CPU and memory demand

(1)

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PÁZMÁNY PÉTER CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

PÁZMÁNY PÉTER CATHOLIC UNIVERSITY SEMMELWEIS

UNIVERSITY

(2)

WORLD OF MOLECULES

MODELING OF ELECTRON AND MOLECULAR STRUCTURE

(Molekulák világa)

(Az elektron-, és molekuláris szerkezet modellezése)

KRISTÓF IVÁN

semmelweis-egyetem.hu

(3)

World of Molecules: Modeling of electron and molecular structure

1. Spectroscopy

2. Absorption spectroscopy 3. Emission spectroscopy

4. Chemical properties of atoms 5. Types of chemical bondings

6. Basic properties of chemical bonds 7. Covalent, ionic and metallic bonds 8. Hydrogen bonds

9. van der Waals forces

Previously – Properties of chemical bonds, spectroscopy

(4)

Previously - Emission in gas discharge tubes

http://commons.wikimedia.org/wiki/File:Gase-in-Entladungsroehren.jpg

Visible emission of different gases in discharge tubes

(5)

• Chemical bonds

• covalent bonds

• polar covalent bond

• non-polar covalent bond

• ionic bonds

• metallic bonding

• Intermolecular forces

• Hydrogen bond

• Van der Waals forces

• dipole –dipole, induced dipole – induced dipole, ...

Previously - Types of chemical bondings

(6)

Previously - Hydrogen bonds in a DNA fragment

(7)

1. Modeling of the molecular and electron structure 2. Different methods

3. MM

4. Hartree-Fock 5. Semi-empirical 6. DFT

7. Møller Plesset 8. Approximations

9. Display options and methods

Applications, aims:

• Displaying/visualizing molecules

• description of the geometry of a molecule (display) – similar to reality

• calculate the energy of a molecule, electron distribution, energy relations

• in case of large molecules the research into

biochemical processes steps are vital to understand nature

• reaction schemes and kinetics, ...

Modeling of the molecular and electron structure

(9)

MM – Molecular Mechanics / force field

HF – Hartree–Fock (first and foremost...)

Semi-empirical (simplified HF)

DFT – Density Functional Theorem (an alternative to HF)

MP – Møller Plesset (post HF method)

Different methods

(10)

Different approximation methods

Schrödinger equation nuclei are fixed

guess electron correlation

electrons are independent, LCAO-MO

parametrization

Density Functional

Models

Møller Plesset Models

Ab initio Hartree-

Fock models Hartree-Fock MO methods AOs don’t interact, parametrization

Semi-empirical Models correct for electron

interaction

(11)

Molecular mechanics (MM, force field method)

• Non quantum chemical method

• Newtonian classical mechanics

• atoms are considered point like particles, with the following interactions

• bonds are modeled as springs (spring constants from experimental bond lengths)

• torsion (rotation along a bond axis)

• dihedral angle changes

• out-of-plain angle changes (molecule inversion)

• interaction of non-bonded atoms (based on Lennard Jones pot.)

• Coulomb interaction

Different approximation methods

(12)

Molecular mechanics (MM, force field method) during the calculations of the MM model, the

following equation is treated and a

minimization (though local) is performed

Different approximation methods

∑

+ +

+

+ +

+

=

−

− Coulomb

AB bonded

non

AB plane

of out

ABCD

dihedral

ABCD bend

ABC stretch

AB

E E

E

E E

(13)

dihedral angle

Molecular mechanics method

http://commons.wikimedia.org/wiki/File:Diederwinkel.svg

(14)

non-bonded atom-atom interaction

Molecular mechanics method

http://en.wikipedia.org/wiki/File:Argon_dimer_potential_and_Lennard-Jones.png

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

−⎛

⎟⎠

⎜ ⎞

⎝

= ⎛

6 12

4 )

(r r r

V ε σ σ

(15)

Available options in most molecular modeling sw.

