dft md part1

Dr. Markus Drees, TU München

Introduction to Computational Chemistry for

Experimental Chemists... (Part 1/2)

Universität

Augsburg

Universität

Regensburg

11th PhD seminar, Garching, September 12th 2008

Outline of today

A) What is computational chemistry?Definitions - Methods - Types of calculations

B) From a Formula to calculated Properties

Graphical interfaces - input files - basis sets - calculation - output analysis

C) How to evaluate Mechanisms with Computational Chemistry

Mechanism theorems - energetic evaluation - transition states and kinetics

D) A Word on Costs and Accuracy

Phrases of prejudice - computational costs - accuracy

A) What is computational chemistry?Definitions - Methods - Types of calculations

A) What is Computational Chemistry? - Definitions

Chemistry, that is investigated in silico

Computational Chem. Cheminformatics

- Storage and analysis of compound

data (databases!)

- Virtual screening (identification of

targets in pharmacy)

- QSAR => quantitative structure-

activity relationship

==> related field: statistics

- Simulation / Modelling of structural

features

- Calculation of molecular properties

and energies

- Ab initio (simulation from the

beginning) and molecular mechanics

-Quantum / Molecular dynamics

==> related field:

theoretical chemistry

A) What is Computational Chemistry? - Method classes (I)

Molecular Mechanics (force field) Ab initio (Latin: from the beginning)

Atoms and bonds as ball and spring

Needs parameters for radii, bonds, angles,

dihedrals and non-bonding contacts from

experiments (force field)

Optimizations in reducing bond-streching

No treatment of electron densities or wave

functions, no direct prediction of properties

Very fast calculations of structures, also

for big systems (atoms > 10,000)

Also suitable for molecular dynamics

(time-dependent structure prediction), but

Difficulties in bond-breaking reactions

Examples: UFF, MM2 - MM4, CHARMM,

AMBER, OPLS

Deals with Electron density and wave

functions as descriptions for the electrons,

no data from experiments needed

Seperates nuclei and electrons

Optimizations in forces and energies,

derived from the electron density

Solvation of the Schrödinger equations

and determination of the suitableHamiltonian (H! = E!)

High computational costs likely (especially

when electron correlation is induced)

Examples: HF, MP2-MP4, CI, CC, (DFT)

(sometimes also semiempric calculations)

A) What is Computational Chemistry? - Method classes (II)

Semiempirics Density functional theory

Mixture of force-field and ab initio

calculation

Reduction of the system (Hueckel => onlyvalid for "-electrons)

Treatment only of valence eletrons

(AM1, MNDO, PM3-PM5)

Simple Hamiltonian used to solve a

simplier Schroedinger equations in

combination with force-field parameters

==> not always suitable for all

configurations of an element

Fast calculations in comparison the ab

initio methods

Also suitable for molecular dynamics

(time-dependent structure prediction),

even for reaction prediction

Correlation between wave function and

electron density (Hohenberg-Kohn)

=> All physical properties derived from

wave functions are also predictable by

electron density

Problem: Exact functional to calculate

electron correlation and exchange is not

exactly known! => All DFT “methods“ are

approximations of this functional

Good treatment of electron correlation

(better than in HF, while with less cost

than in post-HF methods)

Problems with intermolecular interactions

(dispersion, van der Waals-forces)

Examples:B3LYP, PBE, BLYP, BP88,

O3LYP, MPW1K, ...

A) What is Computational Chemistry? - Method classes (III)

QM/MM (or Hybrid Calculation)

Useful for larger molecules (biochemistry!) or solvent systems:

Distribution of the molecule into at least two parts: a QM part and a MM part

QM: normally the reaction center (higher accuracy)

MM: expresses the sterically more demanding sites that play only the role of

spectators.

Example: ONIOM

A) What is Computational Chemistry? - Calculation types (I)

- Geometry optimization of a ground state

Minimizes the energy, while intramoleculare forces are brought to 0.

Needs a structural starting point from the user.

