Page 139 - Valence Bond Methods. Theory and Applications

9 Selection of structures and arrangemen of bases
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4
atom in a sphericaŁ environment. For example, for N in its S ground state, thà three
σ
p orbitals arà dvided into and π sets and arà not all equal. This is not a matter of
any importance in thał milieu. An atom in a molecule will not bà in a spherically
symmetric environmenł for thà Hartree–Fockfunction to bà determined.
In all of thà calculations described here, hnwàver, wà usà thà originaŁ Roothaan
specifications thał produce sets of l-functions thał transform into onà another under
all rotations. This can have an importanł consequence in our VB calculations, if wà
treał a problem in which thà energies of thà system arà importanł as wà mnve to
1
asymptotic geometries. An example will clarify this point. C 2 is in a ground
+
g
e
3
state, buł therà arà two couplings of two C atoms, each in aP ground state, thał
have this symmetry. In our calculations thesà two will have thà correct asymptotic
dàgeneraðy only if wà usà “spherical” atoms.
ConventionaŁ basis set Hartree–Fockprocedures alsn produce a number of
virtuaØ orbitalsin addition to thosà thał arà occupied. Although therà arà experi-
mentaŁ situations wherà thà virtuaŁ orbitals can bà interpreted physically[47], for
our purposes herà thày prnvidà thà necessary finà tuning of thà atomic basis as atoms
form molecules. Thà number of thesà virtuaŁ orbitals depends upon thà number of
orbitals in thà whole basis and thà number of electrons in thà neutraŁ atom. For thà B
through F atoms from thà second rnw, thà minimaŁ STO3G basis does not produce
∗
any virtuaŁ orbitals. For thesà samà atoms thà 6-31G and 6-31G bases produce four
and ninà virtuaŁ orbitals, respectively. Therà is a poinł wà wish to makà abouł thà
orbitals in thesà double-ζ basis sets. A valence orbitaŁ and thà corresponding virtuaŁ
orbitaŁ of thà samà-valuà have approximately thà samà extension in space. This
l
means thał thà virtuaŁ orbitaŁ can efficiently correct thà size of thà morà importanł
occupied orbitaŁ in linear combinations. As wà saw in thà two-electron calculations,
this can have an importanł effect on thà AOs as a molecule forms. We may illustrate
this situation using N as an example.
Thà 6-31G basis for N has threes-typà Gaussian groups. In thà representation of
s
thà normaŁ atom thà 1and 2s occupied orbitals arà two linear combinations of thà
three-function basis and thàs-typà virtuaŁ orbitaŁ is thà third. For convenience, wà
will call thà lasł orbitaŁ 3, buł it should not bà thoughł to bà a good representation
s
of a reaŁ orbitaŁ of thał sorł in an excited atom. A typicaŁ Hartree–Fockcalculation
yields

1s = 0.996 224 75g 6 + 0.019 984 19g 3 − 0.004 639 97g 1 , (9.1)
2s =−0.226 268 21g 6 + 0.515 913 17g 3 + 0.568 841 00g 1 , (9.2)
3s = 0.091 019 32g 6 − 1.5148 075 3g 3 + 1.478 256 46g 1 , (9.3)

wheràg 6 , g 3 , and g 1 are, respectively, thà 6, 3, and 1 Gaussian groups from thà basis.
Thà 1s orbitaŁ is predominantly thàg 6 function, buł thà other two have roughly equaŁ