• MMFF (MMFF94): Merck Molecular Force Field (mostly for organic and biomolecules)

• MMFFaq: (aq = aqeous solution)

solvation energy is added to an MMFF equilibrium geometry result

• SYBYL: alternative and more common implementation of the force field model,

incorporates almost the whole periodic table of elements

• …

Molecular mechanics method

(16)

VMD software

Molecular mechanics method

http://en.wikipedia.org/wiki/File:Vmd_screenshot.png

(17)

PREVIOUSLY:

Schrödinger equation for Hydrogen atom

• simple, since only 1 atom (1 proton)

• only a single electron is involved

• the nucleus is fixed in space (due to the weight difference of proton and electron)

• electron is orbiting in the force field of the nucleus

• we construct the Hamiltonian

• this results in the energy and wave function of the electron at the given orbitals (exact solution!).

Hartree-Fock method

(18)

• the solution of the Schrödinger equation results in the discrete energy levels of electron orbitals, and the

wave functions of these orbitals

• the energy levels

• the lowest energy level (at n=1) and the

corresponding orbital radius from the Bohr model

Previously - Hydrogen atom and hydrogen like atoms

ψ ψ = W H

h ,

8

₀² ² ²

4 2

n e W

_n

mZ

− ε

=

m 0

1 528 , h 0

eV, 6

, h 13

8 e

₁₀

2 0 2 2 1

2 0

4

= = = ⋅

−

⋅

e m π r ε

ε

m

_H

(19)

•

The wave functions of electrons in a general form

•

where are coupled Legendre polynomials

•

and are Laguerre polynomials

Previously - Hydrogen atom and hydrogen like atoms

( ϑ ϕ ) ( ϑ ) ^ϕ

ψ

P ^m

nr L r

nr nr r

r A

r _n ^m

m n

ej 2 cos

e 2 ,

,

1 1

2 1

1 ⎟⎟⋅ ⋅

⎠

⎜⎜ ⎞

⎝

⋅ ⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⋅⎛

−

= ^A₊⁺_A _A

A

( ) ( )

P x₀ 1 P x₁ x P x₂ 1 x² P x₃ x³ x

2 3 1 1

2 5 3

( ) = , ( ) = , ( ) = − , ( ) = −

P

_Am

( )

P x x d P x

l dx

m

m m

l

( ) m( )

= −1 ² ²

1 2 +

+ A n A

L

L x ( )

k

i

k x L x d

dx L x

i

k

i k

i

p p

p i

( ) = − ! ⎛ , ( ) ( )

⎝ ⎜ ⎞

⎠ ⎟ =

∑

=

¹

0

EXACT SOLUTION!

(20)

•

1s orbital

• n=1, l=0, m=0

•

2s orbital

• n=2, l=0, m=0

•

2p orbital

• n=2, l=1, m=-1

•

2p orbital

• n=2, l=1, m=0

•

2p orbital

• n=2, l=1, m=1

• ...

Previously - Hydrogen atom and hydrogen like atoms

1 3

1 100

e 2

1 r

r r

−

= π

ψ

1 1

2 3

1 200

e 2 1 2

4

1 r

r r

−

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

− 2

=

π

ψ

ϕ π ϑ

ψ 1 e 2 sin sin

8

1

₁

1 2 3

1 1

, 1 ,

2

⋅

−

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

−

r

r r

π ϑ

ψ 1 e 2 cos

4

1 ₁

1 2 3

1 0

, 1 , 2

r r r

r r

−

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

ϕ π ϑ

ψ

1 e 2 sin cos 8

1 ₁

1 2 3

1 1

, 1 ,

2 ⋅

−

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛ r

r r

EXACT SOLUTIONS!

(21)

y 2p _z orbital ^y 3d _xy orbital

Previously - Hydrogen atom and hydrogen like atoms

http://commons.wikimedia.org/wiki/File:P2M1.png | http://commons.wikimedia.org/wiki/File:D3M1.png

ORBITALS CALCULATED FROM THE EXACT SOLUTION!

(22)

The Hartree-Fock method is based on the Schrödinger equation (ab initio model)

it is a many-body problem (as in astronomy)

• many (different) nuclei in a molecule

• all of the electrons are present in the force field

• the molecular orbitals are calculated based on the LCAO-MO theory

• the equation system cannot be solved exactly, thus numerical approximations...

