Rearranging summing points

The coefficients a must be so chosen that 2 0,- transforms in the fashion appropriate to the irreducible representation r. Now the important point is that bases for only certain irreducible representations can be constructed out of linear combinations of the vL- To determine which, one ascertains the group characters associated with the transformation scheme, usually reducible, of the original attached wave functions rpi before linear combinations are taken. This step is easy, as the character xd for a covering operation D is simply equal to q, where q is the number of atoms left invariant by D. This result is true inasmuch as D leaves q of the atoms alone, and completely rearranges the others, so that the diagonal sum involved in the character will contain unity q times, and will have zeros for the other entries. The scheme for evaluating the characters is reminiscent of that in the group... 

Equation A7-40 is a differential equation for each MO But as it stands it is not an eigenvalue equation because, instead of regenerating i, we obtain a sum of functions y times the various unknown constants e,y. However, there remains a degree of freedom in the problem that can be used to throw Eq. (A7-40) into eigenvalue form. It is pointed out in Appendix 2 that the value of a determinant is unchanged if any row or column, multiplied by a constant, is added to any other row or column. This means that a Slater determinant of best MOs is unaffected by such internal rearrangements. In other words, if we were to solve Eq. (A7 ) for a set of best MOs, we could form various new orthonormal MOs, (e.g., (j + ) — by mixing them... 

Two premises underlie the study of environmental fate and transport. The first premise is the conservation of mass. A chemical may move from point A in China to point B in California on prevailing westerly winds, and/or its constituent molecules may undergo rearrangement such that one or more chemically distinct compounds result, but mass itself—the sum total of the constituent atoms that comprise the original chemical—is neither created nor destroyed. 

---