Lecture6

MIT10.637

Lecture 6

Transition state searchesand PES sampling

[email protected]

September 23, 2014

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Lecture 6N > 3, General PES features

from H. B. Schlegel, Wayne State U.

We focus on a reaction coordinate defined by critical DOF, not full 3N-6!

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Lecture 6Energy minimizations

Minimize:

Gradient on our PES in terms of all coordinates (internal, cartesian):

For:

Any stationary point (a) local minimum, (b) global minimum, (c) saddle point.

What we usually want!

What we want today!

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Lecture 6Finding saddle points

• Saddle points are more difficult to find than minima. • Interpolation methods: assume reactant and

product minima are known and that a TS is located “between” these two end-points. May only find a geometry very close to the TS, rather than the true TS.

• Local methods: propagate geometry using information about the function, first and second derivatives – do not need to know reactant/product but usually need to have a good estimate of the TS. Once TS is found, can get reaction path by following steepest descent to R and P from TS.

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Lecture 6Saddle points in small systems

For extremely small systems: PES is 3N-6 variables, so it may be possible to map out the complete PES in limited cases.

For simple small/intermediate systems: “coordinate driving” or constrained optimizations.

For cases where there are a few internal degrees of freedom that describe the difference between reactant and product: e.g. torsion angle, bond distances for breaking/forming.

Choose fixed values for selected coordinates and remaining variables are optimized, adiabatically mapping energy as f(reaction variables). TS is geometry where residual gradients for fixed variables are “sufficiently small”.

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Lecture 6Saddle points in small systems

Coordinate driving:

Reaction variables are good if they have large coefficients in the Hessian eigenvector with negative eigenvalue at the TS.

BUT we only know the Hessian of the TS after it has been found, making the reaction coordinate selection very use-rbiased.

If only one or two variables change significantly betweenReactants and products, this method works well,

e.g. Rotation of a methyl group – torsion angle HNC to HCN – HCN angle SN2 reactions X +CH3Y XCH3 + Y – XC and CY distances (pictured)

More than two reaction coordinates becomes difficult and rarely is successful.

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Lecture 6Synchronous transit

Linear synchronous transit: can consider this a coordinate driving method.

All degrees of freedom – Cartesian or internal – are varied linearly between reactant and product.

No optimization is performed.

Assumptions: 1) all variables change at the same rate along the reaction path.2) transition state is the highest energy structure along the interpolation line.

However, assuming all variables changing the same amount all at the same time is a bad approximation. This method rarely leads to a good estimate of the TS.

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R

QST

TS

P

Quadratic synchronous transit (QST): Approximate reaction path by a parabola instead of a straight line.

Find the maximum on the LST, generate QST by minimizing energy perpendicular to the LST path.

Search for maximum on QST path.

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Lecture 6The Hessian/Force matrix

(3N-6)x(3N-6) matrix with elements:

(3N-6)

(3N-6)

Approximate PES around stationary point by harmonic potentials.

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Lecture 6The Hessian/Force matrix

Diagonalize Hessian (eigenvalue problem)

From E. Eliav, Tel Aviv U

minimum maximum saddle point

Eigenvalues: Eigenvectors:normal coordinates

Harmonic frequencies

εk>0 εk<0 εk>0, except one εj<0

ωk all real ωk all imaginary one imaginary ωj on RC

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Lecture 6Eigenvector following

• In the vicinity of transition state, can use information from second derivative (Hessian) matrix.

• Can be expensive to compute all second derivatives for large systems (3N-6x3N-6) so some method to compute only elements needed or to approximate can speed up approach.

• Need good initial guess for the transition state.• Works best if Hessian already has only one negative

eigenvalue – (e.g. augmented-Hessian NR) and need actual Hessian and not just initial approximation! Expensive!

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Improving on LST and QST:

Another class of methods starts with QST solution and then follows eigenvector to the saddle point – “guided QST”

Introduced by Schlegel and coworkers, Synchronous Transit-guided Quasi-Newton (STQN) methods:

-Can use parabola or circle arc for interpolation-Then use tangent to guide search towards the TS.-Once TS region is located, optimization is switched to quasi-Newton-Raphson-Can require either just reactants and products or reactants, transition, state and products.

Note: these methods generally all work for simple systems and coordinates: the interpolation may be nonsensical in Cartesians but meaningful in internals, etc.

