# Single and Triple Differential Cross Sections for DoublePhotoionization of H- Page: 4 of 12

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4

of spherical harmonics is represented in a product ba-

sis of 1D finite element-discrete variable representation

(FEM-DVR) functions, similar to the approach used in

molecular hydrogen DPI referenced above. The FEM-

DVR radial basis is an attractive choice because of the

computational efficiency gained as well as the natural

complementarity for implementing exterior complex scal-

ing [48].

The Coulomb functions in Equation 3 are similarly ex-

panded in partial waves,

1/2 (k , r) ( 2)1 t j ' (r)Y m ( r) r (1 ),

,m k

(9)

where 01 (r) is a radial Coulomb function with asymp-

totic form

0()(r) -> sin(kr + (Z/k) ln 2kr - l/2 + np(k)), (10)

as r -> o and i represents the Coulomb phase

'1(k) argf(l + 1- iZ/k), (11)

with Z equal to the nuclear charge in the one-body po-

tentials of Eq. 4, here Z 1 for H-.

By substituting the partial wave expansions of the full

scattered solution T+ (Eq. 8) and the product of testing

functions 4)z ) (k, r) (Eq. 9) into the expression for the

full amplitude (Eq. 6) and integrating over the angular

coordinates dQ1dQ2 of both electrons, we arrive at an

expression connecting the partial waves of T+ with the

product of Coulomb waves,

f (ki, k2) C 2 (l1+12),2'/1 (ki)+2'q' (k2)

,1 l 2,m2

X TI 1,12,mi,m2 (ki,k2)Yimi (k1)Y2m2 (k2) ,

(12)

due to the orthogonality of the spherical harmonics. The

sum is once again constrained to include only angular

momentum pairs for which M mi + m2 0. Using

the standard theory of rearrangement scattering, com-

bined with a two-potential formalism, we can express the

partial wave amplitudes Y1,12,m1,m2(kI1, k2) appearing in

Eq. 12 as [34]:

1

-T11,12,mi,m2 (ki, k2) = X

k1k2

K Qki (r1)2,k2 (r2) E h- h2 Vlimi,12m2(rl, r2))

k1dk2 J dridr 1)4 (ri)q2 (r2) x

(E -h- h2) m1,12m2((I r2),)

(13)As before, the radial volume integral can be simplified by

application of Green's theorem,

(9(c)11,i (r)9$k2 (r2)IE -h h2 Vimi,2m2 (ri, r2))

1[9 ki(r1 )12k2 (r2) lim1,l2m2(r 1r2)

-limi,12m2(ri, r2) k '1(ri)9Km2 (r2) , dcx

P=Po

(15)

where po defines the hypershere where the partial wave

amplitudes are calculated, usually just inside of the ECS

turning point Ro.

D. Cross Section Evaluation from Reduced

Amplitudes

The partial wave amplitudes evaluated using Eqs. 13

and 15 are then returned to Eq. 12 to construct the full

double photoionization amplitude f(kl, k2). The TDCS

can then be calcuated by Eq. 7.

The single differential cross section (SDCS), describ-

ing the energy sharing between both ejected electrons, is

given by integrating the TDCS over all angles dQ1dQ2 of

electrons 1 and 2. Because of the orthonormality of the

spherical harmonics, cross terms between reduced ampli-

tudes for different angular configurations disappear, thus

the SDCS is simply given bydo, 49r2k2 (2)2

-'

dE1 LJC k2'11,12,m1,m2(ki, k2) 12

11m1 12m2(16)

The total cross section for double photionization is

then given by integrating the SDCS over the energy shar-

ing rangeaE dz dE ,

0 dE1(17)

although the SDCS is sometimes defined to give the to-

tal cross section by integration over half energy range.

Because the SDCS is symmetric about E/2, this simply

redefines the SDCS asdE1 2 dE1

(18)

thus also making the total DPI cross section

E/2 d?

=.Ej dEE.(19)

where hi and h2 are one-electron radial Hamiltonia

1 d2 1(1+1)

2 d,2 + 2r2z

ns

For consistency with our published SDCS results for he-

lium, we have adopted the convention of Equation 19 in

this work.

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Yip, Frank L.; Horner, Daniel A.; McCurdy, C. William & Rescigno,Thomas N. Single and Triple Differential Cross Sections for DoublePhotoionization of H-, article, February 15, 2007; United States. (digital.library.unt.edu/ark:/67531/metadc896689/m1/4/: accessed January 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.