SlideShare a Scribd company logo
1 of 32
Download to read offline
Electron Correlation
Albert DeFusco
March 26, 2015
Hartree-Fock (SCF) methods
What it does well
Geometries
What it doesn’t do well
Energies
No electron correlation
The Fock operator assumes an average electron-electron repulsion
Each orbital energy is independent of another
Average interaction is optimized through the SCF procedure
Perturbation Theory
Corrections to the Hartree-Fock energy and wavefunction
Requires that the corrections be small
HF wavefunction and energy must be pretty good
Møller-Plesset perturbation (MP2, MP3, etc.)
Perturbations are not convergent
Coupled Cluster
CCSD, CCSD(t), CCSDT, CCSDTQ, etc.
Perturbations are guaranteed to converge
Correlated electron motion
He atom
ˆH = −
1
2
2
1 −
Z
r1Z
−
1
2
2
2 −
Z
r2Z
+
1
r1,2
This is not the Hartree-Fock Hamiltonian
There is no analytic solution
We must use hyrdrogen atom solutions as a guide
The Hamiltonian diverges at r1,2 = 0
r1,2 = r1 − r2
r1Z = r1 − RZ
r2Z = r2 − RZ
and at the nuclei ri = 0.
The kinetic energy must also diverge to achieve a finite sum.
A determinant is a wavefunction
Ψ(x1, x2, . . . , xn) =
1
√
n!
χ1(x1) χ2(x1) · · · χn(x1)
χ1(x2) χ2(x2) · · · χn(x2)
...
... · · ·
...
χ1(xn) χ2(xn) · · · χn(xn)
Rows are electron positions and spins
Columns are spin orbitals
Ψ(x1, x2, . . . ) = −Ψ(x2, x1, . . . )
Exchange either two rows or two columns: antisymmetric
Probability density
Ψ(x1, x2) = χ1(x1)χ2(x2)
P(r1, r2) = dω1dω2 |Ψ|2
dr1dr2
The two-electron density
Probability of finding electron 1 at r1 in a volume dr1 while
simultaneously finding electron 2 at r2 in a volume dr2.
The helium atom: ground state singlet wavefunction
Hartree-Fock single determinant for the 11
S state
Minimal 1-electron basis: 2 1s restricted spin orbitals
Ψ(x1, x2) =
1
√
2
ψ1s(r1)α(ω1) ψ1s(r1)β(ω1)
ψ1s(r2)α(ω2) ψ1s(r2)β(ω2)
= 2− 1
2 (ψ1s(r2)β(ω2)ψ1s(r1)α(ω1)
−ψ1s(r2)α(ω2)ψ1s(r1)β(ω1))
|Ψ|2
=
1
2
. . .
= ψ1s(r1)
2
ψ1s(r2)
2
The spatial orbital is identical for α and β spin
Spatial motion in uncorrelated
The helium atom: triplet wavefunction
Hartree-Fock single determinant for the 23
S state
Minimal 1-electron basis: 1s and 2s spin orbitals
Ψ(x1, x2) =
1
√
2
ψ1s(r1)α(ω1) ψ2s(r1)α(ω1)
ψ1s(r2)α(ω2) ψ2s(r2)α(ω2)
= 2− 1
2 (ψ2s(r2)α(ω2)ψ1s(r1)α(ω1)
−ψ1s(r2)α(ω2)ψ2s(r1)α(ω1))
|Ψ|2
=
1
2
. . .
= ψ1s(r1)
2
ψ2s(r2)
2
+ ψ1s(r2)
2
ψ2s(r1)
2
Parallel spin electrons have Fermi correlation
Cusp conditions