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Lecture 6Rational function optimization

Rational function optimization:Recall from geometry optimizations lecture – RFO expands the function in terms of a rational approximation and then have to solve for a shift parameter that makes eigenvalues positive. The quadratic approximation – requires step length to be equal to the trust radius also.

Partial rational function optimization (P-RFO) for TS searches:Now use two shift parameters l and lTS:

Choose lTS such that search will maximize the energy in this direction, but step can be outside trust radius. QA or “TRIM” uses =-l lTS :

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Lecture 6

Problems with eigenvector following

• Need one eigenvector weakly coupled to others to follow (i.e. small 3rd order derivatives).

• NR may fail to converge if the eigenvectors are too coupled and there’s not just one eigenvalue to follow.

• Need a good geometry to start from.

• Hessian is expensive. NR TS search requires explicit Hessian from the start.

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Lecture 6The dimer method

Eliminates the need to calculate the Hessian:

Dimer is oriented towards lowest curvature mode by rotation and translation.

Curvature is estimated by finite difference from only the forces on the two images, avoiding the need for a Hessian calculation.

Once the dimer is aligned on the lowest curvature mode, force parallel to the dimer is inverted from the total force, making the dimer climb up to the saddle point:

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Lecture 6Nudged elastic band

Chain-of-states method: string of interpolated images/geometries is propagated. This is a modification of the elastic band (Elber & Karplus) method.

A chain gang initial state final state guesses(reactants) (products) Springs keep interpolated images separated:

Our chain of states are propagated on the potential energy surface until we find a minimum energy path.

Typically 4-20 images

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Nudged elastic band is a method that propagates a series of interpolated images towards the minimum energy path.

Initial interpolations of the images are usually linear – and if done in Cartesian coordinates (common), one has to be careful as there can be clashing/overlap of atoms.

Spring forces keep the images spaced equally. Otherwise they’d fall downhill to the reactant and product states. We apply the spring force along the tangent (i.e. along the path):

i-1 i i+1

Spring constant

Distance i+1 to i

Distance i to i-1

Tangent unit vector

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i-1 i i+1

The energy is minimized perpendicular to the path – i.e. forces along the path are projected out and replaced with the spring force.

The perpendicular force is the true force minus the component that is parallel to the path:

The total force in NEB is the sum of the spring force along the tangent and the true force perpendicular to the tangent:

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Mueller potential

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Mueller potential

Reactants, intermediates,products

Saddle point

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Mueller potential


Saddle point

minimum energy path

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Mueller potential


Saddle point

minimum energy path

NEB initial guess from interpolation a la LST

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Mueller potential


Saddle point

minimum energy path

NEB initial guess from interpolation

✓NEB image

i

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i

Spring forces: only want component of this force that keeps images separated (along path).

True forces: ignore component that minimizes energy parallel to path, only minimize perpendicularly.

NEB image force:

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Mueller potential


Converged NEB path

NEB image

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The climbing image extension to nudged elastic band ensures that an image converges to the saddle point. While NEB guarantees images to be equally spaced, no guarantee that there’s an image at the top of the barrier – in fact, it’s very unlikely.

Select the highest energy image and invert the force:

This maximum energy image is not affected by the spring forces.

Qualitatively: this image moves up the PES along the band and down in energy perpendicular to the band. Need to have enough images close to the CI to get a good estimate of the path.

Total force Force parallel to the tangent

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Variable spring constants have been introduced to ensure convergence of the most important points along the minimum energy path – around the saddle point.

This is extra important when climbing image is used because you need to have a good estimate of the tangent to the path near the saddle point.

Choose strong forces near the saddle point and weak forces further away.

The force constants are linearly scaled based on the energy of the image:

Higher energy endpoint

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Example of climbing image results vs. standard NEB results

Example of variable springs results vs. standard NEB results

Reaction coordinate

Rel

ati

ve

En

erg

y

Variable springs

Fixed springs

Reaction coordinate

Rel

ati

ve

En

erg

y

Climbing image

Standard

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NEB paths can be difficult to converge: when the force parallel to the MEP is large compared to the force perpendicular, when there are inflection points. Kinks can form on the band and oscillate.

Improved tangent method revises the definition of the tangent from:

A slight improvement ensures equispaced images even in areas of large curvature:

LEPS potential + harmonic oscillator with NEB path in dashed line showing kinks

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Adding in perpendicular spring force when the angle between the vectors between neighboring images (Ri-Ri-1 vs Ri+1-Ri) deviated from zero helps.

However, this leads to corner cutting on the MEP in curved regions. If the saddle point is in a curved region, then the saddle point energy will be overestimated.

