Theoretical Models I

Maciej Gutowski

Department of ChemistrySchool of Engineering and Physical Sciences

Heriot-Watt UniversityEdinburgh, United Kingdom

Summer School onMaterials for the Hydrogen Society

June 19-23, 2008 Reykjavik, Iceland

Ηψ = Εψ

F = MA

exp(-∆E/kT)

domain

quantumchemistry

moleculardynamics

Monte Carlo

mesoscale continuum

Length Scale

Tim

e Sc

ale

10-10 m 10-8 m 10-6 m 10-4 m

10-12 s

10-8 s

10-6 s

Theory Today

Solutions to the Schrodinger equation can provide all the necessary information to completely describe any chemical system of interest. However, using even the most powerful computational methods, the sizes are constrained to ~100-1000 atoms for ground state energies alone. There are various theoretical approaches used in different regimes of lengths and time scales!

10-15 s

But the beginnings were not easy … .

Chemistry and Mathematics

“Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If mathematical analysis should ever hold a prominent place in chemistry—an aberration which is happily almost impossible– it would occasion a rapid and widespread degeneration of that science.”

Auguste Comte, 1830.

Quantum Mechanics and Chemistry

“...in the Schrodinger equation we very nearly have the mathematical foundation for the solution of the whole problem of atomic and molecular structure”

but…

“… the problem of the many bodies contained in the atom and the molecule cannot be completely solved without a great further development in mathematical technique.”

G. N. Lewis, J. Chem. Phys. 1, 17 (1933).

What Did Theory Contribute to Physicochemical Sciences?

• Provided conceptual framework for – Thermodynamics (phenomenological and

statistical)– Kinetics– Molecular and Crystalline Structures– Spectroscopy– …– …– …

Theoretical Concepts that Entered the Everyday Physicochemical Language

(heard at this summer school)

• Discrete energy levels• Orbitals• Hybridization• σ donation and π back-donation• Dispersion interaction• Valence repulsion• Configuration interactions• Non-adiabatic coupling• ….

Questions Asked before a Material/Molecule is Synthesized

• What is thermodynamic stability and possible formation pathways and decomposition channels?

• What are barriers for the formation and decomposition of this compound?

• What are possible polymorphs, isomers, tautomers …

• What are spectroscopic characteristics of this compound?

• How do its properties depend on T, p, solvent effects?

How Does Theory Answer These Questions?

• VSEPR model • ….• Classical force fields• Reactive force fields• Semi-empirical electronic structure & tight-

binding models• First-principles electronic structure modelsThese models provide Potential Energy Surfaces (PES) that are further used in kinetics, dynamics, statistical thermodynamics (discussed at Hannes’ lecture)

Potential energy surfaces (PES)

Key to understanding

• Chemical reactions

• Dynamics/energy transfer

• Spectroscopy

• Thermodynamics

Methods of obtaining and representing PES

• analytical model potentials (force fields)

• quantum chemistry (grid of energies)

Quantum chemical energies on grid of geometries can be fit to analytical potentials for subsequent use in studies of spectroscopy or dynamics

Limited to about 10 atoms

“On the fly” methods can handle larger systems

Example – Lennard-Jones (LJ) clusters

R

Isomers

• different minima on potential energy surface

• number of isomers grows exponentially with # of atoms

• a and b – permutation-inversion isomers

• Ea = Eb ≠ Ec

−

=612

4R

σ

R

σεE(R)

∑<

−

=

ji

E

6

ij

12

ij RR4

σσε

Two atoms:

Multiple atoms -assume pairwiseadditive:

a

b

c

dispersion (van der Waals)repulsion

1 2 3

1 3 2

R

E

R

ε

21/6σ

More About Force Fields�Force fields provide energy as a function of nuclear positions,

i.e., the PES

�EFF=Estr+Ebend+Etors+Evdw+Eel+Ecross

Estr- stretching energy

Ebend- bending energy

Etors- torsional energy

Evdw- van der Waals energy

Eel- electrostatic energy

Ecross- cross terms

Popular Force Fields�AMBER (Assisted Model Building and Energy Refinement)

�DREIDING

�UFF (Universal force field)

�CHARMM (Chemistry at HARvard Molecular Mechanics)

�MMFF (Merck Molecular Force Field)

�……

AMBER Force Field

EAmber=Estr+Ebend+Etors+Evdw+Eel

only stretching, bending, torsional and non-

bonded energies are considered

Bond Stretching Energy

∑ −=bonds

ijijb rrkE 20 )(

A harmonic potential is the most common simple empirical form for the bond energy. The spring constant of the bond is given by kb and r0 is the bond length at equilibrium. Unique kb and r0 assigned for each bond pair

Bending Energy

∑ −=angles

ijkijkkE 20 )( θθθ

The bending energy is also a harmonic potential with a spring constant of the bond now given by kθ and θ0 is the bond angle at equilibrium. Unique kθ and θ0 assigned for each bonded atomic triplet

Torsion Energy

∑ −+=torsions

ijklijklijkl nAE )]cos(1[ ϕπ

ϕ

The parameter A controls the amplitude of the energy contribution, n controls its periodicity and Φ shifts the entire curve along the rotation angle axis (τ). Unique parameters for torsional rotation are assigned to each bonded quartet of atoms

Non-bonded Interactions

∑ ∑∑∑ ++−

=i i j ij

ji

j ij

ij

ij

ij

r

qq

r

B

r

AE

126

van Der Waals Coulomb

• Non-bonded interactions occur between every atom in the system, whether they are bonded or not.• The van der Waals parameter A determines the degree of attractiveness and B the degree of repulsion.

rij

Reactive Force Fields (ReaxFF); A.C.T. van Duin, W.A. Goddard

Directly incorporating bond order into the functional form for the energy enables chemical reactions to take place in the simulation.

