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Prediction of molecular properties (I)
The general question of rational drug design
Wow is the biological space (activity) of a compound
connected to the chemical space (structure) ?
Is it possible to make predictions based on the
molecular structure ?
 QSAR and QSPR
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Prediction of molecular properties (II)
observables
What are molecular properties?
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molecular weight MW (from the sum formula C12H11N3O2)
melting point
boiling point
vapour pressure
solubility (in water)
charge
dipole moment
Directly computable
from the electronic
polarizability
wave function of a
ionization potential
molecule
electrostatic potential
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Prediction of molecular properties (III)
All molecular properties that can be measured by physicochemical methods (so called observables) can also be
comouted directly by quantum chemical methods.
Required: A mathematical description of the electron
distribution e.g. by the electronic wave function of the
molecule
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Electron distribution
Quantum mechanics (QM)
Atomic coordinates
Molecular mechanics (MM)
force fields
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Quantum mechanics (I)
To make the mathematical formalism practically
useable, a number of approximations are necessary.
One of the most fundamental consists in separating
the movement of the atomic cores from that of the
electrons, the so called
Born-Oppenheimer Approximations:
Atomic cores are > 1000 times heavier than the electrons
und thus follow the movements of the faster electrons.
The (electrostatic) interaction between charged particles
(electrons, cores) is expressed by Coulomb‘s law
qi
Vij 
rij
qj
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qi  q j
rij
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Quantum mechanics (II)
As electrons are particles, their movement can be
described by classical mechanics according to
Newtons 2nd law:
F  ma
dV
 2r

m
dr
t
As electrons are also very small particles (quants), they
exhibit properties of particles as well as those of waves:
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particle
wave
galvanic
precipitation
diffraction
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Schrödinger equation (I)
Electrons can be described in the form of a wave
function by the time-dependent Schröder equation

H   i
t
If the Hamilton operator H is time-independent, the timedependence of the wave function can be separated as a
phase factor, which leads to the time-independent
Schrödinger equation. Here, only the dependence from
the coordinates remains.
 r, t    r  e  i E t / 
H  r   E  r 
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The wave function (I)
The wave function is a mathematical expression describing
the spacial arrangement of the (fluctuating) electrons.
The squared wave function holds the propability P to find the
particle (electron) at a given place in space.
P  
P is a so-called observable, whereas the wave function 
itself is no observable, physical quantity.
Thus, integration over the complete space  must yield 1
(= total propability to find it somewhere in space).
    d  1
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The Hamilton operator
The Hamilton operator contains the kinetic (T) and the
potentiell (V) energy of all considered particles i in the system
2 2
T   Ti  
i
i 1
i 1 2mi
N
H TV
N
with the squared Nabla operator
2
2
2





2
i   2  2  2 
  xi  yi  zi 
N
N
V   Vij
i 1 j i
with
Vij 
qi  q j
rij
As a consequence of the Born-Oppenheimer
approximation, also the Hamilton operator can be
sepatared into a core and an electronic part.
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The wave function (II)
Any mathematical expression of the wave function must fulfill
certain criteria to account for the physical nature of the
electrons.
As a simplification the wave function of all electrons in a
molecule is assumed to be the product of one-electron
functions which themselves describe a single electron.
   1  2  ...  N
These function must fulfill some criteria:
• electrons are indistinguisable
• The repell each other
• The Pauli principle (two electrons with different spin can
share a common state (orbital))
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Schrödinger equation (II)
According to the Schrödinger equation there must be several
different energetic levels for the electrons within an atom or
molecule. These (orbital) energies can be obtained by
integration and rearrangement to
2