Improved tangent method:Use only the tangent to the image that’s higher in energy:

Special case where image I is at maximum or minimum:

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Modified tangent NEB converges on the LEPS+H.O. potential

Modified tangent smooths out convergence.

Can also show that there’s a stability criterion for kinking in the path. The following must be satisfied:

F < 2CR

Where F is the perpendicular force to the path and C is the curvature around the MEP. Path will always become unstable for enough images because R goes down for large # images.

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Lecture 6String method

The string method finds a minimum energy path ( *f ) connecting points A+B. The MEP satisfies the constraint:

The force perpendicular to the path is zero.

The string method is similar to the NEB method – it propagates a path of images, e.g.

where a is the normalized arc length between A and B.

There’s a constraint that the parameterization is preserved when the string deforms or the local arc length is constant along the string:

or

This means that the elastic energy in the string is distributed evenly.

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Using the constraint developed for the string along with the definition of the MEP, we can rewrite the expression for the MEP:

Where t is the tangent vector and l is a Langrange multiplier that imposes the constraint.

In order to propagate a string towards the MEP is to carry out steepest descent dynamics in the string space:

The expression for NEB written in the same terminology is:

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Can propagate the string with damped dynamics approach instead to ensure rapid convergence:

where gs is a friction coefficient. Can propagate these with Verlet algorithm:

Where we have a modified friction factor and force:

The values of the Lagrange multipliers can be solved iteratively, based on the fact that the distance between images is constant and the expression for propagation.

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• Main difference between NEB and string is that the images get repositioned at every iteration in the string method.

• Extensions to the string method include:– Growing string method: start two strings – one from

reactants and one from products. Grow them until they join and then propagate towards MEP.

– Frozen string method: Same as growing string but once images are added, they are frozen. More efficient than growing string.

– Finite temperature string: incorporates temperature effects.

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Lecture 6Further considerations

• Good coordinate choices are very important for transition state optimization

• Good coordinate choice can enlarge the convergence region, reducing requirement of a good starting geometry. Poor coordinates, decrease convergence region.

• Eigenvector following methods work for stiff systems – small eigenvalues in Hessians will be a problem.

• Small eigenvalues/soft modes in the Hessian are better suited to path-based methods.

• Transition state may be a minimum for a certain symmetry (vs. the symmetry of the reactants and products).

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Lecture 6Dynamic free energy methods

• For large systems, it may be difficult to optimize transition states via the methods we’ve discussed so far.

• Instead, may want to carry out simulation dynamically.

• There’s also the consideration of free energy (G), which has been excluded so far – only have looked at enthalpy (H).

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Lecture 6Umbrella sampling

Direct dynamics will not sample high-energy regions near the TS if there’s an activation energy.

Need to bias the sampling, use a penalty to the potential to force sampling of high energy regions:

The biasing potential can be made to be sufficiently steep, so the energy far from r0 is very high in energy. Then will only sample near r0.

This method is known as umbrella sampling.

The ensemble is not Boltzmann but it can be deconvoluted.

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Lecture 6Umbrella sampling

Reaction coordinate

Pro

bab

ilit

y

Umbrella sampling locally enhances probability in low-probability regions by altering the potential.

Perform many biased simulations at different positions along reaction path, calculate free energy – called Potential of Mean Force:

this is done by unweighting and stitching together the underlying free energy function.

Weighted histogram-analysis method (WHAM) is a common approach to stitching together the free energy: deconvolute contribution of force constant from curvature of underlying free energy surface.

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Lecture 6Steered molecular dynamics

Steered molecular dynamics is a technique where constant force or velocity is applied to select atoms to move them from one configuration to another.

A set of collective variables is chosen to define progress along a reaction coordinate, e.g. proton transfer in malonaldehyde:

Collective variable: dist(1)-dist(2).

Once reaction coordinate is propagated, can get PMF:

Guiding potential:

Work:PMF:

(1)

(2)

May need to run several (hundreds?) of SMD runs to get accurate PMF.

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Lecture 6Metadynamics

Introduced in 2002 by Laio and Parrinello, metadynamics is a way to speed up dynamics and sample free energy landscapes of complex systems.

Collective variables (up to 3, typically 2) are selected to define the free energy landscape, e.g. coordination number of relevant species participating in a reaction, bond distances or angles, etc. The most challenging aspect of metadynamics is choosing the right collective variables.