Delivers the best structure an its total electronic energy

- Frequency calculation

Delivers the force constants and the vibrational frequencies

Simulations IR and Raman Spectra

Distinguishes a ground state from a transition state (number of negative frequencies)

Delivers thermodynamical data (enthalpy, entropy, free energy, heat capacities, ...)

- Search for a transition state of a chemical reaction

Optimizies the initial structure, while intramoleculare forces are brought to 0.

Follows the most negative frequency of the initial structure

Some algorithms determine the initial TS structure from the geometry of the educt(s)

and product(s)

Determination of barrier heights and kinetics (in combination with frequency

calculation)

To be considered today!

A) What is Computational Chemistry? - Calculation types (II)

- Population analysis

Determines the distribution of the electrons into the molecular orbitals

Delivers dipole moments

Creates graphical expressions of orbitals

- NMR calculation

Simulates the external field of the NMR source as perturbation for the

electron density

Delivers chemical shifts (in combination with a standard) and - very time consuming -

even coupling constants

- UV/VIS calculation

Calculates the most feasible electron excitations

Methods: time-dependent DFT or CIS (configuration interaction - singlets)

- Solvation energies

Implicit methods (SCRF) via correlation of an external reaction field in relation to the

polarity of the solvent, solvent models: PCM, Dipole, ...

Explicit calculations (addition of the solvent molecules to molecular systems) are more

time consuming than implicit methods

To be considered in part 2!

B) From a Formula to calculated Properties Graphical interfaces - input files - basis sets - calculation -

output analysis

B) From a formula to calculated properties - programs

Examples for program suites:

-GAUSSIAN

-NWCHEM

-GAMESS

-TURBOMOLE

-ADF (DFT only)

-MOPAC (Semiempric and HF)

-TINKER (Molecular mechanics)

B) From a formula to calculated properties - graphical interfaces

Graphical interfaces as source for initial geometries:

(GaussView as example)

Also possible is the use of programs like Chem3D or the conversion of crystal

structure data into xyz coordinates or Z-matrix.

B) From a formula to calculated properties - input files

Input files contain: - the level of theory (method and basis sets)

- (initial) geometry of the investigated system

- system requirements (no. of proceesors, memory)

- additional data depending on the type of calculation

Example: Optimization of Ammonia%chk=nh3_zmat.chk

%mem=3800MB

%nproc=2

#opt b3lyp/6-311++G**

Ammonia as ZMat

0 1 N

H 1 B1

H 1 B2 2 A1

H 1 B3 3 A2 2 D1

B1 1.00000000

B2 1.00000000

B3 1.00000000

A1 109.47120255

A2 109.47125080 D1 -119.99998525

%chk=nh3_xyz.chk

%mem=3800MB

%nproc=2

#opt b3lyp/6-311++G**

Ammonia in cart coordinates

0 1 N 0.00000000 0.00000000 0.00000000

H 0.00000000 0.00000000 1.00000000

H 0.94280915 0.00000000 -0.33333304

H -0.47140478 -0.81649655 -0.33333304

Z matrix (left) vs.

cart. coordinates (right)

taken from Gaussian03

B) From a formula to calculated properties - basis sets (I)

Mathematical functions to describe the electrons (wave character)

Mostly used: Linear combinations of gauss functions

Linear combinations of Gauss functions are flexible enough to model

s, p, d and f orbitals!

B) From a formula to calculated properties - basis sets (II)

Minimal basis: STO-NG (N primitive Gauss functions as linear combination

for one orbital each) => Lousy accuracy

Often used: Split valence basis set

(Inner shell orbitals with other numbers of gauss functions than valence shell

orbitals)

3 - 21G

Sum of 3 Gauss functions for inner shell orbitals

Separation for valence shell orbitals into two basis functions consisting of 2

and 1 Gauss functions (carbon: 1s=3, 2s=3, 2p=3: 9 Gaussian functions)

6 - 31G


Separation for valence shell orbitals into two basis functions consisting of 3

and 1 Gauss functions (carbon: 1s=6, 2s=4, 2p=4: 14 Gaussian functions)

6 - 311G


Separation for valence shell orbitals into three basis functions consisting of 3

and a pair of independent gauss func. (C: 1s=6, 2s=5, 2p=5: 16 Gauss func.)

