Calculating Molecular Properties
from molecular orbital calculations
Geometric Properties
 Bond length
 Bond angle
 Dihedral angle
A single lowest energy
equilibrium structure is
generally the result of a
geometry optimization;
actual molecules exist as
an ensemble (mixture) of
conformations which is
temperature dependent.
Experimental measurements of geometry (X-ray, ED,
NMR, ND) measure different aspects of structure.
Molecular Properties
 Many are first, second or third derivatives
of the Hartree-Fock energy (E) with respect
one or more of the following:




external electric field (F)
nuclear magnetic moment (nuclear spin, I)
external magnetic field (B)
change in geometry (R)
Examples…derivatives w/r to:
 external electric field (F):

Raman intensity
d3E/dRdF2
 nuclear magnetic moment (nuclear spin, I)


ESR hyperfine splitting (g)
dE/dI
NMR coupling constant (Jab)
d2E/dIadIb
Examples...
 external magnetic field (B) and (nuclear spin, I)

NMR shielding (s)
d2E/dBdI
 Change in geometry (R)


Energy Gradient
dE/dR
Hessian (force constant; IR vibrational frequencies)
d2E/dR2
Other Properties
 Ionization energy (IP)

Neg. of HOMO energy (Koopmans’ theorem)
Errors due to relaxation and electron correlation CANCEL
 Electron affinity (EA)

LUMO energy
Errors due to relaxation and electron correlation ADD
 UV-Vis spectra

Est. (poorly) by HOMO-LUMO energy difference
UV-Vis Spectra




Can be estimated as the HOMO-LUMO energy
difference
Generally not very accurate because orbital relaxation
and electron correlation effects are ignored, but useful
for relative wavelengths, and to predict trends
Difficult to model effects of solvent, especially on
excited states, about which little is known.
Density functional theory (to be discussed later)
generally does a better job at predicting UV-Vis spectra.
Problems with UV-Vis spectra
 The energy required to promote an electron from
MO i to MO j is not simply equal to the energy
difference e(j) - e(i). The promotion energy E(i-->j)
can be expressed as:
E(i-->j) = e(j) - e(i) - v(i,j)
 The wavefunction |i-->j| of an excited electronic
configuration is not a good approximation to an
eigenfunction of the many-electronic Hamilton
operator H. Excited configurations tend to interact,
and a proper description must include Configuration
Interaction (CI) to account for electron correlation.
Other Properties...
 IR spectra (bond vibrational frequencies)

frequencies are over-estimated by H-F theory; a
scaling factor of 0.89-0.91 must be applied to
reproduce observed values
 Proton affinity (related to basicity, but is
calculated in the gas phase rather than in
aqueous solution)
RNH2 + H
RNH3
Other Properties...
 Acidity
RCO2H
RCO2
+
H
 Gibbs Free energy (G)


Includes Enthalpy (H) and Entropy (S)
A frequency calculation must be performed
on an energy minimized structure to obtain
thermal corrections, which allow calculation of
entropy and other values. (later)
Other Properties...
 Charges on Atoms in Molecules



(topic of a
later lecture)
meaning of charge is ill-defined
value depends on definition
several commonly used charge estimations
•
•
•
•
•
Mulliken
Natural population analysis
Charges fit to electrostatic potential
Atoms in molecules (AIM)
ChelpG
NMR chemical shift calculations
CH3CH2CH2CH3
C1
C2
CH3CH=CHCH3
C1
C2
benzene (C6H6)
(in ppm)
calc.
expt.*
15.9
13.4
23.7
25.2
18.4
124.7
17.6
126.0
128.9
130.9
* in CDCl3 solution
NMR: Effect of Basis Set
Calculated 13C chemical shifts (ppm) of benzene and
difference from gas phase experimental values as a
function of basis set size
HF/3-21G
HF/6-31G
HF/6-31G(d)
HF/6-31G(d,p)
HF/6-31++G(d,p)
(observed)
Shift
119.6
125.8
127.3
128.4
128.9
130.9
Diff.
11.3
5.1
3.6
2.5
2.0
--
IR Frequency Calculations
O
H
C
Formaldehyde
H
C-H bend
C=O stretch
Computed Frequency
Relative intensity
1336 cm-1
0.4
2028 cm-1
150.2
Freq. scaled by 0.89
observed
1189 cm-1
1180 cm-1
1805 cm-1
1746 cm-1
IR Frequencies
-1
(cm ,
gas phase)
Scaled Frequency
Expt.
O
H
C
H
1805
1746
1799
1737
1850
1822
1797
1761
O
CH3
C
CH3
O
CH3
C
Cl
O
CH3
C
OCH3
Zero-point energy
Energy possessed by
molecules because v0,
the lowest occupied
Energy
vibrational state, is
above the electronic
energy level of the
Usual calc’d energy
equilibrium structure.
r0, r1, r2...
v3
v2
v1
v0
zero point
energy
eq. bond length
Distance between atoms
Thermal Energy Corrections
 The following may be derived from the results
of a frequency calculation:
 Zero Point Energy (z.p.e.)
 Free Energy at STP (Gº)
 Free Energy at another Temperature, Pressure
 Entropy (S)
 Enthalpy (H) corrected for thermal contributions
 Constant-volume heat capacity (Cv)
Frequency calculation

