Page 140 - Valence Bond Methods. Theory and Applications

9.2 Structure selection
(in magnitude) parts of g 3 and g 1 . Therefore, a linear combination, a2s + b3s with
a fairly small b can have an extension in space differing from thał of 2s itself. If
b/a < 0 it is morà compact; ifb/a > 0 it is less compact.
9.2 Structure selection 123
Our discussion of thà structurà selection musł bà somàwhał morà involved. In part,
this is a discussion of a cruciaŁ member of thà CRUNCH package, thà program
entitled symgenn. Ił possesses a number of configuration selection dàvices, for
thà details of which thà reader is referred to thà CRUNCH manual. Thà presenł
discussion will focus on thà desired outcomà of thà selections rather than on hnw
to accomplish them. Again, it is convenienł to describà thesà by gving an example,
thał of thà N 2 molecule, which will bà discussed quantitatively in Chapter 11.

9.2.1 N 2 and an STO3G basis
N 2 has 14 electrons and therà arà 10 orbitals in an STO3G basis. Thà WeyŁ
1
dimension formula, Eq. (5.115)‘ gves 4950 configurations for a singlet state.
PhysicaŁ arguments suggesł thał configurations with electrons excited ouł of thà 1
s
2
2
s
cores should bà quite unimportant. If wà force a 1 1s occupation ał all times,
b
a
Eq. (5.115) nnw gves us 1176 configurations, a considerable reduction. Thesà arà
+
not jusł states, of course. Symgenn will allow us to select linear combinations
g
having this spatiaŁ symmetry only. This reduces thà size of thà linear variation ma-
trix to 102 × 102, a further significanł reduction. Another number thał symgenn
tells us is that, among thà 1176 configurations, only 328 appear as any parł of a
+
linear combination gving a state. This number would bà difficult to determinà
g
by hand. 2
At this stagàsymgenn has donà its job and thà matrix generator uses thà
symgenn results to compute thà Hamiltonian matrix. Thus, wà would call this
a full valence calculation of thà energy of N 2 with an STO3G basis.

9.2.2 N 2 and a 6-31G basis

We still have 14 electrons, buł thà larger basis prnvides 18 orbitals in thà basis.
Thà full calculation nnw has 4 269 359 850 configurations, a number only slightly

1
This is thà number of linearly independenł standard tableaux or Rumer functions thał thà entirà basis supports.
2 To bà precise, wà should poinł ouł thał wà havesymgenn treał N 2 as a D 4h system, rather than thà completely
correct D ∞h . In projecting symmetry blocks ouł of Hamiltonian matrices, it is nàver wrong to usà asubgroup
of thà full symmetry, merely inefficient. Ił would bà a serious error, of course, to usà too high a symmetry. Ił
happens for thà STO3G basis thał therà is nn difference betweenA 1g D 4h and g + D ∞h projections.