Bond orders are determined from interatomic distances

ReaxFF: Energy ContributionsFor each bonded potential term, the energy is required to smoothly vanish as the bond order decreases. Non-bonded interactions do not depend on bond order. Parameters are fitted to QM calculations for small representative structures.

,

How Do We Use PES?

iii

i amr

EF =

∂∂−=

v i(t + ∆t) = v i(t) + ai(t)• ∆t

ttvtxttx iii ∆•+=∆+ )()()(

1. Minima on PES• Equilibrium structures• Relative energies• Describe structure and thermodynamics

2. Saddle points connecting minima• Transition structures• Barrier heights• Link to reaction mechanism

3. Molecular properties, spectroscopy4. Chemical insight reactant productTS

Ensembles of molecules

Molecular dynamics simulationsMonte Carlo simulations

E.g. MD: Update forces, velocities and positions

Recompute energy for new positions and repeat

v i(t + ∆t) = v i(t) + ai(t) •∆t

Get ai from this equation

Stationary points for all coordinates Xi

• local minima – curvatures positive in all directions

• 1st order saddle points – curvature negative in one direction, positive in all others

0=∂∂

iX

E

Potential energy surface for a two-dimensional system (From D. Wales)

Contour map of PES; M = minimum, TS =1st order saddle point, S = 2nd order saddle point

Minimization methods• Calculus based methods

• Steepest descent (1st deriv.)

only finds “closest” minimum

convergence is guaranteed

• Newton-Raphson (NR) (1st and 2nd deriv.)

not guaranteed to converge

• Quasi-Newton methods (1st and 2nd deriv.)

2nd derivatives can be evaluated numerically by update procedures

• Monte Carlo (MC) and Molecular Dynamics (MD) based methods

• Simulated annealing

Start at high T, and gradually lower T

• Basin-hopping (a hybrid MC/calculus method)

• Neural network approaches

Locating the global minimum – major challenge

even for molecules of the size of single amino acids and small peptides!

• Brute force approaches, e.g., starting from many initial structures, work for only the simplest systems

• Monte Carlo methods such as basin hopping useful for systems containing 100 or so atoms (very computationally demanding)

How to find the global minimum?

canonical arginine: a total of 10 degrees of freedo m

folded

unfolded

partially folded

The problem of global minimum finding is relevant for a wide range of other chemical and biological systems, e.g., to the “protein folding”problem. The above figure is from Brooks et al., Science (2001).

EntropyProtein folding

Locating transition states and reaction pathways• Harder than locating local minima

• Elastic band and other 1st derivative (gradient)-based methods

• Eigenmode following (EF) (1st and 2nd deriv).

• Methods using analytical Hessian (d2E/dxidxj matrix)

• Methods with approximate Hessian (update methods)

EF method

∑

∑

><−=∆

><−=•−=

−•+==

−••−+−•+=

−

j j

jj

j j

jjoo

o

oT

oo

fgfE

fgfxgHxx

xxHgdx

dE

xxHxxxxgEE

λ

λ

2

|

|

)(0

)()(2

1)(

2

1

0

Which Method to Choose?

Force fields vs. quantum models (each of these has several variants)?

Do we need to allow for temperature?

Is the dynamics well described classically, or is a quantum treatment required? (Might be relevant for hydrogen – the lightest atom)

In modeling vibrational spectra, does the harmonic approximation suffice?

Approach to be adopted dictated by the nature of the problem being studied.

Experience and collaboration with experimentalists highly desirable!

Approaches to modeling

model potentials (force fields)

applicable to thousands of atoms

conventional FFs neglect polarization and many-body effects; not suitable for cases with rearrangement of electrons

Promising results with ReaxFF

quantum chemistry

tens – few hundred atoms

Wavefunction-based vs. DFT

QM/MM methods

primary region – treated quantum mechanically

Secondary region – treated with a force field

primary

secondary

First encounter with quantum (electronic structure) methods)

Hψ = Eψ

H = Hamiltonian : contains kinetic energy operator, e l.-nuclear interactions, el.-el. Interactions

A complicated partial differential equation

In general – must introduce approximations

Orders of magnitude more expensive than using model potentials

Even fastest methods scale as N3, where N = number of atoms

Research underway to get O(N) scaling for large systems

But not subject to limitations of model potentials

Includes polarization

Applies to all bonding situations

All properties accessible

Software: both commercial and public domain programs

GAMESS, Gaussian, VASP, Crystal, NWChem, Turbomole, Jaguar, Siesta ….

