Secular determinant definition

This is known as a secular determinant, where the only unknown is E. The remaining quantities in the determinant, Hy and Sy, may be evaluated readily according to their definitions given in eqs. (3.1.5) and (3.1.6) and using the... 

For formaldehyde (a) Work out the symmetry orbitals for a minimal-basis-set calculation give the symmetry species of each symmetry orbital. (Choose the x axis perpendicular to the molecular plane.) (b) How many <7 and how many v canonical MOs will result from a minimal-basis-set calculation (See the Section 15.10 discussion of ethylene for the definition of <7 and TT MOs.) How many occupied <7 and occupied ir MOs are there for the ground state (c) For each of the eight energy-localized MOs, state which AOs will make significant contributions, (d) What is the maximum-size secular determinant that occurs in finding the minimal-basis-set canonical MOs ... 

If the molecules has some symmetry the solution of the secular determinant can be simplified by the use of symmetry coordinates. Symmetry coordinates are linear combinations of the internal coordinates. After a little experience one can frequently select symmetry coordinates intuitively by taking combinations of symmetry related internal coordinates. The symmetry coordinates picked out are not necessarily unique but must have definite properties. They must be normal, orthogonal, and they must transform properly as will be described below. A symmetry coordinate has the form... 

The secular equation (31) is here of 13th order. Only the 12 terms Ko = Mh are different from zero the other ones vanish because of the forbidden combination Af = 2 and AZ = 0 respectively. The secular determinant has only one line 0 and one column 0 and on the diagonal only five different terms appear. With the definitions Vi = Vio, x = and ... 

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