The mechanism of 4p P-1(1)-->4p P-3(J) intersystem crossing (ISC) following excitation of the 4p P-1(1) level of matrix-isolated atomic zinc is investigated using a pair potentials approach. This is achieved by extending earlier ISC calculations on the Zn . RG(2) and Zn . RG(3) complexes to the square planar Zn . RG(4) and square pyramidal Zn . RG(5) species which are the building blocks of the Zn . RG(18) cluster used to represent the isolation of atomic zinc in the substitutional site of a solid rare-gas host. ISC predictions in these clusters are based on whether crossing of the strongly bound (1)A(1) states, having a 3p P-1(1) atomic asymptote, occurs with the repulsive E-3 states correlating with the 4p P-3(J) atomic level of atomic zinc. Predictions based on (1)A(1)/E-3 curve crossings for E-3 states generated with the calculated ab initio points for the Zn . RG (3)Sigma(p(z)) states do not agree with matrix observations. Based on similar overestimation of ISC in the Zn . RG diatomics, less repulsive Zn . RG (3)Sigma(p(z)) potential curves are used resulting in excellent agreement between theory and observations in the Zn-RG matrix systems. (1)A(1)/E-3 curve crossings do not occur in the Zn-Ar system which shows only singlet emission. Curve crossings are found for the Zn-Xe system which exhibits only triplet emission. The Zn-ES system does not show a crossing of the body mode Q(2), which exhibits a strong singlet emission at 258 nm while the waist mode Q(3), does have a crossing, resulting in a weak singlet emission at 239 nm and a stronger triplet emission-at 312 nm. The efficiency of ISC is determined from Landau-Zener estimates of the surface hopping probabilities between the (1)A(1) and the E-3 states. Differences in the application of this theory in the gas and solid phase are highlighted, indicating that the rapid dissipation of the excited-state energy which occurs in the solid must be included to obtain agreement with observations.