Biomedical Engineering Reference
In-Depth Information
3. The geometry from step 2 is now used in a number of single point calculations start-
ing with MP4/6-311G**. This energy is improved in four distinct stages and these four
improvements are assumed to be additive.
4. As a first correction we add diffuse s and p basis functions at the MP4 level of theory.
These are known to be important for molecules with lone pairs. The correction is obtained
by comparing theMP4/6-311+G** (
−
78.38425
E
h
) and theMP4/6-311G** (
−
78.38239
E
h
)
energies. The correction is therefore
1.86
mE
h
.
5. As a second correction we take account of polarization functions on nonhydrogens.
This correction is found by comparing MP4/6-311G(2df) and MP4/6-311G** energies.
This turns out to be
−
40.07
mE
h
.
6. The third correction allows for the inadequacies of the MP4 treatment. We make an
expensive QCISD(T)/6-311G** calculation. This gives
−
1.75
mE
h
.
7. If we add these three corrections to the MP4/6-311G** energy, we should get an
approximation to a full calculation at the QCISD (T)/6-311G**(2df) level of theory. One
such large calculation has therefore been replaced by four smaller and cheaper ones. This
energy is still incomplete because of remaining deficiencies in the basis set. Amajor diffi-
culty is that the correlation energy between two spin-paired electrons in a typical single bond
converges only slowly as increasing angular momentum quantum numbers are included in
the basis set. G1 theory allows for these effects in an empirical manner and so the fourth
correction is to add terms devised to give agreement between theory and experiment for
the hydrogen atom and dihydrogen. These are taken to be 6.14
mE
h
for each valence elec-
tron pair and 0.19
mE
h
for each unpaired electron, and are referred to as the
higher level
correction
(HLC). In terms of the numbers of α- and β-spin electrons
Δε (HLC) /
mE
h
=−
−
0.19
n
α
−
5.95
n
β
(21.1)
which comes out as
36.84
mE
h
for ethene.
8. Finally, the harmonic frequencies obtained at the HF/6-31G* level of theory are
scaled uniformly by the well-accepted factor of 0.8929 to give a zero-point correction
of
−
48.90
mE
h
.
The total energy G1 (at 0 K) is therefore
+
−
78.414021
E
h
and similar calculations for
H and C give
37.78464
E
h
. Combining these three values gives an ethene
atomization energy of 530.1 kcal mol
−
1
which compares well with the zero-kelvin JANAF
thermodynamic value of 531.9 kcal mol
−
1
.
Final output contains extra useful thermodynamic data and is shown below. A simple
restart gives rapid results for different temperatures and pressures.
Temperature= 298.150000
−
0.5
E
h
and
−
Pressure=
1.000000
E(ZPE)=
0.048894
E(Thermal)=
0.051951
E(QCISD(T))= -78.384142
E(Empiric)=
-0.036840
DE(Plus)=
-0.001864 DE(2DF)=
-0.040069
G1(0 K)=
-78.414021
G1 Energy=
-78.410964
G1 Enthalpy=
-78.410020
G1 Free Energy= -78.435526
21.2 G2 Theory
G1 theory was originally tested against experimental values for a range of simple first-
and second-row molecules. It was observed that G1 theory did badly with ionic molecules,