H

d


E


 d
 H d

E
 d
2
The resulting energies are, however, dependend on the
quality of the applied wave function and thus always
higher or, in the best case, equal to the actual energy.
In the simplest case we chose 1s orbitals as basis set to
describe the wave function
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Molecular Orbital Theory (I)
Molecular orbitals can be constructed as a linear combination
of atomic orbitals (LCAO approach) or other basis functions.
  cA A  cB B  ...  c N N
e.g. for H2
  cA1sA  cB1sB
1sA
1sB
K
Common expression for a MO
with the atomic orbitals 
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 i   c i 
 1
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Molecuar Orbital Theory (II)
Applying the LCAO approach for the wave function we yield
for H2
2

 d   cA A  cB B cA A  cB B  d


  c 2A A2  c 2B B2  2c A c B A B d
 c 2A  A2 d  c 2B  B2 d  2c A c B  A Bd



=1
=1
overlap intergral S
Due to the normalization of the wave function regarding the
complete space:
2
2
2

d


c

c
A
B  2c A c BS

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Molecular Orbital Theory (III)
Common notation of the Sekular equations using matrices:
H  ES  0
The solutions of these Sekular equations for E yield the
energies of the bonding and anti-bonding MOs
E
0
The main numerical effort consists in the iterative
search for suitable coefficients (cA, cB, ...) that
produces reasonable orbital energies
 variational principle
 Hartree-Fock equations
 Self Consistent Field (SCF) method
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Hückel Theory (I)
(1931) Limited to planar,
-orbitale are neglected.
conjugated
-systems,
The original aim was to interpret the non-additive properties
of aromatic compounds (e.g. benzene compared to
“cyclohexatrien”) regarding their heats of combustion.
The -orbitals are obtained as linear combinations of
atomic orbitals (LCAO of pz-orbitals). The -electrons
move in an electric field produced by the -electrons and
the atomic cores.
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Hückel Theory (II)
Example ethene H2C=CH2
E

pz
pz

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Hückel Theory (III)
Within the Hückel theory the Fock matrix contains as many columns,
respectively rows, as atoms are present in the molecule. All diagonal
elements correspond to an atom i and are set to the value . Offdiagonal elements are only non-zero if there is a bond between the
atoms i and j. This resonance parameter is set to  (<0). Values for 
can be obtained experimentally from UV/VIS-spectra (  -4.62 eV).
Example butadiene:
1 2 3 4
2
1
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3
1    0 0 
2    0 
3  0    
4  0 0   
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Hückel Theory (IV)
For a cyclic -system as in benzene, the orbital energies
and orbital coefficients results to
 2k 
 ; with k  0,1,..., N  1
 N 
 2  1k 
1

