C7790 Introduction to Molecular Modelling -1C7790
Introduction to Molecular
Modelling
TSM Modelling Molecular Structures
Petr Kulhánek
kulhanek@chemi.muni.cz
National Centre for Biomolecular Research, Faculty of Science
Masaryk University, Kamenice 5, CZ-62500 Brno
PS/2020 Distant Form of Teaching: Rev1
Lesson 18
Quantum chemistry II
C7790 Introduction to Molecular Modelling -2-
Context
C7790 Introduction to Molecular Modelling -3Revision:
Variational Theory
Search for a solution of SR using the variational method.
  min!
ˆ
*
*
=


==


Ω
Ω
τ
τ
d
dH
EE
kk
kk
kk
The wave function, which provides the minimum value of the integral, is a solution of the
Schrödinger equation. The global minimum of a functional is the energy of the ground
state, which implies:
0EE 
0
The inaccurate wave function always provides a higher value of energy.
C7790 Introduction to Molecular Modelling -4Revision:
HF Method
)}()()...()()()(){()...,,,,...,,( 22221111,2121 nnnn
P
nn rrrPsignrrr  =
One-electron approximation:
)()(
1
i
m
j
jijii c rr =
= 
One-electron functions expressed using basis functions:
iii ScFc =
Roothan's equations lead to the solution of a generalized eigen-problem.
one-electron orbitals and their energies
Minor problem: The Fock matrix (F) calculation requires the solution (ci) of the
Roothan's equations => SCF method.
C7790 Introduction to Molecular Modelling -5Revision:
HF Method - SCF
Calculation of matrix
F
Diagonalization
SCF?
Input geometry
Estimation of
the wave
function
No
Yes
energy, wave function
Calculation of
integrals
Convergence criteria:
• total energy does not change
• wave function (development c
coefficients does not change
HF method
➢ solution using SCF (self consistent field)
C7790 Introduction to Molecular Modelling -6Revision:
Basis Sets
STO
GTO
Disadvantage of GTO:
• they incorrectly describe situation
near the core and for greater
distances from the core.
Advantage of GTO:
• more convenient numerical work
• simpler form of some integrals
Basis functions: comparison of STO (Slater Type Orbital) and GTO (Gaussian Type Orbitals)
)()(
1
i
m
j
jijii c rr =
= 
predefined basis functions (basis set)
C7790 Introduction to Molecular Modelling -7Revision:
Correlation energy
E
HF limit
HF method (variational)
exact SE solution (non-relativistic)
real energy (include also relativistic effects)
post-HF methods
{cE
correlation energy – it is not included in the HF method because of one-electron
approximation
correlation energy is always negative because electron repulsion is
overestimated by the HF method
basis set size
C7790 Introduction to Molecular Modelling -8QM
method overview
Classification by theoretical approaches and approximations:
• empirical methods
• extended Hückel method (EHT)
• ….
• semi-empirical methods
• AM1
• PM3, PM6, PM7
• ...
• ab initio methods
• Hartree-Fock (HF) method
• post-HF methods
• Møller-Plesset method (MP2, MP3, ...)
• coupled-clusters method (CC )
• ...
• density functional theory (DFT)
• LDA
• GGA (BLYP, TPSS, PBE, ...)
• hybrid (B3LYP, M06-2X, ...)
they include the
correlation energy
C7790 Introduction to Molecular Modelling -9Software
for QM calculations
Overview:
http://en.wikipedia.org/wiki/List_of_quantum_chemistry_and_solid-state_physics_software
Paid (commercial, academic license with fee):
● Gaussian (http: // www.gaussian.com/)
● Turbomole (http: // www.cosmologic.de/)
● ADF (http: // www.scm.com/)
● Schrödinger (http: // www.schrodinger.com/)
● Spartan (http: // www.wavefun.com/)
● Hyperchem (http: //www.hyper.com/)
Freely available (free licenses, academic licenses free of charge):
● mopac (http://openmopac.net/)
● orca (https://orcaforum.cec.mpg.de/)
● MPQC (http: // www.mpqc.org/)
● GAMESS-US (http: // www.msg.ameslab.gov/ GAMESS /)
● GAMESS-PC (http://classic.chem.msu.su/gran/gamess/index.html)
● cpmd (http: // www.cpmd.org/)
● cp2k (http://cp2k.berlios.de/)
C7790 Introduction to Molecular Modelling -10System
properties
C7790 Introduction to Molecular Modelling -11-
Overview
Wave function
• population analysis
• electron density
• electrostatic potential
• electric multipole moments (monopole, dipole, quadrupole, ...)
• partial atomic charges
• magnetic properties of molecules (chemical shift, spin-spin interaction constant, ...)
PES curvature and wave function
• vibrations (IR and Raman transitions)
Wave functions of electronic states
• electronic transition (UV/VIS transitions)
Property calculation methods given here are general (they are in no way
limited to the HF method). The quantum chemical method need only
provide the wave function of the ground and possibly excited states and
possibly the curvature of the potential energy surface.
C7790 Introduction to Molecular Modelling -12Wave
function population analysis
Population analysis is a way of studying the wave function that provides the following
information:
• partial contributions of atomic orbitals to molecular orbitals, which is especially
important for frontier orbitals, which are important for the assessment of reactivity
• HOMO - highest occupied molecular orbital (nucleophilic properties)
• LUMO - lowest occupied molecular orbital (electrophilic properties)
• visualization of molecular orbitals
• interaction diagrams
• description of bonds and their quantification (order of bonds)
• partial atomic charges
The most used types of population analyzes:
• Mulliken population analysis (MPA)
• Natural population analysis (NPA)
• Hirshfeld population analysis
http://euch6f.chem.emory.edu/13dipolar.html
More detailed description in
specialized lectures.
C7790 Introduction to Molecular Modelling -13Electron
density
Electron density indicates the density of electrons at a point with coordinates x, y, z. It is
calculated from wave function.
 =
Ω
τRrrRrr dzyx nn ),,...,(),,...,(),,( 11
*