Figure: Hylleraas Hamiltonian for Helium with relative coordinates.(Figure from
Fred Manby)
electron-nucleas cusp
limri →0
∂Ψ
∂ri ave
= −ZΨ(ri = 0)
Exists in Slater orbitals, but approximated in Gaussian orbitals
More angular momentum means a sharper cusp
electron-electron cusp
limr1,2→0
∂Ψ
∂r1,2 ave
= 1
2 Ψ(r1,2 = 0)
Leads to a depletion in in the two-electron density at r1,2
Hylleraas wavefunction
Figure: Hartree-Fock wavefunction and
Hylleraas wavefunction. (Figure from Fred Manby)
One slater determinant is not
enough
To help electrons “avoid” each
other, we can distribute them
among more orbitals using
many determinants
Helium determinants: double zeta basis
Singlet Helium
ψ2s is a “virtual” or unoccupied spin orbital
Ψ2 is a “doubly excited” determinant
CI Singles and Double (CISD)
Why no singles?
Ψ0 = ψ1s
¯ψ1s
Ψ2 = ψ2s
¯ψ2s
Ψ(x1, x2) = K1 ψ1s
¯ψ1s + K2 ψ2s
¯ψ2s
P(r1, r2) = K1ψ1s(r1)ψ1s(r2) + K2ψ2s(r1)ψ2s(r2)
2
lim
r1→r2
P(r1, r2) = K1 ψ1s(r1)
2
+ K2 ψ2s(r1)
2 2
Configuration Interaction
Solving for the determinant coefficients
Ψ = K0Ψ0 + K2Ψ2
H =
Ψ0|H|Ψ0 Ψ0|H|Ψ2
Ψ2|H|Ψ0 Ψ2|H|Ψ2
Ψ2|H|Ψ0 = 1¯1 2¯2 − 1¯1 ¯22
= K12
H
K0
K2
= ECISD
K0
K2
Helium convergence
ERHF = −2.85516 a.u.
Table: FCI/cc-pVQZ
Spatial Orbital CI coefficient
1s 0.995974
2s -0.039200
2p -0.028429
3d -0.005164
3f -0.001472
Table: Helium Full CI total energy J.
Chem. Phys. 127 , 224104 (2007)
Basis Set Energy (a.u.)
cc-pVDZ -2.8875948
cc-pVTZ -2.9002322
cc-pVQZ -2.9024109
cc-pV5Z -2.9031519
Exact Energy -2.9037225
The difference between Full CI and Hartree-Fock energies is
called the correlation energy
Helium cusp
Figure: Helium wavefunction cusps for cc-pVDZ,
cc-pVTZ, cc-pVQZ and cc-pV5Z basis sets from
Martin Sch¨utz
Slater determinant
expansions have even
powers of r1,2
(Helgaker)
Configuration Interaction
Exact when all possible determinants used with a complete basis
set
Full CI
Basis sets
HF has a factorizable two-electron density and requires fewer basis
functions to converge
Correlation methods seek to improve the two-electron density and
require much more functions to converge
Orbitals are frozen at the HF solution
More than two electrons
Excited determinants and matrix elements (Slater Rules)
Single excitations are zero by Brilloun’s theorem
Determinants differening by one spin orbital
Ψ|H|Ψp
m = m ˆh p +
N
n
( mn|pn − mn|np )
Determinants differing by two spin orbitals
Ψ|H|Ψpq
mn = mn|pq − mn|qp
determinants differing by more than two spin orbitals contributes 0
H =


