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Introduction to QM/MM
Emiliano Ippoliti
Forschungszentrum Jรผlich
https://www.cp2k.org/features
โข Energy and Forces
โข Optimisationโข Geometry optimisationโข Nudged elastic bandโข โฆ
โข Molecular Dynamicsโข Born-Oppenheimer MDโข โฆ
โข Propertiesโข Atomic charges (RESP, Mulliken, โฆ)โข Spectraโข Frequency calculationsโข โฆ
DFT+
GPW basis set(Mixed Gaussian and PW)
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Tutorial with
Outline
ยง Why QM?
ยง Elements of Computational Quantum Chemistry
ยง Why QM/MM?
ยง QM/MM couplings
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First part
Second part
QM description
There are cases where the dynamical behavior of the electrons of your systems cannot be neglected and a quantum mechanical description is required:
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1. Chemical reactions (e.g. enzymatic reactions): electron transfer, bond breaking and bond formation
2. Systems with metal atoms (e.g. metallo proteins): no universal parametrization for metal atoms
3. Proton transport (e.g. aerobic generation of ATP and oxygen):Grotthuss mechanism (charge/topological diffusion vs mass diffusion)
4. Spectroscopic analysis and prediction (e.g. absorption and fluorescence spectra)
Schรถdinger Equation (SE)
๐โ๐๐๐กฮจ(๐ฑ!, โฆ , ๐ฑ", ๐ก) = ,โ(๐ฑ!, โฆ , ๐ฑ")ฮจ(๐ฑ!, โฆ , ๐ฑ", ๐ก)
โข ฮจ(๐ฑ!, โฆ , ๐ฑ", ๐ก) = system wavefunction
ฮจ(๐ฑ!, โฆ , ๐ฑ", ๐ก) + โ charge density distribution
O ๐ก = โซโฆโซฮจ ๐ฑ!, โฆ , ๐ฑ/, ๐ก โ 1O ๐ฑ!, โฆ , ๐ฑ" ฮจ ๐ฑ!, โฆ , ๐ฑ/, ๐ก ๐๐ฑ! โฆ๐๐ฑ"
โข !โ(๐ฑ!, โฆ , ๐ฑ") = Hamiltonian operator
e. g. !โ = โโ!โ3
#$4๐!# โ โ%
โ3
#&5๐%# + โ%'(
)3
๐ซ6+๐ซ7โ โ%,!
)3-4๐4+๐ซ6
+ โ!'/)3-4-8๐6+๐7
ri = electronic coordinates, Ri = nuclear coordinates
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e. g. !โ = โโ!โ3
#$4๐!# โ โ%
โ3
#&5๐%# + โ%'(
)3
๐ซ6+๐ซ7โ โ%,!
)3-4๐4+๐ซ6
+ โ!'/)3-4-8๐6+๐7
Time-dependent and time-independent SE
1โ(๐ซ!, โฆ , ๐")ฮจ ๐ซ!, โฆ , ๐" = ๐ธ ฮจ(๐ซ!, โฆ , ๐")
Time-independent Schrรถdinger equation Solving numerical methodsโข Hartree Fock Theoryโข Mรธller-Plesset Perturbation Theoryโข Coupled Clusterโข Generalised Valence Bondโข Complete Active Space SCFโข Density Functional Therory...
Time-dependent Schrรถdinger equation
๐โ๐๐๐ก ฮจ(๐ซ!, โฆ , ๐", ๐ก) =
1โ(๐ซ!, โฆ , ๐")ฮจ(๐ซ!, โฆ , ๐", ๐ก)
Molecular dynamics schemes
Born-Oppenheimer approximation
me <<MIโข Motion of atomic nuclei and electrons
can be treated separatelyโข Nuclear motion can be treated classically
โข Ehrenfestโข Born-Oppenheimerโข Car-Parrinelloโข Kรผhne...
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1โB(๐ซ!, โฆ , ๐")๐0 ๐ซ!, โฆ , ๐ซ" = ๐ธ0๐0(๐ซ!, โฆ , ๐ซ")
๐E๏ฟฝฬ๏ฟฝE ๐ก = โ ๐E minL!๐0 1โB ๐0
Born-Oppenheimer MD scheme
ri = electronic coordinates
RI = nuclear coordinates
!โ = โโ!โ3
#$4๐!# โ โ%
โ3
#&5๐%# + โ%'(
)3
๐ซ6+๐ซ7โ โ%,!
)3-4๐4+๐ซ6
+ โ!'/)3-4-8๐6+๐7
=
= โโ!โ3
#$4๐!# โ !โ)(๐ซ1, โฆ , ๐2)
time-independent SE, RI are here only parameters
classical Newton equation: minL!