nddd rrτ ...2=
it is integrated over all but one
electron coordinate (electrons are
indistinguishable particles)
=
Ω
τdzyxn ),,(
The integral of the electron density over the whole space is equal to the number of
electrons that are present in the system.
Electron density is an important property of the system and plays a central
role in DFT methods (Density Functional Theory, density functional methods).
The electron density can be employed in the calculation of partial atomic charges, it can
be used to define the envelope of a molecule (molecular surface), etc.
C7790 Introduction to Molecular Modelling -14Electrostatic
potential
Electrostatic potential indicates the value of the electrostatic potential at a point with
coordinates x, y, z, which is caused by the electrostatic action of the atomic nuclei of the
molecule and the effective field of electrons. It is calculated from wave function.


 

−
−
=
=
Ω
Ω
τRrRr
τRrRr
Rr d
dV
Z
zyxV
N
i ixyz
i
),(),(
),(),(
),,( *
*
1

ndddddd rrrrrτ ...4321=
it integrates over all
electrons and the entire
space W
Electrostatic potential provides the assessment of electrostatic properties of molecules
and can serve for the calculation of partial atomic charges.
= −
=
n
i ixyz
V
1
1ˆ
rr
action of nuclei action of electrons
Electrostatic potential operator:
C7790 Introduction to Molecular Modelling -15Visualization
of  and V
http://www.nature.com/srep/2014/141020
/srep06678/fig_tab/srep06678_F3.html
Electrostatic potential is mapped
onto isosurface of electron
density (molecular envelope).
convention in chemistry
C7790 Introduction to Molecular Modelling -16Electric
multipole moments
Electric multipole moments are quantities describing electric charge distribution in a
system. They are calculated from wave function.


