View by Category

Loading...

PPT – Perturbation Theory PowerPoint presentation | free to view

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Perturbation Theory

- Due to errors in molecular energies, and other

properties, it is advantageous to attempt to

recover the correlation energy of the molecule - Our second model chemistry is Many-Body

Perturbation Theory (MBPT) - We cannot solve the molecular TISE exactly, but

we can solve a simplified version of it - Leads directly to mathematical field of

perturbation theory - Idea is that the Hamiltonian operator for the

true, unsolvable problem is only slightly

different from the Hamiltonian operator of a

problem we can solve

Perturbation Theory

- Perturbation is applied gradually, to allow for

continuous transformation from solutions of

unperturbed problem to solutions of true problem - Wed like to solve
- Since the Hamiltonian is a function of parameter

l, the wave function yn and the energy En are

also - We can express yn and En as power series in l

Perturbation Theory

- When l0, yn and En go to the unperturbed yn and

En - y n(0), En(0)
- Also, introduce the following notation
- We can write series as

Perturbation Theory

- The hope is that the series converges in few

terms - Plug into TISE and equate terms with like powers

of l

Perturbation Theory

- Before we can solve the l1 problem, well make a

few assumptions - ltyn(0)ym(0)gt dmn
- ltyn(0)yngt 1 By inserting power series of yn

, this leads to a different condition - ltyn(0)yn(k)gt d0,k
- Need to define Hermetian operators
- If ltyAygt ltyAygt for all well-behaved y is

called Hermetian - Expectation value is a real number
- ltHgt is the system energy, a real number, so H is

Hermetian - One trick with Hermetian operators is that
- ltyiAyjgt ltyjAyigt

Perturbation Theory

- Multiply both sides by ym(0) and integrate over

all space

Perturbation Theory

- If mn, we have the 1st order correction to the

energy - If m does not equal n, we can find the 1st order

correction to the wave function - Express yn(1) as a linear combination of the

unperturbed wave functions

Perturbation Theory

- Combining both equations
- We may now also obtain the second-order

correction to the energy

Perturbation Theory

- 2nd order energy correction

Perturbation Theory

- Further refinements to En or yn require much more

complicated expressions - If extended to infinite order, would be exact.
- Always truncated at some small number
- Goes by the name MPx, or xth order Moller-Plesset

theory, when using sum of Fock operators for the

unperturbed Hamiltonian - x refers to the order of the energy correction
- an ith order correction to the wave function is

sufficient to obtain a (2i1)th order correction

to the energy - For most of the life of computational chemistry,

this was the most popular post H-F method to

attempt to recover electron correlation - E(0) E(1) is EH-F, so E(2) is first

correction to the H-F energy

Perturbation Theory

- Performance of MP2 Dipole moments
- Better description of electron clustering

(electron correlation) gives rise to better

predictions of the dipole moments - All calculations performed at optimum geometry

for given method and basis set - This is not a geometric phenomenon

Perturbation Theory

- Performance of geometry optimizations
- Water
- More General
- MP2/6-31g

Perturbation Theory

- Generally MP2 gives better geometries than does

H-F for sufficiently large basis sets - Small basis sets used with correlation methods

still give rise to large errors - For some compounds, additional computational

expense is not worth it - H-F gives good geometries much cheaper
- Harmonic Frequencies
- Water

Perturbation Theory

- Molecular Energies Water
- Reaction energies

Perturbation Theory

- In general, where H-F does well, MP2 may not be a

significant improvement (geometries, isodesmic

reactions) - When H-F fails, MP2 can be a significant

improvement (dipoles, dissociation reactions)

Perturbation Theory

- Conclusions
- MPx corrects most of the flaws in H-F theory
- MPx has good quantitative accuracy even for the

first term of the series, MP2 - Price you pay is much more computationally

intensive than H-F theory (formal N5 scaling) - May not be practical to apply to the system of

interest - Subsequent terms in the series (MP3, MP4,...)

scale even more severely - MP2 most frequently used post-H-F correlation

method - In practice, MP3, MP4, ... are used only to

evaluate energies (geometries and frequencies at

some lower level)

Perturbation Theory

- There is no accepted scaling factor for MP2

frequencies - Relative timings and disk usages
- All calculations use 6-31g basis set, on a

RS/6000 - Numbers should not be taken as absolute, as MP2

has many different algorithmic flavors to run in

differing amounts of disk space. (Minimum is

7N4/16 words, N of basis functions, 1

word8 bytes) - Disk space is usual limiting factor

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Page of

CrystalGraphics Sales Tel: (800) 394-0700 x 1 or Send an email

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2014 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2014 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Perturbation Theory" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!

Committed to assisting Juniata University and other schools with their online training by sharing educational presentations for free