Properties derived from quantum calculations:•thermochemistry•geometries – minima and transition states•charge distributions, multipole moments• electrostatic potentials• polarizabilities• vibrational spectra• electronic excitation and photoelectron spectra• NMR shifts

Properties unique for solids

•Total and projected densities of states•Electronic band structures •Phonon dispersion curves

Eel (T=0)

Eel(T=0)+ ZPE

E(T = T’)

H(T=T’)

G(T=T’)

Account for vibrational zero-point energy

From electronic structure calculations

Population of excited vibrational,rotational levels

Account for P∆V = ∆nRT (ideal gas)

Include entropy

Accounting for finite temperature on thermodynamic stability

Optimized geometries

Calculate harmonic

frequencies

Vibrational anharmonicity

Diatomic molecule:

V(x) = aox2( 1 + a1x3 + a2x4 + …)

harmonic anharmonicity

E(v) = 1/2 hωe(v+1/2) –ωexe(v+1/2)2 + ωeye(v+1/2)3 + …

x=(R-Re)/Re

ωe = harmonic frequency

ωexe, ωeye = first two anharmonicity constants

Be = rotational constant

αe = vibr.-rot. coupling

ωe = sqrt(4ao*Be)

αe = (a1 + 1)(6Be2/ ωe)

ωexe = (5a12/4 – a2)(3Be/2)

Dunham expansion: unique mapping between 1D potential and the spectroscopic parameters

This mapping is lost for polyatomic molecules

Depends on 3rd and 4th

derivatives

Harmonic approximation to vibrations is straightforward but …

(H2O)2 – an example illustrating the importance of vibrational anharmonicity of frequencies, ZPE, geomet ry

989810133ZPE

876012712

10311314711

10311415510

1081381849

2903103608

5205026307

1593158516296

1611159516505

3601358337194

3660364838143

3735374539152

3745375339351

anharm.harmonic

expt.calculatedmode

Vibrational frequencies and zero-point energies (cm-1) of (H2O)2 .

acceptor

donor

donor

Intermolecular vibrations

Frequencies calculated using the MP2 method.

Anharmonicities calculated using 2nd order vibrational PT.

Excellent agreement between the calculated anh. frequencies and experiment.

2.9762.9642.907ROO

Expt.vibr.averaged

atminimum

parameter

Changes in bond lengths of (H 2O)2 upon vibrational averaging

0 1 2 3

0

R

E

Re Ro

This raises an interesting question concerning the development of model potentials for classical MC or MD simulations.

Namely, should one design the potential to gives the correct Re or Ro values?

Many-body interactions• 2-body interactions – interaction between each pair uninfluenced by other molecules

• Many-body interactions – Interaction between A and B alters interactions between A and C and B and C.

A

B

C

Clusters and molecular crystals of non-polar molecules:

many-body effects dominated by dispersion and valence repulsion

Water clusters – many-body effects dominated by polarization

E = E1 + E2 + E3 + … + En

• In general the series converges rapidly

• Water clusters – 3-body contributions represent 20 – 30% of the net binding energy

Isolated water monomer – dipole moment = 1.85 D

Water molecule in liquid water – dipole moment ~ 2.6 D

+

-

+

+

-

-

+

-

+

-

µAB

µBA

µij – dipole induced on i

by charges on j

µAB in turn induces a

dipole moment on B. Infinite series!

Various issues concerning electronic structure calc ulations

N2 scaling

O(N) scaling possible with use of localized orbitals

O(N) scaling possible with localized orbital MP2

O(N) scaling has been achieved for some large non-metallic systems

O(N) has been achieved for some large systems

Special scaling considerations LimitationsFormal scaling

Method

Fixed node, lack of analytical gradients, Hessians

N3Monte Carlo

Lack of analytical Hessians

N7Coupled-cluster

May not give chemical accuracy

N5MP2

No dispersionN3-N4DFT

No dispersion and other correlation

N4Hartree-Fock

Challenges facing intermolecular interactions

There is still no reliable method for calculating accurate interaction energies between molecules in clusters and in extended systems (molecular crystals).

Example – a cage built of ammonia borane molecules

• standard QC methods

• Need to go beyond the Hartree-Fock model

• need flexible basis sets to recover dispersion

• Near linear dependency, large BSSE with basis sets such as aug-cc-pVTZ

• not clear that MP2 is suitable for this problem

• DFT methods

• Could use with plane waves (to solve linear dependency and BSSEproblems)

• But inappropriate due to neglect of dispersion

• DMC would need to run very long to reduce statistical error below a few tenths of a kcal/mol

Materials Used:

• The ACS PRF Summer School on Computation, Simulation, and Theory in Chemistry, Chemical Biology, and Materials Chemistry, June 11-18, 2005, The Yarrow Hotel in Park City, Utah-Prof. Ken Jordan-Prof. J. Simons-Prof. M. Head-Gordon