B) From a formula to calculated properties - basis sets (III)

More flexiblity via polarized and diffuse functions

6-31++G** Polarized functions: * add polarized functions per element

(except H, He)

** add polarized functions on all elements

(incl. H, He)Diffuse functions:

+ add diffuse functions on all elements (except H,He)

++ add diffuse functions on all elements (incl. H, He)

Polarized functions: add functions for orbitals with higher angular momentum

(6 d functions summed into p orbitals of elements Li-Ca, 10 f functions are

added for Sc to Zn, 3 p functions summed into s orbitals of H and He, ...)

=> Required at least for the heavier elements (=*), because of the electric field

that is polarizing the calculated AOs in the non-spherical MO environment

Diffuse functions: add small amounts (small orbital exponent) of

functions of the same angular momentum (distributes electron density far

away from the nuclei) => Suggested for lone pair compounds, anions,

radicals (+), and hydrogen bridges and transition states with H

abstraction(++)

B) From a formula to calculated properties - basis sets (III)

More flexiblity via polarized and diffuse functions

6-31++G** Polarized functions: * add polarized functions per element

(except H, He)

** add polarized functions on all elements

(incl. H, He)Diffuse functions:

+ add diffuse functions on all elements (except H,He)

++ add diffuse functions on all elements (incl. H, He)

Diffuse functions: add small amounts (small orbital exponent) of

functions of the same angular momentum (distributes electron density far

away from the nuclei) => Suggested for lone pair compounds, anions,

radicals (+), and hydrogen bridges and transition states with H

abstraction(++)

B) From a formula to calculated properties - basis sets (IV)

What about heavier atoms (transition metals)?

Description of metals via ECPs (Effective core potential)

#Valence shell orbitals via basis functions

#Inner shell electrons summed up into an averaged field

#Relativistic effects easily incorporated

Examples: LANL2DZ / Hay-Wadt; Stuttgart-Dresden-(Köln)-ECPs

Suggestions for suitable basis sets (number of atoms < 90):

6-31G* is regarded to be the minimum requirement for publications

Smaller systems (number of atoms < 20): 6-311+G** might be useful

Anions, Radicals: 6-31+G* might be required

B) From a formula to calculated properties - calculations

From the basis set to a valid wave function: SCF cycle

Complex evaluation of Integrals!

SCF = self-consistent field

B) From a formula to calculated properties - calculations (II)

Geometry Optimization: also a calculation cycle

Initial

structure

SCF procedure

(Determination of the wave

function and the energy)

Calculation of

orbital energies

Check for reaching the thresholds in

forces and changes in coordinates

and SCF energy down to 0?

Algorithm for determing the

geometry for the next point on

energy surface

Final summary of optimized geometry

End of optimzation run

First derivative of the SCF energy

with respect to the vector of location

For TS searches: also evaluation of

the second derivatives (Hessian

matrix)

YES

NO

B) From a formula to calculated properties - calculations (III)

Geometry Optimization: Minimum versus Transition state

Reactant(s) (Local Minimum)

Transition state

(1st order saddle point)Product(s) (Local Minimum)

Potential Energy Surface (PES)

B) From a formula to calculated properties - calculations (IV)

Transition state searches - Strategies

The search for transition states is one of the toughest businesses in

computational chemistry.

=> Mostly no automaticity, human interaction is needed

Strategies:

A) Draw your TS structure guess as accurate as you think.

Most negative eigenvector is followed. This might be also an unwanted

motion in the molecule (e.g. methyl rotation).

B) Pre-optimization

Draw the molecule with a reasonable geometry especially at the reaction

center. Freeze the reaction coordinate(s) and make a normal geometry

optimization (to avoid other unwanted modes).

Open the frozen coordinates and search the TS.