Ψ0 H Ψ0 0 Ψ0 H Ψ2 0 0 · · ·
Ψ1 H Ψ1 Ψ1 H Ψ2 Ψ1 H Ψ3 0 · · ·
Ψ2 H Ψ2 Ψ2 H Ψ3 Ψ2 H Ψ4 · · ·
Ψ3 H Ψ3 Ψ3 H Ψ4 · · ·
Ψ4 H Ψ4 · · ·
...


Expansions
Slow convergence with excitation
Scaling problem
Full CI grows factorially
Many billion determinant Full CI is possible, but only small
molecules
Truncation CI expansions
CISD: CI Singles and Doubles
CISDT: With triple excitations
CISDTQ: With quadruple excitations
QCISD is closely related to coupled-cluster
What is still missing?
Bond breaking
Including transition states
Diradicals
Degenerate orbital pictures
H2 dissociation
UHF
Not smooth due to spin
contamination
RHF
MP2 likely to fail
H2 dissociation determinants
A single determinant has a spurious J integral at long range
ψσ = φA + φB
ψσ∗ = φA − φB
Eeq = 2hσσ + Jσσ
E∞ = 2hσσ + Jσσ = 2hσ∗σ∗ + Jσ∗σ∗
H2 dissociation determinants
Full CI minimum basis
Separate correlation
Dynamic correlation
Short range cusp conditions
Perturbation theories are very efficient
Requires large basis sets
Static correlation
Degeneracies in orbitals or determinants
“chemical intuition”
Full CI does both
Diradicals
What is the Lewis structure of ozone?
Ozone
What are the molecular orbitals?
Each oxygen supplies 1s, 2s and 3 2p orbitals
Hartree-Fock will only fill the first 12 orbitals
The HOMO energy is -0.4810 a.u.
The LUMO energy is -0.0299 a.u.
Ozone
Ozone
Distribute 4 electrons in the 3 π orbitals
Full CI = CISD
0.929
-0.224
-0.224
-0.166
-0.066
0.066
Determinant weights for Hartree-Fock orbitals
Ozone
Multi-configurational Self Consistent Field (MCSCF)
Improve the energy by minimizing the orbitals
New definition of Fock matrix
Fully Optimized Reaction Space (FORS)
Complete Active Space Self Consistent Field (CAS)
Full CI within a “chosen” orbital space
MCSCF does not imply CASSCF
Ozone
Determinant weights for CASSCF optimized orbitals
0.864
-0.240
-0.240
-0.295
-0.159
0.159
What about the orbitals?
Hartree-Fock orbitals have little physical meaning
DFT orbitals even less
Orbital energies may be helpful to pick active space
Determinants define electron configurations
Natural orbitals have populations
Diagonalize the density matrix
U are orbitals and n are electron occupations
Ψ =
k
Ak Ψk
P = Ψ∗
Ψ
PU = nU
Ozone
Electron populations in natural orbitals for the active space
1.993 1.598 0.409
Dynamic correlation
CASSCF has no dynamic correlation
Similar to Hartree-Fock
Multi-reference CI (MR-CISD)
Very accurate for excitation energies
Still not size-consistent
Alternatives to MR-CI expansions, perturbation theory
MRPT (MRMP), CASPT, MCQDPT
MRCC: Multiple types
Good for potential energy surfaces
Pople Diagram
Figure: From Chem. Soc. Rev., 2012,41, 6259-6293

More Related Content

What's hot

Density Functional Theory
Density Functional TheoryDensity Functional Theory
Density Functional TheoryWesley Chen
 
Quantum Theory of Spin and Anomalous Hall effects in Graphene
Quantum Theory of Spin and Anomalous Hall effects in Graphene Quantum Theory of Spin and Anomalous Hall effects in Graphene
Quantum Theory of Spin and Anomalous Hall effects in Graphene Mirco Milletari'
 
5 introduction to quantum mechanics
5 introduction to quantum mechanics5 introduction to quantum mechanics
5 introduction to quantum mechanicsSolo Hermelin
 
History of Quantum Mechanics
History of Quantum MechanicsHistory of Quantum Mechanics
History of Quantum MechanicsChad Orzel
 
Intro. to quantum chemistry
Intro. to quantum chemistryIntro. to quantum chemistry
Intro. to quantum chemistryRawat DA Greatt
 
Band theory of solids
Band theory of solidsBand theory of solids
Band theory of solidsutpal sarkar
 
Fermi dirac basic part 1
Fermi dirac basic part 1Fermi dirac basic part 1
Fermi dirac basic part 1kiranmadhale1
 
