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
Lesson 12
Quantum Chemistry I
PS/2021 Present Form of Teaching: Rev2
C7790 Introduction to Molecular Modelling -2-
Context
microworldmacroworld
equilibrium (equilibrium constant)
kinetics (rate constant)
free energy
(Gibbs/Helmholtz)
partition function
phenomenological thermodynamics
statistical thermodynamics
microstates
(mechanical properties, E)
states
(thermodynamic properties, G, T,…)
microstate ≠ microworld
Description levels (model chemistry):
• quantum mechanics
• semiempirical methods
• ab initio methods
• post-HF methods
• DFT methods
• molecular mechanics
• coarse-grained mechanics
Structure
EnergyFunction
Simulations:
• molecular dynamics
• Monte Carlo simulations
• docking
• …
C7790 Introduction to Molecular Modelling -3Method
overview (model chemistry]
QM (Quantum mechanics) MM (Molecular mechanics) CGM (Coarse-grained mechanics)
)(RE )(RE )(RE
R - position of atom nuclei R - position of atoms R - position of beads
C7790 Introduction to Molecular Modelling -4Quantum
chemistry I
Multi-electron systems
Approximations:
➢ Born-Oppenheimer approximation
➢ One-electron approximation
➢ Basis functions
C7790 Introduction to Molecular Modelling -5Chemical
system
 = = == =
+−+−=
n
i
n
ij ij
N
i
n
j ij
i
N
i
N
ij ij
ji
n
i
i
rr
Z
r
ZZ
m
H
11 111
2 1
2
1ˆ
Hamiltonian of chemical system consisting of N nuclei of mass M and charge Z
and n electrons is given by:
kinetic energy operator potential energy
electrons electron-electron
electron-nucleus
nucleus-nucleus
Nuclei motion is not considered in the BO
approximation.
),()(),(ˆ RrRRr kkk EH  =
Schrödinger equation:
C7790 Introduction to Molecular Modelling -6Finding
the solution
),()(),(ˆ RrRRr kkk EH  =
SE is complicated differential equation:
(the devil is a kinetic operator)
...210  EEE
➢ Instead of searching for all possible solutions, lets focus on a ground state.
➢ The ground state is the state with the lowest energy E0.
➢ Since Ek are electronic states, it can be expected that for the most systems that the
ground state will determine the essential behavior of the system.
The ground state can be found be a variational method.
Alternative: a perturbation method.
C7790 Introduction to Molecular Modelling -7Variational
method
The variational method employs variational calculus. The method essence is
to find the local extreme of functional, which is the way how to transform
functions on real numbers. Such a representation is the relationship between
energy and wave function expressed in integral form:




=
Ω
Ω
τRrRr
τRrRr
R
d
dH
E
),(),(
),(ˆ),(
)( *
*
ndddddd rrrrrτ ...4321=
integrates across all electrons and
the entire space W
),()(),(ˆ RrRRr kkk EH  = differential form
integral form (fully equivalent to SE)
C7790 Introduction to Molecular Modelling -8Variational
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 -9-
Hartree-Fock
method
C7790 Introduction to Molecular Modelling -10Finding
a wave function
Finding of wavefunction is practically impossible because it is a function of all electron
positions. Possible simplification is one-electron approximation:
)()...()(),...,,( 221121 nnn rrrrrr =
Hartree's method
Similar approach was employed for:
➢ time-independent SE
➢ BO approximation
C7790 Introduction to Molecular Modelling -11Finding
a wave function
Finding of wavefunction is practically impossible because it is a function of all electron
positions. Possible simplification is one-electron approximation:
)()...()(),...,,( 221121 nnn rrrrrr =
Hartree's method
Hartree's method does not consider important properties of multi-electron systems.
Electrons are indistinguishable fermions (particles with half spin), which they must comply
with Pauli Exclusion Principle:
no two indistinguishable fermions can be in the same quantum state
The wave function of the system must be antisymmetric. Antisymmetric wave functions
can be obtained by all permutations between one-electron functions and spatial and spin
coordinates.
)...,,,,...,,(),...,,,,...,,( ,12212121 nnnn rrrrrr  −=
)...,,,,...,,(),...,,,,...,,( ,21122121 nnnn rrrrrr  −=
Antisymmetric wave function:
spatial coordinates of electrons spin coordinates of electrons (z-component of spin)
C7790 Introduction to Molecular Modelling -12One-electron
approximation
Alternative notation: Slater determinant
)()()()()()(
)()()()()()(
)()()()()()(
)...,,,,...,,(
2211
2222222121
1112121111
,2121
nnnnnnnn
nn
nn
nn
rrr
rrr
rrr
rrr



 =
)}()()...()()()(){()...,,,,...,,( 22221111,2121 nnnn
P
nn rrrPsignrrr  =
all permutations z-component of spin (spin coordinate)
spin part of one-electron function
Hartree-Fock method
C7790 Introduction to Molecular Modelling -13One-electron
functions
)()(
1
i
m
j
jijii c rr =
= 
➢ Even with one-electron approximation, finding one-electron functions is difficult.