Hartree-Fock method

(23)

Approximations

• Born-Oppenheimer approximation

the protons are heavier then electrons, thus consider the nuclei immobile (for the electron calculations)

• relativistic electron effects are neglected

• the LCAO-MO is using a certain set of atomic orbitals to combine the molecular orbitals

• ther orbitals are calculated using single electrons, thus the electron-electron interaction is missing from the terms

Hartree-Fock method

(24)

Solution steps of the Hartree-Fock method

1. guess the geometry of the molecule

2. choose the basis set for the atoms (simplified mathematical formula for the description of th behavior of atomic orbitals)

3. first approximation of LCAO-MO coefficients 4. solve Schrödinger’s equation for bonds, electric

charge, molecula energy, ...

5. calculate more exact coefficients for LCAO-MO 6. check convergence, (if insufficient back to 3-4.)

Hartree-Fock method

(25)

Basis sets (commonly used)

• STO-3G, where STO = Slater type

orbital approximative function

3G = the atomic orbitals are

approximated using 3 Gaussian functions (GTO)

Hartree-Fock method

( )

radius -

exponent orbital

-

5 , 0

3

r

e

r

^r

STO

ζ π

ζ

₋_ζ_⋅

= Ψ

( )

radius -

exponent orbital

-

2

²

75 , 0

r

e

r

^r

GTO

χ π

χ

₋_χ_⋅

=

Ψ

(26)

Basis sets...

• 3-21G: approximate the s,p,d,f orbitals using 3 = the closed shells are approximated by 3 Gaussian approximative function

all atomic orbitals are described by 2 STOs

2 = the 1

^st

STO is expressed by 2 Gaussian functions

1 = the 2

^nd

STO is expressed by 1 Gaussian function

Hartree-Fock method

( )

₂

(

₁

)

₂

(

₂

)

2

, ,

:

e.g. Ψ

_s

r = Ψ

^STO_s

r ζ + c ⋅ Ψ

^STO_s

r ζ

(27)

Basis sets...

• **6-31G: = includes polarization components of the p+d** orbital interactions

6 = the closed shells are approximated by 6 Gaussian approximative function

3 = the 1

^st

STO is expressed by 3 Gaussians 1 = the 2

^nd

STO is expressed by 1 Gaussian

• 6-31G: = includes polarization components of the s+p orbital interactions

6 = the closed shells are approximated by 6 Gaussian approximative function

3 = the 1

^st

STO is expressed by 3 Gaussians 1 = the 2

^nd

STO is expressed by 1 Gaussian

Hartree-Fock method

(28)

Basis sets...

• 6-31+G*

+ = contains orbital corrections for heavy atoms, diffuse functions

• 6-31 1G*

1 = contains further general improvements on the orbital approximations

• 6-31 1+G**

1+ = contains more orbital corrections, diffuse functions and polarization than 1

Hartree-Fock method

(29)

Basis set # basis functions relative time

STO-3G 26 0.05

3-21G 48 0.2

6-31G* 72 1

6-311G* 90 3

6-311+G** 130 25

Hartree-Fock method

A comparison of calculations with different basis sets applied

(30)

• Based on the Schrödinger equation,

• using Hartree-Fock approximations,

• most of the ab initio calculations are substituted by look up tables from empirical (spectroscopic) data

• electron correlation is included in empirical data, so that is taken into account most of the times

• furthermore, the Hückel approximation is valid:

• only valence electrons are taken into account

• only π orbitals and σ orbitals are considered

Semi-empirical method

(31)

common types

• MNDO: modified neglect of differential overlap,

neglects the repulsion effect between two electrons during calculations

• AM1: Austin Model 1,

generalization of MNDO method with different

parametrization and more utilization of spectroscopic data

• PM3: Parametrized Model #3,

similar method to AM1, but fewer parameters, also includes transition metals

• ...

Semi-empirical method

(32)

• also based on the Schrödinger equation

• alternative to the Hartree-Fock approximation

• difference is in the simplification of Schrödinger equation with the many body problem

• originally developped to solid state physics

• after this it has been generalized and adapted to molecular dynamics simulation

• the electron-electron interaction is also taken into account

• electrons have a probability between 0 and 2 of occupying a molecular orbital (non integer numbers also allowed here)

Density Functional Theory

(33)

common types

• BP: Becke type electron interaction is applied, also a Perdew-type electron

correlation

• BLYP: Becke-Lee-Yang-Parr type approximation (widely used model)

• B3LYP: Becke, 3 parameter Lee-Yang-Parr type, electron correlation and interaction is considered

• ...