Quick and Dirty Introduction to Mott Insulators
Quick and Dirty Introduction to Mott InsulatorsQuick and Dirty Introduction to Mott Insulators
Quick and Dirty Introduction to Mott InsulatorsABDERRAHMANE REGGAD
 
Electronic structure of strongly correlated materials
Electronic structure of strongly correlated materialsElectronic structure of strongly correlated materials
Electronic structure of strongly correlated materialsABDERRAHMANE REGGAD
 
Physics barriers and tunneling
Physics barriers and tunnelingPhysics barriers and tunneling
Physics barriers and tunnelingMohamed Anwar
 

What's hot (20)

Origin of quantum mechanics
Origin of quantum mechanicsOrigin of quantum mechanics
Origin of quantum mechanics
 
Density Functional Theory
Density Functional TheoryDensity Functional Theory
Density Functional Theory
 
Quantum Theory of Spin and Anomalous Hall effects in Graphene
Quantum Theory of Spin and Anomalous Hall effects in Graphene Quantum Theory of Spin and Anomalous Hall effects in Graphene
Quantum Theory of Spin and Anomalous Hall effects in Graphene
 
5 introduction to quantum mechanics
5 introduction to quantum mechanics5 introduction to quantum mechanics
5 introduction to quantum mechanics
 
History of Quantum Mechanics
History of Quantum MechanicsHistory of Quantum Mechanics
History of Quantum Mechanics
 
NANO266 - Lecture 4 - Introduction to DFT
NANO266 - Lecture 4 - Introduction to DFTNANO266 - Lecture 4 - Introduction to DFT
NANO266 - Lecture 4 - Introduction to DFT
 
Tight binding
Tight bindingTight binding
Tight binding
 
THE HARTREE FOCK METHOD
THE HARTREE FOCK METHODTHE HARTREE FOCK METHOD
THE HARTREE FOCK METHOD
 
Intro. to quantum chemistry
Intro. to quantum chemistryIntro. to quantum chemistry
Intro. to quantum chemistry
 
Dft presentation
Dft presentationDft presentation
Dft presentation
 
Ising model
Ising modelIsing model
Ising model
 
CHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics IICHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II
 
Exercises with DFT+U
Exercises with DFT+UExercises with DFT+U
Exercises with DFT+U
 
Band theory of solids
Band theory of solidsBand theory of solids
Band theory of solids
 
Fermi dirac basic part 1
Fermi dirac basic part 1Fermi dirac basic part 1
Fermi dirac basic part 1
 
Quick and Dirty Introduction to Mott Insulators
Quick and Dirty Introduction to Mott InsulatorsQuick and Dirty Introduction to Mott Insulators
Quick and Dirty Introduction to Mott Insulators
 
Lecture7
Lecture7Lecture7
Lecture7
 
Hartree fock theory
Hartree fock theoryHartree fock theory
Hartree fock theory
 
Electronic structure of strongly correlated materials
Electronic structure of strongly correlated materialsElectronic structure of strongly correlated materials
Electronic structure of strongly correlated materials
 
Physics barriers and tunneling
Physics barriers and tunnelingPhysics barriers and tunneling
Physics barriers and tunneling
 

Viewers also liked

Isotop, isobar dan isoton
Isotop, isobar dan isotonIsotop, isobar dan isoton
Isotop, isobar dan isotonABINUL HAKIM
 
ISOTOP, ISOTON DAN ISOBAR
ISOTOP, ISOTON DAN ISOBARISOTOP, ISOTON DAN ISOBAR
ISOTOP, ISOTON DAN ISOBARHome
 
Atomic theory timeline
Atomic theory timelineAtomic theory timeline
Atomic theory timelineramatboy
 
Chapter 22.2 : Radioactive Decay
Chapter 22.2 : Radioactive DecayChapter 22.2 : Radioactive Decay
Chapter 22.2 : Radioactive DecayChris Foltz
 
