Perturbation Theory

1
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

2
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

3
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

4
Perturbation Theory

• The hope is that the series converges in few
  terms
• Plug into TISE and equate terms with like powers
  of l

5
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

6
Perturbation Theory

• Multiply both sides by ym(0) and integrate over
  all space

7
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

8
Perturbation Theory

• Combining both equations
• We may now also obtain the second-order
  correction to the energy

9
Perturbation Theory

• 2nd order energy correction

10
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

11
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

12
Perturbation Theory

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

13
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

14
Perturbation Theory

• Molecular Energies Water
• Reaction energies

15
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)

16
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)

17
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