 

−=
=
Ω
Ω
τRrRr
τRrRr
d
dQ
RRRZM
N
i
m
iz
l
iy
k
ixizyx mlk
),(),(
),(),(
*
*
1
,,,

contribution of nuclei
electron contribution
=
=
n
i
m
iz
l
iy
k
ix rrrQ
1
,,,

Operator of electric multipole:
The sum of k+l+m determines the type of multipole moment component:
• monopole (0) - scalar number, total charge of the molecule
• dipole (1) - vector (three components)
• quadrupole (2) - tensor (3x3 components)
Component of moment:
C7790 Introduction to Molecular Modelling -17Electric
dipole moment
Electric dipole moment describes the asymmetrical distribution of the electric charge in the
system. It is a vector quantity.
Size of the vector is independent of the position and rotation of the system of atoms only in
electrically neutral systems!
For electrically charged systems, the value must be calculated for the system geometry in
standard orientation (may depend on the program used).
Unit: Debye (D), a unit of dipole moment not belonging to the SI system, is defined as 10-18
sC·cm, where sC is a statcoulomb. Relationship of debye to the SI unit is
1 D = 3.33564x10-30 C·m
zyx MMM ,,=μ
Attention (sign convention):
physics chemistry (sometimes)
- + - +
222
zyx MMM ++=
C7790 Introduction to Molecular Modelling -18Partial
atomic charges
Partial atomic charge expresses the property of the atom that is caused by uneven
distribution of electrons in molecules.
For qualitative expression, the d+ and d- designation is used.
Quantitative calculation of partial atomic charges is possible, however there is no uniform
or best approach or method for the calculation. Calculation and most importantly analysis
of calculated charges must always be done in the context of the used method.
Use of partial atomic charges:Electron density:
• estimation of chemical reaction mechanism
C7790 Introduction to Molecular Modelling -19Partial
atomic charges - classes
Classes of charges:1
• I. class (class I charges) - charges are not determined from quantum chemical
calculations but based on intuitive or other approaches using experimental data
(electronegativity, dipole moments).
• II. class (class II charges) - charges are derived based on the distribution of a wave
function using some scheme based on orbitals
• III. class (class III charges) - charges are derived based on the distribution of a
physically observable quantity derived from a wave function (e.g., electron density)
• IV. class (class IV charges) - charges are derived based on semiempirical mapping of
precursors of charges from II. or III. classes in order to reproduce experimental data
(electronegativity, dipole moments)
(1) Cramer, CJ Essentials Of Computational Chemistry: Theories And Models;
John Wiley & Sons, 2004.
C7790 Introduction to Molecular Modelling -20Most
often used types of charges
Population analysis of the wave function (Class II):
• Mulliken charges from Mulliken population analysis (MPA - Mulliken Population
Analysis)
• their values greatly depend on the size and type of basis set
• they do not have a clear CBS limit
• natural charges from natural population analysis (NPA - Natural Population
Analysis)
Electron density analysis (class III):
• Bader charges (from AIM analysis [Atoms In Molecules])
• Hirschfeld charges
• VDD charges (Voronoi Deformation Density)
Charges derived from electrostatic potential (Class III):
• ChelpG charges
• Merz-Singh-Kollman charges (MSK or only Merz-Kollman [MK])
C7790 Introduction to Molecular Modelling -21Bader
charges (AIM charges)
http://aim.tkgristmill.com/screenshots1aaa.html
surface delimiting individual atoms
0. = n
W
−=
k
dZq kk rr)(
normal surface vector
electron density gradient
Partial charge:
integration over the space of
the atom defined by surfaces
Derived from electron density
analysis (wave functions).
C7790 Introduction to Molecular Modelling -22ESP
charges
ESP charges (ElectroStatic Potential) are charges derived from electrostatic potential. The
principle of charge calculation consists of two steps:
1. calculation of electrostatic potential VQM from wave function on discretized
molecular envelope (set of points)
2. finding point atomic charges that create electrostatic potential VPC which is in the
best agreement with the quantum mechanical potential (least squares method)
pQMV ,→ kpPC qV ,
( ) min!
2
,, =−p
pPCpQM VV
is searched by least
squares method
http://biomodel.uah.es/Jmol/surfaces/inicio.htm
ESP charges and their derivatives are used in molecular
mechanics because, by their nature, they describe well
electrostatic properties of molecules/system.
C7790 Introduction to Molecular Modelling -23Magnetic
properties
It is necessary to use special methods and analyzes.
Calculated properties:
• hyperfine splitting constants (EPR)
• chemical shift (NMR)
• spin-spin interaction constant (NMR)
• NICS (Nuclear Independent Chemical Shielding) (NMR)
NICS
C7790 Introduction to Molecular Modelling -24-
Vibrations
When searching for vibrational states, the following SR is solved:
)(
1
2
ˆ
1
2
2
RE
M
H e
N
i
i
i
R +−= =

