Horizontal recurrence relations

Using the horizontal recurrence relation (9.10.10), a similar relation may be written down for increments in j, replacing XpA with Xpu. For completeness, we give also the recurrence relation for increments in k for the second electron ... 

From our discussion of the Obara-Saika scheme three techniques emerge, differing in the use of the electron-transfer and the horizontal recurrence relations see Table 9.5. In the OS4 scheme, no use is made of the transfer and horizontal recurrences - the primitive Cartesian integrals... 

The recurrence relation (9.10.9) is identical to (9.10.8) except that XpA s replaced by XpB and is most easily obtain from the horizontal recurrence... 

We have now succeeded in setting up a set of recurrence relations by means of which the two-electron Cartesian integrals may be obtained from the Boys function. The resulting expressions are rather complicated, however, involving as many as eight distinct contributions. Unlike the McMurchie-Davidson scheme, the Obara-Saika scheme does not treat the two electrons separately since the recurrences (9.10.24) and (9.10.25), for example, affect the indices of all four orbitals. In Section 9.10.3, we shall see how the Obara-Saika recurrences may be simplified considerably when used in conjunction with two other types of recurrence relations the electron-transfer recurrences and the horizontal recurrences. 

In this way, we may build up the full set of Cartesian integrals by the use of three sets of recurrence relations, each of which is considerably simpler than the full Obara-Saika scheme. The horizontal recurrences (9.10.28) and (9.10.29) do not involve the orbital exponents and may therefore be applied cfier the transformation of the integrals to the contracted basis, as will be discussed later. In passing, we note that this procedure may also be used for the one-electron integrals, replacing the first step by the similar recurrence relation... 

Successful BNCT depends on three factors shallow seating of the tumor in the brain limited tumor infiltration and location of tumor outside of the eloquent cortex (those areas of the brain which, if removed, will result in loss of sensory processing or linguistic ability, or paralysis). Therapeutic depth is limited, because thermal neutrons undergo a Maxwellian distribution in the brain, fiuence decreases three dimensionally along the incident axis (Figure 11.4). Consequently, related to the incidental axis, both vertically and horizontally, the neutron dose exponentially decreases. If tumor cells are present beyond the therapeutic isodose contour, the absorbed dose will not be sufficiently lethal, and peri-lesional recurrence is likely. 

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