➢ Thus, one-electron functions are expressed using a linear combination of basis
functions.
➢ This description is exact if we use complete system of basis functions (infinitely large
set of orthonormal functions).
➢ The problem is then reduced to finding linear coefficients c, which determine extent of
given basis functions to searched one-electron functions.
pre-defined basis functions
searched numbers
one-electron function
(molecular/atomic orbital)
for multielectron atoms
C7790 Introduction to Molecular Modelling -14Basis
functions
)()(
1
i
m
j
jijii c rr =
= 
pre-defined basis functions
From a practical (numerical) point of view, it is necessary to use limited number of basis
functions.
Basis function choice affects speed of calculation and accuracy of achieved results.
Basis set types:
➢ atomic orbitals (atom centered) are usually derived from simplified SE solution for
hydrogen atom
➢ GTO - Gaussian Type Orbital
➢ STO - Slater Type Orbital (more accurate but more complicated)
➢ plane waves - solid state physics
C7790 Introduction to Molecular Modelling -15-
)()(
1
i
m
j
jijii c rr =
= 
HF method
The HF method tries to find such c, which minimizes energy functional.
  min!00 == EE
)}()()...()()()(){()...,,,,...,,( 22221111,21210 nnnn
P
nn rrrPsignrrr  =
occupied orbitals
one-electron molecular orbitals
system WF
variational method
C7790 Introduction to Molecular Modelling -16RHF
method
RHF - (Restricted Hartree-Fock Method) is applicable to closed systems (closed shell
systems), where each molecular orbital contains exactly two electrons with opposite spin.
Using the variational method, one-electron approximation and linear combination of basis
functions, the solution can be found solving a generalized eigenproblem:
The solution is m (basis set size) of eigenvalues e and vectors c. Eigenvalues e represent the
energy of one-electron functions (orbitals).
n/2 (n - number of electrons) of the occupied orbitals (solution with
the lowest energy ei)
m - n/2 virtual orbitals
the occupied orbitals determine the ground state energy and its
wave function
iii ScFc e=
LUMO
HOMO
C7790 Introduction to Molecular Modelling -17-
Summary
➢The HF method is starting point for other QM methods
➢HF is a variational method
➢It uses two approximations:
➢one-electron approximation (very bad approximation)
➢correlation energy (post-HF methods)
➢finite number of basis functions
➢correction is possible by extrapolation to a complete basis
C7790 Introduction to Molecular Modelling -18Correlation
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 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 -19Quantum
chemical
methods
C7790 Introduction to Molecular Modelling -20Method
classification
HF
post-HF
DFT SCF
Ec
hybrid
DFT
 kk EE = kk EE =
variational
variational + perturbation
empirical
Andrew Gilbert
C7790 Introduction to Molecular Modelling -21Quantum
chemistry
),()(),(ˆ RrRRr kkke EH  =
time-independent Schrödinger equation
Methods
Formal scaling HF CI methods MP methods CC methods
N4 -> N2 -> N1 HF,DFT
N5 MP2 CC2 (iterative)
N6 CISD MP3, MP4(SDQ) CCSD (iterative)
N7 MP4 CCSD(T), CC3 (iterative)
N8 CISDT MP5 CCSDT
N9 MP6
N10 CISDTQ MP7 CCSDTQ (iterative)
scaling, time complexity: http://en.wikipedia.org/wiki/Time_complexity
HF - Hartree–Fock method, DFT - density theory functionals,
CI - configuration interaction method, MP - Møller–Plesset perturbation method,
CC - coupled-clusters method, N - number of basis functions
Jensen, F. Introduction to computational chemistry; 2nd ed.; John Wiley & Sons:
Chichester, England; Hoboken, NJ, 2007.
C7790 Introduction to Molecular Modelling -22QM
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, ...)
C7790 Introduction to Molecular Modelling -23Method
designation
MP2/aug-cc-pVTZ // HF/def2-SVP
HF/def2-TZVP
method (model chemistry)
basis
geometry optimizationenergy and property calculations
energy and property calculations are
performed on the geometry
(structure), which was obtained by
geometry optimization at the same
level of theory