Types of radioactive decay
Types of radioactive decayTypes of radioactive decay
Types of radioactive decayDavid Graff
 
Isotopes And Radioactivity 09
Isotopes And Radioactivity 09Isotopes And Radioactivity 09
Isotopes And Radioactivity 09Paula Mills
 
Radioactivity + isotopes lect.1,2,
Radioactivity + isotopes lect.1,2,Radioactivity + isotopes lect.1,2,
Radioactivity + isotopes lect.1,2,kamal2200
 
isotop, isobar, isoton
isotop, isobar, isotonisotop, isobar, isoton
isotop, isobar, isotonberlianablue
 
Chapter 2 basic structure concepts
Chapter 2  basic structure conceptsChapter 2  basic structure concepts
Chapter 2 basic structure conceptsSimon Foo
 
Isotope isotone isobar
Isotope isotone isobarIsotope isotone isobar
Isotope isotone isobarFendi Ard
 
Isotopes, isotones, and isobars
Isotopes, isotones, and isobarsIsotopes, isotones, and isobars
Isotopes, isotones, and isobarsEvina Nathalia
 
Option C Nuclear Physics, Radioactive decay and half life
Option C Nuclear Physics, Radioactive decay and half lifeOption C Nuclear Physics, Radioactive decay and half life
Option C Nuclear Physics, Radioactive decay and half lifeLawrence kok
 
Models of the Atom
Models of the AtomModels of the Atom
Models of the AtomOhMiss
 
Nuclear Power Taja
Nuclear Power TajaNuclear Power Taja
Nuclear Power TajaRosalmara
 
DIFFERENT ATOMIC MODELS
DIFFERENT ATOMIC MODELSDIFFERENT ATOMIC MODELS
DIFFERENT ATOMIC MODELSShahn Tee
 
Atomic Theory power point cscope
Atomic Theory power point cscopeAtomic Theory power point cscope
Atomic Theory power point cscopeJenny Dixon
 
Nucleus Ppt
Nucleus PptNucleus Ppt
Nucleus Pptiyss0104
 

Viewers also liked (20)

Isotop, isobar dan isoton
Isotop, isobar dan isotonIsotop, isobar dan isoton
Isotop, isobar dan isoton
 
ISOTOP, ISOTON DAN ISOBAR
ISOTOP, ISOTON DAN ISOBARISOTOP, ISOTON DAN ISOBAR
ISOTOP, ISOTON DAN ISOBAR
 
Atomic theory timeline
Atomic theory timelineAtomic theory timeline
Atomic theory timeline
 
Chapter 22.2 : Radioactive Decay
Chapter 22.2 : Radioactive DecayChapter 22.2 : Radioactive Decay
Chapter 22.2 : Radioactive Decay
 
Types of radioactive decay
Types of radioactive decayTypes of radioactive decay
Types of radioactive decay
 
Isotopes And Radioactivity 09
Isotopes And Radioactivity 09Isotopes And Radioactivity 09
Isotopes And Radioactivity 09
 
Radioactivity + isotopes lect.1,2,
Radioactivity + isotopes lect.1,2,Radioactivity + isotopes lect.1,2,
Radioactivity + isotopes lect.1,2,
 
isotop, isobar, isoton
isotop, isobar, isotonisotop, isobar, isoton
isotop, isobar, isoton
 
Nuclear Materials
Nuclear MaterialsNuclear Materials
Nuclear Materials
 
Chapter 2 basic structure concepts
Chapter 2  basic structure conceptsChapter 2  basic structure concepts
Chapter 2 basic structure concepts
 
Isotope isotone isobar
Isotope isotone isobarIsotope isotone isobar
Isotope isotone isobar
 
Isotopes, isotones, and isobars
Isotopes, isotones, and isobarsIsotopes, isotones, and isobars
Isotopes, isotones, and isobars
 