ci 
exp 

N
N


 i    2 cos 
E


0


This also yields the Hückel rule, where a system
of [4n+2] -electrons is aromatic.
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Hückel Theory (V)
• Application of the Hückel method to predict and interpret
UV/VIS spectra
• Different parameters  for different atoms (C,N,O) allow
the application of the Hückel theory to further compounds
• Orbital energies can be determined experimentally by
photo electron spectroscopy (PES) and thus also  (the
respective ionization potential) and 
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Hartree-Fock based methods
H = E
Born-Oppenheimer approximation
one-determinant approach
ZDO-approximation
valence electrons
parameters
Hartree-Fock-equations
optimized basis sets
RHF
semiempirical methods
with minimal basis set
all electron
ab initio methods with
limited basis set
multi-determinant approaches
UHF
Valence electrons
ECP
spin (,)
space
semiempirical C.I. methods
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CI
MCSCF
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CASSCF
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Semiempirical methods (I)
The problem of ab initio calculation is their N4 dependence from
the number of two-electron integrals. These arise from the
number of basis functions and the interactions between electrons
on different atoms.
In semiempirical methods the numerical effort is strongly reduced
by assumptions and approaches:
1. Only valence electrons are considered, the other electrons and
the core charge are described by an effective potential for each
atom (frozen core).
2. Only a minimal basis set is used (one s and three p-orbitals per
atom), but using precise STOs that are orthogonal to each other.
3. More or less stringent use of the Zero Differential Overlap
(ZDO) approach.
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Semiempirical methods (II)
Since 1965 a series of semiempirical methods has been
presented from which still some are in use today for the simulation
of electromagnetic spectra: CNDO/S, INDO/S, ZINDO
Following methods have shown to be particularly successful in
predicting molecular properties:
MNDO (Modified Neglect of Diatomic Overlap) Thiel et al. 1975,
AM1 (Austin Model 1) Dewar et al. 1985 und
PM3 (Parameterized Method 3) J.P.P. Stewart 1989
This partly also due to their availability of the wide spread
MOPAC program package and its later commerical sucessors.
All three method are based on the same NDDO approach and
differ in the respective parameterization of the respective
elements.
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AM1 (Austin Model 1)
Dewar, Stewart et al. J.Am.Chem.Soc. 107 (1985) 3902
Advantages compared to MNDO:
+ better molecular geometries esp. for hypervalent
elements (P, S)
+ H-bonds (but with a tendency towards forking)
+ activation energies for chemical reactions
Deficiencies of AM1 (and all other methods based on NDDO):
- hypervalent elements in general, because no d-orbitals
- compounds with lone electron pairs (esp. anomeric effect)
- NO2 containing compounds
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PM3 (Parameterized Method 3)
J.J.P. Stewart J.Comput.Chem. 10 (1989) 209
Parametrization was performed more rigerously using errror
minimization than in previous methods.
Advantages compared to AM1:
+ better molecular geometries for C, H, P and S
+ NO2 containing compounds better
Disadvantages compared to AM1:
- All other nitrogen containing compounds worse
- higher atomic charges lead to a more polar character of the
molecules
- Not all parameterized elements (e.g. Mg and Al) yield
reliable results for all substance classes
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Molecular properties from
semiempirical QM calculations (I)
In contrast to ab initio calculations the semiempirical methods
MNDO, AM1, and PM3 were calibrated to reproduce
experimental data:
• heats of formation [Bildungswärmen]
• molecular geomtries (bond lengths, bond angles)
• dipole moments
• ionization potentials
The results of semiempirical methods regarding these
properties are therefore often better than that of ab initio
calculations at low level (with comparable computational
effort)
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Heats of formation
Computation of heats of formation at 25° C
H f o (molecule )  E elec (molecule ) 
atoms
E
elec
(atoms) 
atoms
 H
fo
(atoms)
atomization Heats of formation
energies
of the elements
Experimentally known
Only the electronic energy has to be computed
O
H
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O
atomization
H
H
Eelec(molecule)
H
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O
H
H
25
Comparison of the methods
Calculated heats of formation at 25° C for different compounds
Average mean error (in kcal/mol)
Number of compounds
method
(C, H, N, O, and)
MNDO
AM1
PM3
Al (29)
22.1
10.5
16.4
4.9
Si (84)
12.0
8.5
6.0
6.3
P (43)
38.7
14.5
17.1
7.6
S (99)
48.4
10.3
7.5
5.6
Cl (85)
39.4
29.1
10.4
3.9
Br (51)
16.2
15.2
8.1
3.4
I (42)
25.4
21.7
13.4
4.0
Zn (18)
21.0
16.9
14.7
4.9
Hg (37)
13.7
9.0
7.7
2.2
Mg (48)
9.3
15.4
12.0
9.3
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MNDO/d
26
New semiempirical methods since 1995
MNDO/d
Thiel & Voityuk J.Phys.Chem. 100 (1996) 616
Expands the MNDO methods by d-obitals and is “compatible” with
the other MNDO parameterized elements
PM3(tm), PM5
d-orbitals for transition elements (transition metals)
SAM1 Semi ab initio Method 1
Certain integrals are thouroghly computed, therefore also
applicable to transition metals (esp. Cu and Fe)
AM1*
Winget, Horn et al. J.Mol.Model. 9 (2003) 408.
d-orbitals for elements from the 3rd row on (P,S, Cl)
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Electronic molecular properties (I)
Besides the structure of molecules all other electronic properties
can be calculated. Many of those results as response of the
molecule to an external disturbance:
Removal of one electron  ionization potential
In general a disturbance by an electric field can be expressed in
the form of a Taylor expansion. In the case of an external
electrical field F the induced dipole moment ind is obtained as:
ind   o   F  12  F 2  ...
o
permanent dipol moment of the molecule (if present)

