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 17
Reaction Energy II (QM specific)
C7790 Introduction to Molecular Modelling -2Revision:
QM method classification
HF
post-HF
DFT SCF
YEc
hybrid
DFT
 kk EE Y= kk EE =
variational
variational + perturbation
empirical
Andrew Gilbert
C7790 Introduction to Molecular Modelling -3Reaction
Energy
E(x)
xreaction coordinate
(usually not known)
REMEMBER: This is 1D projection of E(R), which is a function of 3N variables (N-number of atoms).
reactant
configuration
product
configuration
E(R)
states
reactant
state
product
state
𝐸(𝜉 𝑃)
𝐸(𝜉 𝑅) 𝐸 𝑅
𝐸 𝑃
Δ𝐸𝑟
Δ𝐸𝑟 = 𝐸 𝜉 𝑃 − 𝐸 𝜉 𝑅 = 𝐸 𝑃 − 𝐸 𝑅
reaction energy the sign convention:
always use the thermodynamics convention
difficult description
(rarely used in
practice)
only important states
are described
C7790 Introduction to Molecular Modelling -4QM
- two approches
I. supermolecular approach:
➢ each component is characterized individually
Δ𝐸𝑟 = 𝐸 𝜉 𝑃 − 𝐸 𝜉 𝑅 = 𝐸 𝑃 − 𝐸 𝑅
II. energy decomposition (EDA):
➢ only interaction energy is accessible for non-covalent interactions
➢ no deformation energy is available
➢ typical methods:
➢ SAPT (Symmetry-adapted perturbation theory)
𝐸𝑖𝑛𝑡
𝑆𝐴𝑃𝑇0
C7790 Introduction to Molecular Modelling -5Supermolecular
approach
It seems to be SIMPLE. But it is
NOT because …
I. supermolecular approach:
➢ each component is characterized individually
Δ𝐸𝑟 = 𝐸 𝜉 𝑃 − 𝐸 𝜉 𝑅 = 𝐸 𝑃 − 𝐸 𝑅
C7790 Introduction to Molecular Modelling -6Supermolecular
approach, cont
Problem 1: Small numbers from big numbers
bambus[6]uril/anion interaction
(139 atoms)
BU6/I(-) -4152.181032604 Hartree
BU6 -3854.321084579 Hartree
I(-) -297.740268591 Hartree
---------------
-0.119679434 Hartree
~ -75.1 kcal/mol
RI-BLYP-d3/def2-TZVPP (vacuum)
chemistry of interest
Requirements:
➢ robust (numerically stable) algorithms
➢ well optimized geometries
➢ well converged WF and energy
C7790 Introduction to Molecular Modelling -7Supermolecular
approach, cont.
Problem 2: Basis set superposition error
Cause:
➢ As the atoms of interacting molecules (or different parts of the same molecule)
approach one another, their basis functions overlap.
➢ Each monomer "borrows" functions from other nearby components, effectively
increasing its basis set and improving the calculation of derived properties such as
energy.
This error is consequence of finite atom centered basis sets and variational nature of
employed theory (HF, DFT). It also influences non-variational post-HF methods.
e-
e-
bb
b
b
b
b
e-
e-
bb
b
b
b
b
long-distance separation
monomermonomer
complex
https://en.wikipedia.org/wiki/Basis_set_superposition_error
C7790 Introduction to Molecular Modelling -8Basis
set superposition error (BSSE)
Types of basis set superposition error:
➢ intramolecular (conformation changes)
➢ intermolecular (interaction)
B
B+
A
A
intermolecular BSSEintramolecular BSSE
A
C1
C2
BSSE
C7790 Introduction to Molecular Modelling -9BSSE
corrections
BSSE can be avoided or corrected by:
➢ the chemical Hamiltonian approach (CHA) - a priori correction
➢ the counterpoise method (CP) - a posteriori correction
➢ extrapolation to CBS limit - a posteriori correction
➢ space centered basis functions such as plane waves - a priori correction
Counterpoise method
e-
e-
bb
b
b
b
b
complex
e-
bb
b
b
b
b
monomer A
ghost B (only basis functions,
no nuclei, no electrons)
e-
bb
b
b
b
b
monomer Bghost A
Δ𝐸𝑟 = 𝐸𝐴𝐵 − (𝐸 𝐴 𝐵 + 𝐸 𝐵 𝐴 )
the same error is on both sides and thus it
vanishes in the interaction energy
CP is only applicable to non-covalent
interactions (intermolecular BSSE)
C7790 Introduction to Molecular Modelling -10Supermolecular
approach, cont.
Problem 3: Size consistency
Size consistency is a concept relating to how the behavior of quantum chemistry
calculations changes with size.
Size consistency (or strict separability) is a property that guarantees the consistency of the
energy behavior when interaction between the involved molecular system is nullified (for
example, by distance).
𝐸(𝐴 + 𝐵) = 𝐸 𝐴 + 𝐸(𝐵)
large separation of A and B individual A and B
For example:
The Restricted Hartree–Fock model (RHF, a single reference method) is not able to
correctly describe the dissociation curves of H2 and therefore all post HF methods that
employ HF as a starting point will fail in that matter.
The solution would be to use multi-reference methods, which are however more
computationally demanding.
C7790 Introduction to Molecular Modelling -11Supermolecular
approach, cont.
Problem 4: Deficient description of long-range interactions
HF and DFT method provide no or incomplete treatment of dispersion interaction.
Dispersion interaction is a weak attractive long-range force.
This can be problematic for studying systems, in which these forces dominates:
➢ noble gases interaction
➢ supramolecular and biomolecular systems
Solution:
dispersion corrected methods:
➢ HF-3c
➢ DFT-D3, DFT-D4, etc.
Further readings:
Grimme, S.; Hansen, A.; Brandenburg, J. G.; Bannwarth, C. DispersionCorrected
Mean-Field Electronic Structure Methods. Chem. Rev. 2016,
116 (9), 5105–5154. https://doi.org/10.1021/acs.chemrev.5b00533.
C7790 Introduction to Molecular Modelling -12Symmetry-adapted
perturbation theory
SAPT (Symmetry-adapted perturbation theory)
➢ only interaction energy is accessible for non-covalent interactions
➢ no deformation energy is available
➢ it employs the perturbation theory (the method is not variational)
➢ interaction energy is composed from several contributions:
➢ electrostatic
➢ exchange repulsion
➢ induction
➢ dispersion
➢ and other contributions …
➢ no susceptible to BSSE, but the accuracy strongly depends on basis set
➢ accuracy might depend on cancellation of errors (low order SAPT + specially tuned
basis sets)
➢ high accuracy requires higher orders (SAPT2, …), which are computationally
demanding
𝐸𝑖𝑛𝑡
𝑆𝐴𝑃𝑇0
= 𝐸𝑒𝑙𝑒
(1)
+ 𝐸𝑒𝑥𝑐ℎ
(1)
+ 𝐸𝑖𝑛𝑑
(2)
+𝐸𝑒𝑥𝑐ℎ−𝑖𝑛𝑑
(2)
+ 𝐸 𝑑𝑖𝑠𝑝
(2)
+ 𝐸𝑒𝑥𝑐ℎ−𝑑𝑖𝑠𝑝
(2)
C7790 Introduction to Molecular Modelling -13-
Summary
➢ While QM provides very sophisticated methods, their use is practically difficult due to
several QM specific problems:
➢ numerical stability
➢ basis set dependence of calculated properties including energy
➢ size consistency
➢ deficiency in proper description of long-range interactions
➢ Therefore, a special care must be taken when QM is utilized for calculation of reaction,
binding, and interaction energies.