Electron Correlation Wave Funtion Methods

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Electron Correlation Wave Funtion Methods
Computational Chemistry 5510 Spring 2006 Hai Lin
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Electron Correlation Energy

• Electrons avoid each other

Probability of finding another electron
Intra-orbital correlation
r(x)
Inter-orbital correlation
x
For Coulomb hole For Fermi hole

• Electron correlation energy

Ecorr Eexact - EHF

• HF calculations do not account for electron
  correlation.
• We only discuss RHF (and ROHF) if not otherwise
  indicated.

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Excited Slater Determinants

• Replace occupied MO by unoccupied MO in the
  determinant in the HF calculation.

fj(x1)
fk(x1)
fi(x1)
fj(x2)
fk(x2)
fi(x2)
F(x1, x2, , xN ) (1/N!)½

fj(xN)
fk(xN)
fi(xN)
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Multi-determinant Trial Wave Function

• More determinants, a better many-electron wave
  function

Complete CI expansion Exact solution to
electronic Schrödinger equation
Full CI expansion Best many-electron WF
Number of determinants
Complete basis set limit Best one-electron MO
HF
Basis set size
Minimum Basis set
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How Many Determinants?

• Number of determinants goes dramatically.
• H2O with 6-31G(d) basis 10 electrons in 38
  orbitals
• Total of determinants 38! / 10!(38-10)!
• Total of singlet (S ½) determinants 3?107
• H2O with 6-311G(2d,2p) basis 10 electrons in 82
  orbitals
• Total of determinants 82! / 10!(82-10)!
• Total of singlet (S ½) determinants 1?1011
• C2H4 with 6-31G(d) basis 16 electrons in 76
  orbitals
• Total of determinants 76! / 16!(76-16)!
• Total of singlet (S ½) determinants 3?1014

E 0

• Frozen core and frozen virtuals approximations

O 1s2 2s2 2p4 3s0 3p0 3d0 4s0 4p0
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Electron Correlation Methods

• How many and which determinants to be included?
• How to construct those determinants?
• How to determine the expansion coefficients ai?
• Configuration interaction (CI)
• Multi-configuration self-consistent field (MCSCF)
• Many-body Moller-Plesset perturbation theory
  (MPn)
• Coupled cluster (CC)

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Configuration Interaction

• Determine the expansion coefficients ai
  variationally.
• Full-CI is feasible only for small system with
  small basis sets.
• Truncated CI at singly and doubly excitations
  (CISD) is the most popular CI method.

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MCSCF

• Variationally not only optimize the expansion
  coefficients ai, but also optimize the MOs (F)
  themselves like HF does. That is, optimize the
  coefficients both between and within F.
• Which MOs to be included in the expansion
  requires your chemical insights.

E 0

• Complete active space self-consistent field
  (CASSCF) performs MCSCF calculations with a
  full-CI expansion for a selected active space.
• n,m-CASSCF n electrons in m orbitals of the
  active space

Full-CI expansion for the active space
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MCSCF vs. HF

• Both HF and MCSCF optimize MOs.
• HF uses single determinant, while MCSCF uses many
  determinants.
• Advanced electron correlation methods
  constructing wave function based on
• HF optimized MOs single-reference methods,
  e.g., CI.
• MCSCF optimized MOs multi-reference methods,
  e.g., MR-CI.
• When single-determinant HF fails to provide a
  qualitatively correct zero-order description, you
  should try MCSCF.

O
O
?O
O
O?
O-
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Perturbation Theory

• Devide the exact Hamiltonian H into two parts An
  unperturbed reference H0 and a pertubation H'.

H H0 H'
For the reference Hamiltonian operator H0, the
solution is known.
H0 Fi0 Ei0 Fi0, (i 0, 1, 2, ...)
For the exact Hamiltonian operator H, the
solution can be obtained
H Yi ei Yi
Yi Fi0 Fi1 Fi2 ...
Corrections
ei Ei0 Ei1 Ei2 ...
The corrections can be expressed in terms of H',
Fi0 and Ei0.
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MPn

• Devide the exact electronic Schrödinger
  Hamiltonian H into two parts

H H0 H'
Sum of the Fock operators SFi
All the remaining terms ? Fluctuation potential
Corrections to the n-th order are known as MPn
H Yi ei Yi
Yi Fi0 Fi1 Fi2 ...
Corrections
ei Ei0 Ei1 Ei2 ...

• Not a variational method ? the energy can be
  lower than the exact energy.
• Not necessarily converge with increasing n.

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Coupled Cluster Methods

• Most advanced methods
• Rather reliable if work with large basis sets
  (e.g., cc-pVQZ)
• Commonly used CCSD, CCSD(T)
• Often used as reference calculations to assess
  and/or calibrate other less-expensive methods
  (e.g., DFT)

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Size Consistency

• 2EA ? EB

CIS two excitations
CIS one excitation
100 Å
? 2

• Truncated CI methods (e.g., CIS) are not size
  consistent.
• Full-CI is size consistent.
• HF, DFT, MPn, and CC methods are size consistent.

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Which Method to Use?

• When single-determinant HF fails to provide a
  qualitatively correct zero-order wavefunction,
  you should try MCSCF.
• Truncated CI is not size consistent, and is
  usually not recommended if other methods are
  feasible.
• Excited states are difficult to treated by MPn
  and CC methods, and MCSCF and CI (especially
  MR-CI) are the choice.
• CCSD(T) with sizeable basis sets may reach
  chemical accuracy ( 1kcal/mol).

MP2
CISD, CCSD
MP4
CCSD(T)
HF
Accuracy computational cost
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Summary

• Electron Correlation
• Excited Slater determinants
• Multi-determinant trial wave function
• Electron Correlation Methods
• CI
• MCSCF
• MPn
• CC
• Size-consistency

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Your Homework

• Read the slides.
• Read textbook (Take notes when you read.)
• 4 pages 98 102, 117 122, 144 146
• Skip the discussion on UHF, natrual orbital, and
  localized orbitals methods.
• Questions
• What is a S-type excited Slater determinant?
  D-type?
• What is a frozen core approximation?
• What is the major difference between CI and
  MCSCF?
• What is a single-reference method and what is a
  multi-reference method?