Formaldehyde was optimized and a
frequency calculation performed in
Gaussian 98 at NCSC.
Zero-point correction (all in Hartrees/Particle) =
Thermal correction to Energy=
Thermal correction to Enthalpy=
Thermal correction to Gibbs Free Energy=
Sum of electronic and zero-point Energies=
Sum of electronic and thermal Energies=
Sum of electronic and thermal Enthalpies=
Sum of electronic and thermal Free Energies=
0.028987
0.031841
0.032785
0.007373
-113.840756
-113.837902
-113.836958
-113.862370
Frequency calculation...
Heat capacity
TOTAL
ELECTRONIC
TRANSLATIONAL
ROTATIONAL
VIBRATIONAL
E (Thermal)
Kcal/mol
19.980
0.000
0.889
0.889
18.203
CV
cal/mol-Kelvin
6.260
0.000
2.981
2.981
0.298
Entropy
S
cal/mol-Kelvin
53.483
0.000
36.130
17.303
0.050
Gº = H º - TS º
-113.862370 = -113.836958 - 298.15 * 53.483 / 627.5095 * 1000
(in Hartrees)
(kcal/mol per Hartree)
Dipole Moment
(in Debyes)
MMFF
AM1
PM3
HF /
HF /
MP2 /
3-21G* 6-311+G** “
NH3
2.04
1.85
1.55
1.75
1.68
1.65
1.47
H2O
2.46
1.86
1.74
2.39
2.12
2.08
1.85
P(CH3)3
2.06
1.52
1.08
1.28
1.44
1.31
1.19
thiophene
1.32
0.34
0.67
0.76
0.80
0.47
0.55
Expt.
(note that none are very accurate; this reflects two factors:
equilibrium geometry is only one of several, even many, in
an ensemble of conformations, and charges are ill-defined.
Conformational Energy Difference
(in kcal/mol)
Good:
Generally poor:
Sybyl
MMFF
AM1
HF /
HF /
MP2 /
PM3 3-21G* 6-311+G** “
Expt.
acetone
(trans/gauche) 0.6
0.8
0.7
0.5
0.8
1.0
0.6
0.8
N-Me formamide
(trans/cis)
-1.8
2.6
0.4
-0.5
3.9
2.7
2.7
2.5
1,2-diF ethane
(gauche/anti) 0.0
-0.6
-0.5
1.4
-0.9
0.3
0.8
0.8
1,2-diCl ethane
(anti/gauche) 0.0
1.2
0.8
0.6
1.9
1.9
1.3
1.2
Equilibrium Bond Length
(in Å)
Sybyl
propane
(C-C single)
MMFF
AM1
PM3
HF /
HF /
MP2 /
3-21G* 6-311+G** “
Expt.
1.551 1.520 1.501 1.512 1.541 1.525 1.526 1.526
propene
(C=C double) 1.334 1.334 1.331 1.328 1.316 1.316 1.336 1.318
1,3-butadiene
(C=C double) 1.338 1.338 1.335 1.331 1.320 1.320 1.342 1.345
propyne
(CC triple)
1.204 1.201 1.197 1.192 1.188 1.181 1.214 1.206
Log P
Log of the octanol/water partition coefficient; considered
a measure of the bioavailability of a substance
Log P = Log K (o/w) = Log [X]octanol/[X]water



most programs a use group additivity approach
(discussed later, with QSAR)
some use more complicated algorithms, including
the dipole moment, molecular size and shape
subject to same limitations as dipole moment
Conclusions
 Many useful molecular properties can be
calculated with reasonably good accuracy,
especially if methods including electron
correlation and large basis sets are used.
 Some properties (charges on atoms, dipole
moments, UV-Vis spectra) are not well modeled,
even by high level calculations.
 Some of the errors are because of problems
defining the property (e.g., charge); others are
because of limitations of the method (orbital
relaxation and electron correlation).