polarizability

(first) hyperpolarizability
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Electronic molecular properties (II)
Selection of properties that can be computed from the n-th derivative
of the energy according to external fields
electr. magn. nuc.spin coord.
property
0
1
0
0
0
2
3
0
1
1
0
0
energy
electric dipol moment
magentic dipol moment
hyperfine coupling constant (EPR)
energy gradient (geom.optimization)
electric polarizability
(first) hyperpolarizability
harmonic vibration (IR)
IR absorption intensities
circular dichroisms (CD)
nuclear spin coupling const. (NMR)
nuclear magnetic shielding (NMR)
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0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
1
0
0
0
0
0
0
2
1
0
0
0
0
1
0
0
2
1
0
0
0
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Molecular electrostatic potential (I)
Due to the atomic cores Z and the electrons i of a molecule a
spacial charge distribution arises. At any point r the arising
potential V(r) can be determined to:
VESP r  
cores

A
 ri 
2
ZA

dri
r  RA
r  ri
While the core part contains the charges of the atomic cores only,
the wave function has to be used for the electronic part.
Remember: In force fields atomic charges (placed on the atoms)
are used to reproduce the electric multipoles and the charge
distribution.
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Molecular electrostatic potential (II)
To determine the MEP at a point r the integration is
practically replaced by a summation of sufficiently
small volume elements.
For visualization the MEP is projected
e.g. onto the van der Waals surface.
Other possibilities are the representation of surfaces
with the same potential (isocontour)
From: A. Leach,
Molecular
Modelling,
2nd ed.
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Molecular electrostatic potential (III)
Knowledge of these surface charges enables computation of
atomic charges (e.g. for use in force fields)
 ESP derived atomic charges
These atomic charges must in turn reproduce the electric
multipoles (dipole, quadrupole,...).
Therefore the fitting procedures work iteratively.
literature:
Cox & Williams J.Comput.Chem. 2 (1981) 304
Bieneman & Wiberg J.Comput.Chem. 11 (1990) 361
CHELPG approach
Singh & Kollman J.Comput.Chem. 5 (1984) 129
RESP approach  atomic charges for the AMBER force field
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Quantum mechanical descriptors (selection)
atomic charges (partial atomic charges) No observables !
Mulliken population analysis
electrostatic potential (ESP) derived charges
E
dipole moment
LUMO
polarizability
HOMO
HOMO / LUMO
of the frontier orbitals
WienerJenergies
(Pfad Nummer)
given in eV
Donor
Akzeptor
covalent hydrogen bond acidity/basicity
difference of the HOMO/LUMO energies compared
to those of water
Lit: M. Karelson et al. Chem.Rev. 96 (1996) 1027
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Molecular properties from
semiempirical methods (II)
Which method for which purpose ?
strutural properties only (molecular geometries):
PM3
esp. for NO2 compounds
electronic properties:
MNDO for halogen containing compounds (F, Cl, Br, I)
AM1
for hypervalent elements (P,S), H-bonds
Do not mix descriptors computed from different
semiempirical methods !
E.g. PM3 for NO2 containing molecules and AM1 for
the remaining compounds in the set.
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Prediction of Molecular Properties,
Examples
Descriptors from semiempirical methods