GF method (Wilson's method) solves the above equation in the limit of classical mechanics
in the harmonic approximation (rotation and translation are not considered).
The method considers only normal vibration modes when all atoms move with the same
frequency and phase. The atomic system has 3N-6 (5) linearly independent normal modes
of frequencynk.
kkk cGFc = kk n = 
−
=
=
63
1
0,
2
1 N
k
kv hE n
!!!! analyzed geometry must be a stationary point !!!
zero-point vibrational energy (ZPVE)matrix of force constants
Hessian
mass of atoms
C7790 Introduction to Molecular Modelling -25Infrared
spectroscopy
http://www.chm.bris.ac.uk/webprojects1997/RogerEC/welcome.htm
Infrared Spectroscopy (IR) - the absorption of infrared radiation is measured. Absorption
leads to excitation between adjacent vibrational levels. Active are only those transitions
which results into the change of the dipole moment of the molecule (this can be
determined from the analysis of the wave function).
C7790 Introduction to Molecular Modelling -26Raman
spectroscopy
Active are only those transitions which leads to
the change of the polarization of the molecule
(this can be determined from the analysis of
the wave function).
(1) Dietzek, B .; Deckert, V .; Popp, J. Raman Spectroscopic Instrumentation, Experimental
Considerations. In Encyclopedia of Biophysics; Roberts, GCK, Ed .; Springer Berlin
Heidelberg: Berlin, Heidelberg, 2013; pp 2173–2178.
C7790 Introduction to Molecular Modelling -27Type
of stationary point on PES
kkk cHc =
eigenvectors
eigenvalues
Nk 3,...,1=
• 6 (5) eigenvalues ​​are zero - this corresponds
to the translation and rotation of the system
• remaining eigen numbers:
• all positive - local minimum
• one negative, others positive - first
order saddle point
• two negative, the other positive - saddle
point of the second order
• .....
• all negative - local maximum
kkk cGFc =
FΗ =
eigenvalues
eigenvectors
• 6 (5) frequencies are zero - this corresponds
to the translation and rotation of the system
• remaining frequencies:
• all positive - local minimum
• one imaginary, others positive - first
order saddle point
• two imaginary, the others positive - a
saddle point of the second order
• .....
• all imaginary - local maximum
Analysis of Hessian Vibrational analysis
kk n =
!!!! analyzed geometry must be a stationary point !!!
C7790 Introduction to Molecular Modelling -28Electronic
transitions
http://www.nature.com/nchem/journal/v2/n11/fig_tab/nchem.838_F4.html
It is necessary to use special methods (time-dependent DFT or time-dependent HF; CI;
CASSCF). The transition probability is given by the strength of the oscillator.
C7790 Introduction to Molecular Modelling -29-
Summary
➢ Quantum chemical methods provide powerful tools to study small to middle size
models.
➢ QM describes behavior of electrons and nuclei. QM is then suitable for studying all
possible chemical transformations (conformational changes, interactions, reactivity,
etc.)
➢ Solution of Schrodinger equation (SR) is not only potential energy but also
wavefunction (WF).
➢ WF can provide additional properties of system, which can further improve our
knowledge about the system.