Chemistry 6440 / 7440
Model Chemistries
Electronic Structure Theory
Schrodinger equation HY = EY
The many-electron CI expansion
Y ( R , r ) 
C


( R )  ( R , r )

  ( R , r )     Mi O ( R , r )
i
The one-electron LCAO expansion
AO
 MO
(
R
,
r
)

C
(
R
)

 ik
i
k (r )
k
Model Chart
Minimal
STO-3G
HF
MP2
MP3
MP4
QCISD(T)
...
Full CI
Split-Valence
3-21G
Basis
Polarized
6-31G*
6-311G*
Diffuse
6-311+G*
High Ang. Mom.
6-311+G(3df,p)
…

Schrödinger
Equation
Model Chemistries
• A theoretical model chemistry is a complete algorithm for
the calculation of the energy of any molecular system.
• It cannot involve subjective decisions in its application.
• It must be size consistent so that the energy of every
molecular species is uniquely defined.
• A simple model chemistry employs a single theoretical
method and basis set.
• A compound model chemistry combines several theoretical
methods and basis sets to achieve higher accuracy at lower
cost.
• A model chemistry is useful if for some class of molecules
it is the most accurate calculation we can afford to do.
Development of a
Model Chemistry
• Set targets
– accuracy goals
– cost/size goals
– validation data set
• Define and implement methods
– Specify level of theory for geometry optimization,
electronic energy, vibrational zero point energy
• Test model on validation data set
Evaluating Model Chemistries
(Page 147 in Exploring Chemistry)
Largest Errors
Model Chemistry
B3LYP/6-311+G(2d,p) // B3LYP/6-311+G(2d,p)
BLYP/6-311+G(2d,p) // BLYP/6-311+G(2d,p)
BLYP/6-31+G(d,p) // BLYP/6-31+G(d,p)
B3LYP/6-31+G(d,p) // B3LYP/6-31+G(d,p)
B3LYP/6-31G(d) // B3LYP/6-31G(d)
MP2/6-311+G(2d,p) // MP2/6-311+G(2d,p)
MP2/6-31+G(d,p) // MP2/6-31+G(d,p)
PM3 // PM3
SVWN5/6-311+G(2d,p) // SVWN5/6-311+G(2d,p)
AM1 // AM1
SVWN/6-311+G(2d,p) // SVWN/6-311+G(2d,p)
HF/6-31+G(d,p) // HF/6-31+G(d,p)
HF/6-31G(d) // HF/6-31G(d)
HF/3-21G(d) // HF/3-21G(d)
HF/STO-3G // HF/STO-3G
MAD
3.1
3.9
3.9
3.9
7.9
8.9
11.4
17.2
18.1
18.8
24.9
46.7
51.0
58.4
93.3
StdDev
3.0
3.2
3.2
4.2
9.5
7.8
8.1
14.0
19.8
16.9
19.2
40.6
41.2
50.1
66.3
Positive Negative
-19.7
13.6
-15.9
14.3
-15.2
15.2
-33.8
17.6
-54.2
12.2
-39.2
18.3
-44.0
15.6
-69.9
49.6
-10.1
81.0
-95.5
47.8
-10.4
89.3
-179.8
10.1
-184.2
11.5
-215.2
19.5
-313.9
101.3
Compound Model Chemistries:
G2 and G2(MP2)
• Proposed by J. Pople and co-workers
• Goal: Atomization energies to 2 kcal/mol
• Strategy: Approximate QCISD(T)/6311+G(3df,p) by assuming that basis set
and correlation corrections are additive
• Mean absolute error of 1.21 kcal/mol in 125
comparisons
REQUIRED ACCURACY FOR CHEMISTRY
Components of the calculated dissociation energy (mEh) for 02.
SCF
MP2
MP3,4 QCISD(T)
Core
ZPE
O2 3 g
-149,691.91 -496.12
6.92
-16.48
-118.16
4.22
2O 3Po
Do
-149,637.86 -337.96
-23.14
-6.77
-116.81
0.
-30.06
9.70
1.35
-4.22
CPU time
N2.4
54.05
158.17
N4
N6
N7
N5
For 1 millihartree accuracy:
•Need SCF energy to six figures
•MP2 Contribution to three figures
•Higher order correlation to two figures
•Core and ZPE contribution to one figure
*Octerski, Joseph W., G.A. Petersson, and J.A. Montgomery, “A complete basis set
model chemistry. V. Extensions to six or more heavy atoms”, J.Chem. Phys., 104,2598, (1996).
N3.4
Cost-Effective Models
• Small basis set, low-order geometry and ZPE
• Large SCF basis set
– Medium MP2 basis set
– Small MP3, MP4(SDQ), CCSD(T) basis sets
• Cancellation of errors in 3-body effects (CBS-4)
• CBS2 extrapolation of MP2 energy reduces the
need for large MP2 basis set
• Size consistent empirical corrections
CBS Extrapolation
The slow, N-1, convergence
of the correlation energy vs
the one-electron basis set
expansion is the result of
the universal cusp in wave
functions as interelectronic
distances, rij  0 .
Thus, we can reasonably
expect the N-1 form to also
be universal.
SCF
MP2
MP4(SDQ) MP4(SDTQ) QCISD(T)
FCI
6-31G
631G†
6-31+G†
6-31+G††
6-311G(d,p)
6-31+G(d(f),d,p)
6-311+G(d,p)
6-311G(2df,p)
6-311+G(2df,p)
6-311+G(3df,2p)
6-311+G(3d2f,2df,p)
6-311++G(3d2f,2df,2p)
[6s6p3d2f,4s2p1d]
CBS
Exact
RM S Error: G 2 test set (kcal/m ol)
3.0
2.5
C BS-4
C BS-q
2.0
G 2(MP2)
G2
1.5
C BS-Q B3
1.0
C BS-Q C I/APNO
0.5
0.0
0
5
10
15
20
25
M axim um N um ber of H eavy Atom s
30
Compound Model Chemistries:
Thermochemistry
The CBS-QCI/APNO model shows the best performance with this test
set of any general theoretical model proposed to date, with a mean
absolute deviation (MAD) from experiment of 0.53 kcal/mol. Of the
models defined for both the first-and second-row elements, the CBSQB3 model gives the greatest accuracy (MAD=0.87 kcal/mol), followed
by G2 theory (MAD=1.21 kcal/mol), G2(MP2) theory (MAD=1.59
kcal/mol), the CBS-q model (MAD=1.71 kcal/mol) and finally, the CBS4 model (MAD=1.98 kcal/mol). All are at or under the accuracy of ~2
kcal/mole required for meaningful thermochemical predictions.