B) From a formula to calculated properties - output analysis

Analysis of output files: text file or graphical tools (GaussView, MOLDEN)

Geometry optimization:

Intermediate step: SCF Done: E(RB+HF-LYP) = -760.849791059 A.U. after 14 cycles

Convg = 0.4573D-08 -V/T = 2.0696 S**2 = 0.0000

Item Value Threshold Converged?

Maximum Force 0.061492 0.000450 NO

RMS Force 0.005240 0.000300 NO

Maximum Displacement 0.188036 0.001800 NO

RMS Displacement 0.044821 0.001200 NO

Predicted change in Energy=-1.175625D-02

B) From a formula to calculated properties - output analysis (II)

Optimization accomplished: Item Value Threshold Converged?

Maximum Force 0.000011 0.000450 YES

RMS Force 0.000001 0.000300 YES

Maximum Displacement 0.001232 0.001800 YES RMS Displacement 0.000269 0.001200 YES

Predicted change in Energy=-1.834641D-09

Optimization completed.

-- Stationary point found.

----------------------------

! Optimized Parameters !

! (Angstroms and Degrees) !

-------------------------- --------------------------

! Name Definition Value Derivative Info. !

-------------------------------------------------------------------------------- ! R1 R(1,2) 1.7311 -DE/DX = 0.0 !

! R2 R(1,3) 1.7724 -DE/DX = 0.0 !

! R3 R(1,4) 2.3156 -DE/DX = 0.0 !

! R4 R(1,20) 2.4801 -DE/DX = 0.0 !

! R5 R(1,21) 2.5229 -DE/DX = 0.0 !

! R6 R(1,22) 2.5034 -DE/DX = 0.0 !

! R7 R(1,23) 3.156 -DE/DX = 0.0 !

! R8 R(1,24) 2.4604 -DE/DX = 0.0 !

! R9 R(1,25) 3.163 -DE/DX = 0.0 ! ! R10 R(1,26) 2.4485 -DE/DX = 0.0 !

....

103,-0.4177548259\H,2.9027772668,-1.9957406754,1.1866693856\H,2.109005 4958,-2.5938312578,-0.2920838803\\Version=AM64L-G03RevC.02\State=1-A\H

F=-760.883834\RMSD=4.495e-09\RMSF=5.242e-06\Dipole=-1.8589791,0.386761

,-0.5495466\PG=C01 [X(C10H18Mo1O4)]\\@

Electronic energy! (atomic units: 1 au = 627.50956 kcal/mol)

B) From a formula to calculated properties - output analysis (III)

Graphical analysis of the profile of a geometry optimization (TS search)

in MOLDEN (left) and GaussView (right)

B) From a formula to calculated properties - output analysis (IV)

Frequency calculations:

Summary of vibrational modes

Harmonic frequencies (cm**-1), IR intensities (KM/Mole), Raman scattering

activities (A**4/AMU), depolarization ratios for plane and unpolarized incident light, reduced masses (AMU), force constants (mDyne/A),

and normal coordinates:

1 2 3

A A A

Frequencies -- -540.2082 25.3888 35.6969

Red. masses -- 8.7207 3.1003 3.7200

Frc consts -- 1.4994 0.0012 0.0028

IR Inten -- 79.5409 0.5030 0.7834

Atom AN X Y Z X Y Z X Y Z 1 42 0.00 0.01 0.00 0.00 0.01 0.00 -0.02 0.03 -0.03

2 8 0.02 0.00 -0.01 -0.04 0.03 0.01 -0.04 0.11 -0.06

3 8 -0.04 0.03 0.02 0.03 0.00 0.02 0.02 -0.03 -0.02

4 8 0.39 0.12 -0.11 0.00 0.01 0.05 0.01 0.01 0.05

5 6 -0.05 0.05 0.24 0.00 -0.01 -0.02 0.01 -0.04 0.04

6 6 0.03 0.08 0.08 0.00 -0.04 0.14 0.02 -0.05 0.16

7 6 0.07 0.01 0.13 0.10 -0.13 -0.13 0.04 -0.13 -0.03

8 6 -0.18 -0.03 0.07 -0.08 0.13 -0.11 -0.02 0.05 0.01

One true imaginary frequency => transition state

B) From a formula to calculated properties - output analysis (IV)

Frequency calculations: Thermodynamic data

-------------------

- Thermochemistry -

-------------------

Temperature 298.150 Kelvin. Pressure 1.00000 Atm.