Density Functional Theory

http://en.wikipedia.org/wiki/File:C60_isosurface.png

(34)

• also based on the Schrödinger equation

• it is a post-Hartree-Fock approximation

• because it contains an electron correlation term

• it is using Rayleigh-Schrödinger perturbation theory

• using a simple structure, and slightly perturbing it results in information about a more complex system

• the different types of MP are based on the degree of perturbation

Møller Plesset model

(35)

common types

• MP2: contains a second order perturbation

• R1-MP2: a variation of MP2 with optimized and extended basis set,

results in lower

computational costs

Møller Plesset model

http://commons.wikimedia.org/wiki/File:PhI-F5PhI.png

(36)

Molecular size and computational demand correlation

Ab initio methods HF, MP, DFT

Semi-empirical

Molecular mechanics

molecule size

CPU and memory demand

(37)

Molecular mechanic methods

• no electron structure calculations

• wide range of molecule sizes

• dependent on parameters (accuracy)

Semi-empirical methods

• wide range of molecule sizes

• quite fast

• electron structure based

• not always reliable

Ab initio methods (HF, MP, DFT)

• closest to exact molecular and electron structure

• electronic details

• reactions, intermediate and transition states

• highest computational costs

Use of different approximations

(38)

task MM

SYBYL

SE PM3

HF 3-21G

HF 6-31G*

DFT B3LYP

MP MP2 geometry (organic)

geometry (metal) -

transition state geometry - conformation

thermochemistry -

computation time low high

Comparison of different tasks

adapted from: Wavefunction, and Incorporated. 2006. Spartan Physical Chemistry Edition - Tutorial and Activities. Wavefunction, Incorporated, January 1.

good with caution poor results

(39)

Using a chemical modeling software the display options can be (not exclusively)

• molecules with different models

• atomic properties

• isomer properties

• structural information

• repetitive structures

• inter-, or intramolecular forces

• electron structure

Display options

(40)

Displaying molecules with different models

(41)

displayed as labels

• atomic labels (identification)

• atomic mass

• atom types (element)

• R/S isomer (in case of active centers)

• electrostatic charge (local value)

• Millikan charge

• normal charge

• Chemical shift

• ....

Displaying atomic properties

(42)

Displaying atomic label

(43)

Displaying atomic mass

(44)

Displaying atom types (element)

(45)

Displaying isomer information

(46)

Displaying electrostatic charge

(47)

in proteins or polypeptides

• α-helical structures

• β-sheet structure in DNA, RNA or

polynucleotides

• α-helical structures

• β-helical structures

Displaying structural information

CRYSTAL STRUCTURE OF HUMAN

INSULIN-DEGRADING ENZYME IN COMPLEX WITH INSULIN

(48)

Displaying repetitive structures (α-helices)

SEMI-SYNTHETIC ANALOGUE OF HUMAN INSULIN PROB26-DTI

(49)

Displaying repetitive structures (α-helices)

(50)

Displaying repetitive structures (α-helices)

(51)

Displaying intermolecular forces (Hydrogen bonds)

(52)

Displaying electron structures

• atomic and molecular orbitals (HOMO, LUMO)

• electron density surface (e.g. at 90%)

• Van der Waals surface (molecular volume)

• covalent bond electron density surface

• electrostatic potential map

• ionization potential map

• combination of any two...

(53)

Displaying HOMO and LUMO electron orbitals

(54)

Displaying electron density surfaces

from atomic positions to molecular shape sized electron density surfaces

0.08 e^-/au³ 0.002 e^-/au³ VdW surface

a density of 0.002 e^-/au³ represents ~ 90% of the electron density of a molecule

(55)

Displaying electron density surfaces

simple

space filling model of benzene

Van der Waals electron density surface of benzene

(56)

electrostatic potential

• interaction of a positive charge and the charge

distribution of the molecule (nuclei and electrons)

• negative electrostatic potentials indicate slightly positive local charge

• positive electrostatic potentials indicate negative local charges

• the software calculates the energy required to move a proton from infinity near the molecule and displays an isosurface of this map

(e.g. IsoVal:-20 equals -83.68 kJ/mol energy)

Displaying electrostatic potential surfaces

(57)

Displaying electrostatic potential surfaces

PABA EP surface IsoVal: -20

E = -83.68 kJ/mol

PABA EP surface IsoVal: -6.6

E = -27.62 kJ/mol

(58)

