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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

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