Atom 1 has atomic number 42 and mass 97.90550





Atom 6 has atomic number 6 and mass 12.00000 Atom 7 has atomic number 6 and mass 12.00000


...

Zero-point correction= 0.265624 (Hartree/Particle) Thermal correction to Energy= 0.284234

Thermal correction to Enthalpy= 0.285178

Thermal correction to Gibbs Free Energy= 0.218585

Sum of electronic and zero-point Energies= -760.618210

Sum of electronic and thermal Energies= -760.599600

Sum of electronic and thermal Enthalpies= -760.598656 Sum of electronic and thermal Free Energies= -760.665249

E (Thermal) CV S

KCal/Mol Cal/Mol-Kelvin Cal/Mol-Kelvin

Total 178.360 68.622 140.157

Electronic 0.000 0.000 0.000

Translational 0.889 2.981 42.993

Rotational 0.889 2.981 32.417

Vibrational 176.582 62.661 64.747

Adjustable isotope pattern, temperature, pressure

#H enthalpy

#G Gibbs free energy

B) From a formula to calculated properties - output analysis (V)

Graphical analysis of the vibrational modes of the transition state

C) How to evaluate Mechanisms with

Computational Chemistry

Mechanism theorems - energetic evaluation -

transition states and kinetics


What is a reaction mechanism?

- Collection of elementary processes (also called elementary steps or

elementary reactions) that explains how the overall reaction proceeds.

- A proposal from which you can work out a rate law that agrees with the

observed rate laws.

- Rationalization of a chemical reaction

But: Is the explanation of experimental (and computational) results alone a

proof of the correctness of a mechanism?

Inductive approach:

Formulate hypothesis based on known facts

Execute experiments and/or

calculations to test this hypothesis

Results and hypothesis consistent?

All features tested?

Proven mechanism!!

No!

Yes! No!

Yes!



Case study: Hydrosilylation of benzaldehyde

Possible mechanistic pathways!

Drees and Straßner, Inorg. Chem. 2007, 10850-9


How to check mechanistic pathways in computational chemistry?

1. Optimize all structures of the pathways (starting material, intermediates,

transition states, products) - be aware of isomers!

2. Frequency calculations in order to distinguish from minima structure

and transition state and to obtain the thermodynamical data

3. Choose a point on the PES to scale all the energies to (zero point).

Non-catalytic reaction: sum of reactants.

Catalytic cycle: starting point or the resting state

4. Substrate the energies of the zero-point from the energy of the desired

calculated point

5. Important: Do not loose atoms or fragments, even when compounds

are added or eliminated during the reaction!

Energy evaluation approach:


Further approaches:

- If kinetic data is available (rate constants, derived activation energies), a

comparison with the calculated barriers of the rate-determining step is

useful.

- Kinetic isotope effects could also be evaluated computationally

(=>isotope pattern at the frequency calculation!)

!

KIE = e"#G

$(D )"#G

$(H )

RT

- Simulation of spectroscopy (NMR, IR, UV) and comparison with

measurements - although with higher absolute errors than the energy

approach

derived from

!

KIE =k(H)

k(D)


Excursion: Transition states and kinetics

!

k = A " e#Ea

RT

!

k =kT

h" e

#$G

%

RT =kT

h" e

#$S

%

R " e#H

%

RT

Arrhenius equation

=> Simplest approach to calculate reaction rates or activation barriers

Transition state theory (Eyring)

=> Incorporation of the laws of thermodynamics to have access to

activation enthalpy and entropy


Meaning of the activation energies: Free energy of activation $G‡

Downside: Temperature- and concentration-dependent But: the easiest parameter for evaluation

Correlated via kinetic measurements.