Option C Nuclear Physics, Radioactive decay and half life
Option C Nuclear Physics, Radioactive decay and half lifeOption C Nuclear Physics, Radioactive decay and half life
Option C Nuclear Physics, Radioactive decay and half life
 
Models of the Atom
Models of the AtomModels of the Atom
Models of the Atom
 
RADIOACTIVE DECAY AND HALF-LIFE CONCEPTS
RADIOACTIVE DECAY AND HALF-LIFE CONCEPTSRADIOACTIVE DECAY AND HALF-LIFE CONCEPTS
RADIOACTIVE DECAY AND HALF-LIFE CONCEPTS
 
Nuclear Power Taja
Nuclear Power TajaNuclear Power Taja
Nuclear Power Taja
 
Radioactivity.ppt
Radioactivity.pptRadioactivity.ppt
Radioactivity.ppt
 
DIFFERENT ATOMIC MODELS
DIFFERENT ATOMIC MODELSDIFFERENT ATOMIC MODELS
DIFFERENT ATOMIC MODELS
 
Atomic Theory power point cscope
Atomic Theory power point cscopeAtomic Theory power point cscope
Atomic Theory power point cscope
 
Nucleus Ppt
Nucleus PptNucleus Ppt
Nucleus Ppt
 

Similar to Introduction to Electron Correlation

Similar to Introduction to Electron Correlation (20)

Atomic Structure-21092023.pdf
Atomic Structure-21092023.pdfAtomic Structure-21092023.pdf
Atomic Structure-21092023.pdf
 
Atomic structure part 3/3
Atomic structure part 3/3Atomic structure part 3/3
Atomic structure part 3/3
 
Atomic structure part 2
Atomic structure part 2Atomic structure part 2
Atomic structure part 2
 
structure of atom crash course .pptx
structure of atom crash course .pptxstructure of atom crash course .pptx
structure of atom crash course .pptx
 
Pcv ch2
Pcv ch2Pcv ch2
Pcv ch2
 
NANO266 - Lecture 2 - The Hartree-Fock Approach
NANO266 - Lecture 2 - The Hartree-Fock ApproachNANO266 - Lecture 2 - The Hartree-Fock Approach
NANO266 - Lecture 2 - The Hartree-Fock Approach
 
Poster Icqc
Poster IcqcPoster Icqc
Poster Icqc
 
Intro-QM-Chem.ppt
Intro-QM-Chem.pptIntro-QM-Chem.ppt
Intro-QM-Chem.ppt
 
Intro-QM-Chem.ppt
Intro-QM-Chem.pptIntro-QM-Chem.ppt
Intro-QM-Chem.ppt
 
NANO266 - Lecture 3 - Beyond the Hartree-Fock Approximation
NANO266 - Lecture 3 - Beyond the Hartree-Fock ApproximationNANO266 - Lecture 3 - Beyond the Hartree-Fock Approximation
NANO266 - Lecture 3 - Beyond the Hartree-Fock Approximation
 
Chapter 7 notes
Chapter 7 notes Chapter 7 notes
Chapter 7 notes
 
Gnp ch103-lecture notes
Gnp ch103-lecture notesGnp ch103-lecture notes
Gnp ch103-lecture notes
 
Part i
Part iPart i
Part i
 
Ece5318 ch4
Ece5318 ch4Ece5318 ch4
Ece5318 ch4
 
EM_Theory.pdf
EM_Theory.pdfEM_Theory.pdf
EM_Theory.pdf
 
Hartree fock theory
Hartree fock theoryHartree fock theory
Hartree fock theory
 
IR Spectroscopy - MC.pptx
IR Spectroscopy - MC.pptxIR Spectroscopy - MC.pptx
IR Spectroscopy - MC.pptx
 
M sc sem iv u iii
M sc sem iv u iiiM sc sem iv u iii
M sc sem iv u iii
 
Limitations of Bohr's theory
Limitations of Bohr's theoryLimitations of Bohr's theory
Limitations of Bohr's theory
 
(S.h.m & waves).(reflection & refrection).(interferenc) prove s & important ...
(S.h.m & waves).(reflection & refrection).(interferenc)  prove s & important ...(S.h.m & waves).(reflection & refrection).(interferenc)  prove s & important ...
(S.h.m & waves).(reflection & refrection).(interferenc) prove s & important ...
 