Displaying electrostatic potential surfaces

acetic acid EP surface IsoVal: -6.6

E = -27.62 kJ/mol

benzene EP surface IsoVal: -6.6

E = -27.62 kJ/mol

pyridine EP surface IsoVal: -6.6

E = -27.62 kJ/mol

(59)

The local ionization potential displays the energy required for electron removal (“ionization”) at any location around a molecule.

e.g. a surface of “low” local ionization potential for sulfur tetrafluoride (SF

₄

) shows the areas that are easily ionized, the non-bonding (lone)

electron pair can be identified.

furthermore, a combination: mapping the local ionization potential onto the electron density

surface shows regions from which electrons are most easily removed.

Displaying ionization potential surfaces

(60)

Displaying ionization potential surfaces (and combinations)

ionization potential surface ionization potential mapped onto the electron density surface

(61)

• Any potential can be mapped to any surface

• e.g. electrostatic potential mapped to the electron density surface

• HOMO or LUMO mapped onto the electron density surface

• these combinations give us insight into the

• molecular level electron distribution and

• the reactivity of the molecule,

• also protonation, or

• ionization information can be extracted.

Displaying mapped surfaces

(62)

Mapping electrostatic potential to the electron density surface

PABA

EP on electron density surface red: low EP values

blue: high EP values

(63)

benzene pyridine

Displaying electrostatic potential mapped onto the electron density surfaces

(64)

Displaying electrostatic potential mapped onto the electron density surfaces

(65)

CPU and memory demand

WORLD OF MOLECULES

KRISTÓF IVÁN

1. Spectroscopy

2. Absorption spectroscopy 3. Emission spectroscopy

4. Chemical properties of atoms 5. Types of chemical bondings

6. Basic properties of chemical bonds 7. Covalent, ionic and metallic bonds 8. Hydrogen bonds

9. van der Waals forces

Previously – Properties of chemical bonds, spectroscopy

Previously - Emission in gas discharge tubes

• Chemical bonds

• covalent bonds

• ionic bonds

• metallic bonding

• Intermolecular forces

• Hydrogen bond

• Van der Waals forces

Previously - Types of chemical bondings

Previously - Hydrogen bonds in a DNA fragment

1. Modeling of the molecular and electron structure 2. Different methods

3. MM

4. Hartree-Fock 5. Semi-empirical 6. DFT

7. Møller Plesset 8. Approximations

9. Display options and methods

Table of Contents

Applications, aims:

• Displaying/visualizing molecules

• description of the geometry of a molecule (display) – similar to reality

• calculate the energy of a molecule, electron distribution, energy relations

• in case of large molecules the research into

biochemical processes steps are vital to understand nature

• reaction schemes and kinetics, ...

Modeling of the molecular and electron structure

MM – Molecular Mechanics / force field

HF – Hartree–Fock (first and foremost...)

Semi-empirical (simplified HF)

DFT – Density Functional Theorem (an alternative to HF)

MP – Møller Plesset (post HF method)

Different methods

Different approximation methods

Molecular mechanics (MM, force field method)

• Non quantum chemical method

• Newtonian classical mechanics

• atoms are considered point like particles, with the following interactions

• bonds are modeled as springs (spring constants from experimental bond lengths)

• torsion (rotation along a bond axis)

• dihedral angle changes

• out-of-plain angle changes (molecule inversion)

• interaction of non-bonded atoms (based on Lennard Jones pot.)

• Coulomb interaction

Different approximation methods

Molecular mechanics (MM, force field method) during the calculations of the MM model, the

following equation is treated and a

minimization (though local) is performed

Different approximation methods

∑

∑

∑

∑

∑

∑

+ +

+

+ +

+

=

E E

E

E E

E E

dihedral angle

Molecular mechanics method

non-bonded atom-atom interaction

Molecular mechanics method

Available options in most molecular modeling sw.

• MMFF (MMFF94): Merck Molecular Force Field (mostly for organic and biomolecules)

• MMFFaq: (aq = aqeous solution)

solvation energy is added to an MMFF equilibrium geometry result

• SYBYL: alternative and more common implementation of the force field model,

incorporates almost the whole periodic table of elements

( ϑ ϕ ) ( ϑ ) ^ϕ

¹