calculate HOMO and LUMO eigenstates and 3D isosurface

[YCLin]

According to the tutorial of ATK 2008.10, we can extract the HOMO and LUMO eigenstate of water molecule as well as generate a 3D plot.
http://quantumwise.com/documents/manuals/ATK-2008.10/chap.complexmol.html
In the ATK 10.8.2 version, I am wandering if it is same function to the "Analysis->eigenstate".
Referring to the tutorial, the quantum number 3 and 4 stand for HOMO and LUMO state.
But, what are the meaning of the quantum numbers 0, 1, 2, 5, 6,...?
I would like to calculate the charge transfer of a water molecule adsorbed on graphene, and identify the electron or hole transfer to the water molecule from the overlap of graphene pi orbitals with HOMO or LUMO states.
The calculation is similar to these two papers "PRB 77, 073404 (2008)" and "PRB 76, 195406 (2007)".  
Also, I am not really understand the meaning of "effective potential", "electron density", and "electrostatic difference potential" and how to analysis the data from the 3D isosurface.

Thank you

[zh]

In the ATK 10.8.2 version, I am wandering if it is same function to the "Analysis->eigenstate".

Yes, it is same with the older version.

The quantum numbers 0, 1, 2, ....  are the index of eigenstates. Since in your case the fourth and fifth eigenstates (i.e., corresponding to the eigenstates with quantum numbers 3 and 4, respectively) are the HOMO and LUMO states, respectively, the eigenstates with quantum number 0, 1,2, 5, 6 will be the HOMO-3, HOMO-2, HOMO-1, LUMO+1, LUMO+2 states, respectively.

The charge transfer may be obtained by a charge population analysis, e.g., the Mulliken charge population, which is implemented in ATK.

For the physics meaning of "effective potential", "electron density", and "electrostatic difference potential", the best way is to refer to the following paper:
J. M. Soler, E. Artacho, J. D. Gale, A. García, J. Junquera, P. Ordejón, and D. Sánchez-Portal, J. Phys. Condens. Matter 14, 2745 (2002)

[YCLin]

Thank you so much. 

[YCLin]

I have tried the calculation of eigenstate for water molecular and I found that the quantum number 3 and 4 represent LUMO and HOMO, but not HOMO and LUMO in ATK 10.8.2, respectively. Why?
Is the quantum number 3, 4 indicated HOMO, LUMO in different molecular also, or it depends? How to choose the correct number?
The other question is, I am wondering if it is possible to calculate graphene pi-orbital isosurface at specific energy?
I have also calculated the Mulliken charge population, but could you kindly tell me how the extract the amount of charge transfer?

Thank you

[zh]

Of course, the quantum numbers for the HOMO and LUMO of a molecule are different to those of other one.  A very simple rule of thumb is that the quantum number for HOMO can be counted by the Pauling principle: each level can be occupied with 2 electrons.  Assuming the total electrons of a molecule is N, if N is odd, the quantum number for HOMO in ATK will be (N+1)/2; if N is even, the quantum number for HOMO in ATK will be N/2.  Remind that the quantum number for the first molecule level is defined as zero, i.e., the index for the molecule levels in ATK starts from zero.

Your first question may be wrong. The quantum number for HOMO   + 1 = The quantum number for LUMO.  

For the second question, your description is not clear. But one can obtain the isosurface plot of an eigenstate that from pi-orbital. First, do the projected density of states (DOS) calculations and find out the energy regions (or energy position) for the states with only from the pi-orbital.  And then calculate the eigenstate for such energy position (or molecule level).

For the third question,  the Mulliken charge population can give the number of  electrons for each atom.  For instance, if the electrons for a carbon atom given by the Mulliken charge population is 3.5, the charge transfer between this carbon atom and its nearest neighbors will be 0.5 e since an isolated carbon atom has 4 electrons, and its neighbors transfer 0.5 e to this carbon atom.

[YCLin]

Thanks for the answers.