๐0 1โB ๐0 is the potential felt by the nuclei
At each time step:โข Solving the time-independent SEโข Use ๐0 to find the forces on nucleiโข Move the nuclei
Algorithm
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Electronic Structure Methods
1โ(๐ซ!, โฆ , ๐")ฮจ ๐ซ!, โฆ , ๐" = ๐ธ ฮจ(๐ซ!, โฆ , ๐")
Time-independent Schrรถdinger equation
Many numerical methods to solve it approximately:
โข Hartree Fock Theoryโข Mรธller-Plesset Perturbation Theoryโข Coupled Clusterโข Generalised Valence Bondโข Complete Active Space SCFโข Density Functional Theory...
Best compromisebetween accuracy and
computational cost
Very suitable for large systems
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Density Functional Theory (DFT)
HohenbergโKohnTheorems
1. The ground state energy is a unique functional of theelectronic density:
๐ธ = ๐ธ[๐]
2. The functional for the energy ๐ธ[๐] is variational:
min3!
๐0 !โ) ๐0 = min4๐ธ[๐]
Benefit:
Drawback:
๐0(r1,...,rn)
๐(r)function of 3 x number of electron coordinates (n)
function of 3 coordinates
functional ๐ธ[๐] is not known
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Kohn-Sham DFTKohn and Sham define a fictitious system of non-interacting particles with a local potential that generates the same density as the density of the real fully-interacting system:
๐ ๐ซ =VW
๐W ๐ซ + Y๐๐ซ ๐Wโ ๐ซ ๐Z ๐ซ = ๐ฟWZ๐ธ ๐ = ๐ธ\] ๐W =
=VW
Y๐๐ซ ๐Wโ ๐ซ โโ+
2๐B๐+ ๐Z ๐ซ + Y๐๐ซ Vbcd ๐ซ ๐ ๐ซ +
12Y๐๐ซ๐๐ซโฒ
๐ ๐ซ ๐โฒ ๐ซ๐ซ โ ๐ซโฒ + ๐ธcf[๐]
๐ธgh/ ๐ ๐ธbcd ๐ ๐ธi ๐
minj๐ธ[๐] โ
12๐
+ + Vbcd ๐ซ + Y๐๐ซk๐ ๐ซโฒ๐ซ โ ๐ซโฒ +
๐ฟ๐ธcf[๐]๐ฟ๐ ๐ซ ๐W ๐ซ = ๐๐๐W ๐ซ
1 x N-electronequation
N x one-electron equationsNOT KNOWN
Kohn-Sham equations
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Finding Kohn-Sham orbitals ๐Wby iterative procedure
For a given (fixed) ionic configuration RN:
1. Solve KS equations for an initial or a previous density ฯ:
โ12๐
+ + Vbcd ๐ซ + Y๐๐ซk๐โฒ ๐ซ๐ซ โ ๐ซโฒ +
๐ฟ๐ธcf[๐]๐ฟ๐ ๐ซ ๐W ๐ซ = ๐๐๐W ๐ซ
2. Determine the updated density:
๐nBo ๐ซ =VW
๐W ๐ซ +
3. Check if convergence is reached:
Y๐๐ซ ๐nBo ๐ซ โ ๐ ๐ซ < ThresholdYes: exit
No: back to step 1
๐ ๐ซ = ๐nBo ๐ซ
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Basis set
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โ12๐
+ + Vbcd ๐ซ + Y๐๐ซk๐โฒ ๐ซ๐ซ โ ๐ซโฒ +
๐ฟ๐ธcf[๐]๐ฟ๐ ๐ซ ๐W ๐ซ = ๐๐๐W ๐ซ KS equations
๐W ๐ซ โ Vxy!
z
๐x ๐๐ ๐ซDiscretization: {๐x ๐ซ } = Basis set
โฏโฎ โฑ โฎ
โฏ
๐!โฎ๐z
= ๐๐๐!โฎ๐z
LocalizedGTOs
NonlocalPW
+ Atomic orbital-like + Few functions needed+ Analytic integration for
many operators+ Optimal for regular grids+ Finite extend- Non-orthogonal- Linear dependences for large basis set- Complicated to generate- Basis set superposition error
+ Ortogonal+ Independent from
atomic positionยฑ Naturally period- Many function needed
Gaussian and Plane Waves method (GPW)
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Fast Fourier Trasform
(FFT)
Real space Reciprocal (G) space
Atom-centeredGaussian-type basis + Auxiliary
Plane-wave basis
๐W ๐ซ โ Vxy!
z
๐x ๐๐ ๐ซ ๐ ๐ซ โ1ฮฉV๐y๐
๏ฟฝ#$%
๐ ๐ exp[๐๐ ๏ฟฝ ๐ซ]
12๐
+ < ๐ธf๏ฟฝd
CP2K can scale linearlywith the system size
Biological system sizes
Largest systems investigated at full QM level โฒ 10,000 atoms
Typical biological system sizes โซ 10,000 atoms
Combining different levels of theory and resolutions
Multiscale approach: QM/MM
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QM/MM Approach
The system is separated into two parts:
A small QM part comprises the chemically/photophysically active region treated by computationally demanding electronic structure methods.
The remainder MM part is described efficiently at a lower level of theory by classical force fields.
QM/MM Interface
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