 [6] 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

asT v* f= (XerfV K*d .

(31)

Note that this expression differs slightly from the re-

sult of Domcke (ref. [6], 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)and therefore

(- 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

discrete statep -(r) = (4 (r))*,

(34)

we can simplify the matrix element between electronic

wave functions in (30) as

(0-IPHIQd) = (Kf-HejlId) -- (OdHeJllf,) = Vtf

(35)

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 [6] should be replaced by

V - under the assumption that Od is real, otherwise

V must be used.

dk

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).

where

J (R) = dro*(r; R)Ji (r).

(37)

(38)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 functionT+(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)(39)

(40)= 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 )(33)

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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.