A biasing Gaussian potential is added to the real potential:

New Gaussian is added at every tG in space of collective variables, S are the coordinates, s is the collective variable, w is the height of the Gaussian, ds is the width of the Gaussian.

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Lecture 6Metadynamics

The potential wells are gradually filled up minima until the full landscape is traversed.

The free energy surface is obtained by taking the negative image of all of the Gaussians.

If Gaussians are too large or added too frequently, free energy surface will be inaccurate, Gaussians too small or infrequently added – will take too long for simulation to finish.

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Lecture 6Summary

• For small systems with simple reaction coordinates: coordinate driving, linear synchronous transit, quadratic synchronous transit, or synchronous transit-guided-Quasi-Newton can be good approaches.

• For intermediate systems: string method or nudged elastic band

• For large systems with many DOF or to compute free energies: steered MD, metadynamics, umbrella sampling

• Coordinate selection and initial guess for a reaction pathway are almost always challenging.

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Lecture 6Survey!

bit.ly/lec-6

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Lecture 6Transition state theory

Reaction coordinate

Ene

rgy

Ea

kBT

ReactantsProducts

Transition State

Activation energy is one piece of the puzzle – full exploration of TST allows us to calculate absolute rates, through prefactor calculations.

A potential energy surface connects the reactants and products: this is a QM concept!

The activated complex or transition state is at the maximum potential energy. Once configuration is achieved, fast decay to products.

Concentration of the transition state is dependent upon equilibrium with reactants, vibration along the path.

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Lecture 6The Eyring equation

Entropy of activation term

Enthalpy of activation term

Macroscopic Eyring equation:

Microscopic Eyring equation:

These are the molecular partition functions of A and B and the activated complex, AB.

The Eyring equation is a rigorous method of obtaining kinetics from free energy of activation and may be derived from statistical mechanics (not here):

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Lecture 6The Eyring equation and TST

In the derivation of the Eyring equation (not shown), one vibrational frequency is stripped out of the molecular partition function for the activated complex.

This is assumed to be the mode that carries the complex over the barrier to the products.

Also: note that TST assumes once the activated complex is formed, it readily proceeds to products and does not cross back, which can be inaccurate.

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Lecture 6Interpretation of

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The potential energy surface is represented by changes in the internal coordinates of the molecule(s). These correspond to vibrational modes that alter internal coordinates or translation of the reactants with respect to each other.

At the TS, we have one “molecule” and motion is characterized by the vibrational modes.

One imaginary vibrational frequency characterizes the transition state and correlates to the attempt frequency in our derivation of the Eyring equation.

Imaginary frequency of the transition state for HCNHNC isomerization

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Lecture 6Partition function review

#

Maxwell-Boltzmann

E

Fraction molecules in state i.

b= 1/kBT

energy of the state

Boltzmann distribution

Molecular partition function

sum over

j states:

sum over

i levels:

1 2 3 4 5 6 1 2 3 4

5

6 g=5

g=1

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Lecture 6Molecular partition function

contributions are derived from solving eigenvalue problems in quantum mechanics.

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Lecture 6Molecular degrees of freedom

Linear Nonlinear

Translational 3 3

Rotational 2 3

Vibrational 3N-5 3N-6

For zero field, potential energy derived only from vibrational coordinates.

e.g. H2O: 3 normal modes

symmetric stretch asymmetric stretchbend

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Lecture 6Vibrational zero-point energy

From Q.M., molecules have zero point energy:

ZPE

Reaction coordinate

+ B P

AB‡

A

E0

Ea

Ene

rgy

Enthalpy of activation:

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Lecture 6Contributions to prefactors

DOF form f(TX)

if

if

Order

½

½

0

11

10

108 cm-1

53

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Lecture 6

Prefactors from quantum chemistry

• Must have structures that are at minima or saddle points – vibrationally characterize them to get zero point energy and vibrational partition function.

• ZPE corrects the enthalpy of activation.• Rotational partition function may also be calculated

and its contribution is added to the partition functions.

• Electronic terms are neglected.• Translational terms only depend on temperature.

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Lecture 6Summary

• It is possible to use a saddle point structure and minimum energy structures to obtain absolute rates of reaction by calculating prefactors from statistical mechanics.

• However, relative rates, where the prefactors are assumed to be the same for different reactions are typically preferred unless the method is extremely accurate.

Lecture6

Science

true ts

saddle points saddle

reaction coordinates

fixed variables

remaining variables

freaction variables

small systems

approximate reaction