Photosynthesis: What Powers the Splitting of Water?
Catalysis is about reducing the free energy barrier (aka. activation energy) of a reaction, so it does not require any energy. In photolysis (e.g. splitting water) you get the energy from the absorbed photons.

The exact process is called the Joliot-Kok cycle:




Figure 1 - Joliot-Kok cycle - source


So the photon separates the charges on the P680, after that the activated P680 activates the Yz intermedier, which forces the enzyme to the next step (Sx) in the reaction. 


2012 - Transmembrane Electric Potential Difference in the Protein–Pigment Complex of Photosystem 2  
2006 - The Manganese-calcium oxide cluster of Photosystem II and its assimilation by the Cyanobacteria



  The overall process comprises three types of reaction sequences: (a)
  light-induced charge separation leading to formation of the radical
  ion pair P680+QA(-) ; (b) reduction of plastoquinone to plastoquinol
  at the QB site via a two-step reaction sequence with QA(-) as
  reductant and (c) oxidative water splitting into O2 and four protons
  at a manganese-containing catalytic site via a four-step sequence
  driven by P680+ as oxidant and a redox active tyrosine YZ acting as
  mediator.



2007 - Photosystem II: structure and mechanism of the water:plastoquinone oxidoreductase.


So the process does not involve ATP or NAD or something like that just the "redox active tyrosine YZ". ATP and NADPH are created after the PS2 part of photosynthesis.




Figure 2 - Light-dependent reactions of photosynthesis - electron transport chain - source


The photolysis of water is about storing the energy coming from the photons (light) in ATP and NADPH. In the Calvin cycle the cells use the stored energy to reduce CO2 into carbohydrates. For instance glucose can be delivered to the cells in the root, which can use it as food.