Recently uploaded

Kochi ❤CALL GIRL 84099*07087 ❤CALL GIRLS IN Kochi ESCORT SERVICE❤CALL GIRL
Kochi ❤CALL GIRL 84099*07087 ❤CALL GIRLS IN Kochi ESCORT SERVICE❤CALL GIRLKochi ❤CALL GIRL 84099*07087 ❤CALL GIRLS IN Kochi ESCORT SERVICE❤CALL GIRL
Kochi ❤CALL GIRL 84099*07087 ❤CALL GIRLS IN Kochi ESCORT SERVICE❤CALL GIRLkantirani197
 
American Type Culture Collection (ATCC).pptx
American Type Culture Collection (ATCC).pptxAmerican Type Culture Collection (ATCC).pptx
American Type Culture Collection (ATCC).pptxabhishekdhamu51
 
SAMASTIPUR CALL GIRL 7857803690 LOW PRICE ESCORT SERVICE
SAMASTIPUR CALL GIRL 7857803690  LOW PRICE  ESCORT SERVICESAMASTIPUR CALL GIRL 7857803690  LOW PRICE  ESCORT SERVICE
SAMASTIPUR CALL GIRL 7857803690 LOW PRICE ESCORT SERVICEayushi9330
 
module for grade 9 for distance learning
module for grade 9 for distance learningmodule for grade 9 for distance learning
module for grade 9 for distance learninglevieagacer
 
Pests of cotton_Borer_Pests_Binomics_Dr.UPR.pdf
Pests of cotton_Borer_Pests_Binomics_Dr.UPR.pdfPests of cotton_Borer_Pests_Binomics_Dr.UPR.pdf
Pests of cotton_Borer_Pests_Binomics_Dr.UPR.pdfPirithiRaju
 
Labelling Requirements and Label Claims for Dietary Supplements and Recommend...
Labelling Requirements and Label Claims for Dietary Supplements and Recommend...Labelling Requirements and Label Claims for Dietary Supplements and Recommend...
Labelling Requirements and Label Claims for Dietary Supplements and Recommend...Lokesh Kothari
 
Justdial Call Girls In Indirapuram, Ghaziabad, 8800357707 Escorts Service
Justdial Call Girls In Indirapuram, Ghaziabad, 8800357707 Escorts ServiceJustdial Call Girls In Indirapuram, Ghaziabad, 8800357707 Escorts Service
Justdial Call Girls In Indirapuram, Ghaziabad, 8800357707 Escorts Servicemonikaservice1
 
Pests of mustard_Identification_Management_Dr.UPR.pdf
Pests of mustard_Identification_Management_Dr.UPR.pdfPests of mustard_Identification_Management_Dr.UPR.pdf
Pests of mustard_Identification_Management_Dr.UPR.pdfPirithiRaju
 
Chemical Tests; flame test, positive and negative ions test Edexcel Internati...
Chemical Tests; flame test, positive and negative ions test Edexcel Internati...Chemical Tests; flame test, positive and negative ions test Edexcel Internati...
Chemical Tests; flame test, positive and negative ions test Edexcel Internati...ssuser79fe74
 
TEST BANK For Radiologic Science for Technologists, 12th Edition by Stewart C...
TEST BANK For Radiologic Science for Technologists, 12th Edition by Stewart C...TEST BANK For Radiologic Science for Technologists, 12th Edition by Stewart C...
TEST BANK For Radiologic Science for Technologists, 12th Edition by Stewart C...ssifa0344
 
Nightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43b
Nightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43bNightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43b
Nightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43bSérgio Sacani
 