(ionization potential, dipole moment ...)
along commonly used variables in QSAR equations
and classification schemes.
Often much more qualitative experimental data than
quantitative date are available.
• in vitro mutagenicity of MX compounds
• Blood-brain distribution (logBB)
• CNS permeability of substances
• QT-interval prolongation (hERG channel blockers)
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Quantum QSAR
Generation of molecular properties as descriptors for QSARequations from quantum mechanical data.
Example: mutagenicity of MX compounds
Cl2CH
Cl
R2
H
R1
H
O
ln TA100 [ revertants/nmol ]
O
HO
10
O
R3
O
8
6
O
O
4
2
0
-2
ln(TA100) = -13.57 E(LUMO) –12.98 ; r = 0.82
-4
-6
-8
-0.3
-0.5
-0.7
-0.9
-1.1
-1.3
-1.5
E(LUMO) AM1 calculation [ eV ]
Lit.: K. Tuppurainen et al. Mutat. Res. 247 (1991) 97.
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BBB-model with 12 descriptors
Descriptors mainly from QM calculations: electrostatic
surface, principal components of the geometry,
H-bond properties
2.5
r2=0.866, adj. r2=0.844, se=0.308, n=90
predicted logBB
1.5
0.5
-0.5
-1.5
-2.5
-2.5
-1.5
-0.5
0.5
1.5
2.5
observed logBB
Lit: M. Hutter J.Comput.-Aided.Mol.Des. 17 (2003) 415.
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CNS Permeability
95%
CNS–
91%
CNS+
82%
hlsurf
99%
72%
vxbal
99%
qsum+
qsumo
ar5
96%
96%
qsum+
qsum+
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89%
100%
100%
100%
100%
88% qsum–
100%
pcgc
100%
qsum–
mpolar
79%
100%
89%
77%
cooh
83% dipdens
89% qsum–
80%
hbdon
size & shape
99% qsum+
86%
dipm
electrostatic
H-bonds
mde34
100%
100%
100%
100%
100%
100%
100%
kap3a
92%
mde13
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Lit.: C.Andres & M.Hutter QSAR Comb.Sci. 25 (2006) 305.
100%
94%
38
Decision tree for QT-prolonging drugs
89%
size & shape
88%
electrostatic
75%
73%
H-bonds
hacsurf
86%
71%
99%
QT+
QT–
t1e
88%
Level of accuracy in %
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MR
mde23
100%
100%
100%
logP
100%
100%
qsumn
100%
100%
MR
100% 92%
83% MR
100%
t2e
99%
93%
MR
100%
96% mpolar
sgeca
95%
87% dipdens
82%
89%
96%
89%
SMARTS
90%
100%
chbba
hlsurf
hy
mghbd
logP
MR
Modern Methods in Drug Discovery WS06/07
96%
qsumn
100%
100%
100%
100%
39
Common structural features
of QT-prolonging drugs
F
NH2
Cl
F
O
H
N
Cl
N
O
HO
O
N
N
H
OH
N
O
N
O
N
N
N
HO
N
F
O
N
N
O
N
N
O
Astemizole
H
Sertindole
H
F
Terfenadine
Cisapride
Grepafloxacin
Derived common substructure expressed as SMARTS string
Lit.: M.Gepp & M.Hutter Bioorg.Med.Chem. 14 (2006) 5325.
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Molecular properties from force fields
As as principal consequence force fields show an even
more emphasized dependence from the underlying
parameterization.
Thus only predictions regarding structure ( docking),
dynamics ( molecular dynamics) and, rather limited,
about spectra (vibrational Infra Red) can be made.
Due to the low computational effort, force fields are well
suited to allow conformational searches.
 4D-QSAR (different docked conformations,
e.g. in cytochrome P450)
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Molecular properties from molecular
dynamics simulations
Binding affinities (actually free energies of binding)
G for ligand binding to enzymes from free energy
perturbation calculations
Advantage: quite precise predictions
Disadvantage: computationally very demanding, thus
only feasible for a small number of ligands
Lit.: A.R. Leach Molecular Modelling, Longman.
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