A fair comparison: correlation of $G‡ with $G0 in a linear relationship.

#Example: Hammett equation

Correlation between kinetic data from a test reaction and substituent effects

!

logk

k0

"

# $

%

& ' = ()

%>0: substituent more electron withdrawing than H

%<0: substituent less electron withdrawing than H

&>0: reaction accelerated by electron withdrawing substituent

&<0: reaction accelerated by electron donating substituent

Connection to $G‡:

!

" =

#$G%

#&

'

( )

*

+ ,

#$G

#&

'

( )

*

+ ,

=

T

-.1

2.303RT/#$H%

#&


Meaning of the activation energies: Activation enthalpy $H‡

Experimental value: from the gradient in the Eyring plot (ln(k/T) vs. 1/T))

!

lnk

T

"

# $ $

%

& ' ' = (

)S*

R+ ln

k

h

"

# $ $

%

& ' ' (1

T+H

*

R

Temperature independent

#Fair comparison between theory and experiment

Interpretation as the heat that is needed to reach the reaction barrier


Meaning of the activation energies: Activation entropy $S‡

Sign of $S‡ in unimolecular reactions:

$S‡ <0 !loss of degrees of freedom in the transition state

(e.g. hydrogen transfer, cleavage of a C-X bond via HX eliminiation,

rearrangements, cyclic transition states, system crossings)

$S‡ >0 !gain of degrees of freedom in the transition state

(e.g. C-C cleavage, peroxide cleavage, biradical formations without

system crossings)

D) A Word on Costs and Accuracy Phrases of prejudice - computational costs - accuracy


„Regardless of what the experimental result would be, the computational

chemist tries to find the level of theory, where the experiment would be verified.“

(Or if the computational chemist is not your friend, but your enemy: „Regardless

of what the experimental result would be, the computational chemist tries to find

the level of theory, where the computational approach delivers a more feasible

mechanism that also fits the experimental result.“)

„Four computational chemists working on one problem would result in five

different computational results for the same experiment.“

„A computational study always needs more computational runs than initially

thought.“ (Murphy’s law for impossible time-management)

„A model system (e.g. smaller substituents) always has different behaviors

than the desired system, but saves a lot of computer time“

Well-known prejudice phrases:


Comparison of various levels of

theory in calculating the G2 test suite.

(Hundreds of molecules incl. reaction

energies)

taken from the Gaussian manual

Accuracy considerations

Level 2 // Level 1:

Geometry optimization at Level 1,

Energy calculation at level 2 for the

optimized geometry from level1.


Cost considerations

CPU = computer time

Memory = RAM memory

Disk = Disk space needed

Direct: solve integrals in the memory without storing data on the hard disk

Conventional: solve the integrals using a lot of temporary files on disk

N = number of basis sets

O = number of occupied orbitals

V = number of virtual orbitals


Cost meets accuracy

SCF Cycle for methane

SCF Cycle for pentane

The computational chemist has always to consider the need for a

specific accuracy in the focus of the higher computational costs

that this might cause.

Further reading....

Computational:

I.N. Levine - Quantum Chemistry - Prentice Hall

(broad overview)

W. Koch, M. C. Holthausen - A Chemist‘s Guide to Density Functional

Theory - Wiley-VCH

(as the name indicates, focus on DFT ,but also with a lot of physical formulas)

A. Szabo, N. S. Ostlund - Modern Quantum Chemistry: Introduction to

Advanced Electronic Structure Theory - Dover

(partly hardcore derivation of all the theory - HF and post-HF only)

Mechanisms:

B.K. Carpenter - Determination of Organic Reaction Mechanisms -

Wiley&Sons

(rather old book, focuses on kinetics and thermodynamical values)

... To be continued!

Thank you very much for your kindest attention!

dft md part1

Documents

computational costs

forcefield parameters

related field

electron densitysolvation

definitions methods

ab initiocalculationreduction

types of calculationsa

abinitio methodsalso