High Profile 🔝 8250077686 📞 Call Girls Service in GTB Nagar🍑
High Profile 🔝 8250077686 📞 Call Girls Service in GTB Nagar🍑High Profile 🔝 8250077686 📞 Call Girls Service in GTB Nagar🍑
High Profile 🔝 8250077686 📞 Call Girls Service in GTB Nagar🍑Damini Dixit
 
GBSN - Biochemistry (Unit 1)
GBSN - Biochemistry (Unit 1)GBSN - Biochemistry (Unit 1)
GBSN - Biochemistry (Unit 1)Areesha Ahmad
 
Presentation Vikram Lander by Vedansh Gupta.pptx
Presentation Vikram Lander by Vedansh Gupta.pptxPresentation Vikram Lander by Vedansh Gupta.pptx
Presentation Vikram Lander by Vedansh Gupta.pptxgindu3009
 
Hire 💕 9907093804 Hooghly Call Girls Service Call Girls Agency
Hire 💕 9907093804 Hooghly Call Girls Service Call Girls AgencyHire 💕 9907093804 Hooghly Call Girls Service Call Girls Agency
Hire 💕 9907093804 Hooghly Call Girls Service Call Girls AgencySheetal Arora
 
Formation of low mass protostars and their circumstellar disks
Formation of low mass protostars and their circumstellar disksFormation of low mass protostars and their circumstellar disks
Formation of low mass protostars and their circumstellar disksSérgio Sacani
 
SCIENCE-4-QUARTER4-WEEK-4-PPT-1 (1).pptx
SCIENCE-4-QUARTER4-WEEK-4-PPT-1 (1).pptxSCIENCE-4-QUARTER4-WEEK-4-PPT-1 (1).pptx
SCIENCE-4-QUARTER4-WEEK-4-PPT-1 (1).pptxRizalinePalanog2
 
GBSN - Microbiology (Unit 2)
GBSN - Microbiology (Unit 2)GBSN - Microbiology (Unit 2)
GBSN - Microbiology (Unit 2)Areesha Ahmad
 

Recently uploaded (20)

Kochi ❤CALL GIRL 84099*07087 ❤CALL GIRLS IN Kochi ESCORT SERVICE❤CALL GIRL
Kochi ❤CALL GIRL 84099*07087 ❤CALL GIRLS IN Kochi ESCORT SERVICE❤CALL GIRLKochi ❤CALL GIRL 84099*07087 ❤CALL GIRLS IN Kochi ESCORT SERVICE❤CALL GIRL
Kochi ❤CALL GIRL 84099*07087 ❤CALL GIRLS IN Kochi ESCORT SERVICE❤CALL GIRL
 
American Type Culture Collection (ATCC).pptx
American Type Culture Collection (ATCC).pptxAmerican Type Culture Collection (ATCC).pptx
American Type Culture Collection (ATCC).pptx
 
SAMASTIPUR CALL GIRL 7857803690 LOW PRICE ESCORT SERVICE
SAMASTIPUR CALL GIRL 7857803690  LOW PRICE  ESCORT SERVICESAMASTIPUR CALL GIRL 7857803690  LOW PRICE  ESCORT SERVICE
SAMASTIPUR CALL GIRL 7857803690 LOW PRICE ESCORT SERVICE
 
module for grade 9 for distance learning
module for grade 9 for distance learningmodule for grade 9 for distance learning
module for grade 9 for distance learning
 
Pests of cotton_Borer_Pests_Binomics_Dr.UPR.pdf
Pests of cotton_Borer_Pests_Binomics_Dr.UPR.pdfPests of cotton_Borer_Pests_Binomics_Dr.UPR.pdf
Pests of cotton_Borer_Pests_Binomics_Dr.UPR.pdf
 
Labelling Requirements and Label Claims for Dietary Supplements and Recommend...
Labelling Requirements and Label Claims for Dietary Supplements and Recommend...Labelling Requirements and Label Claims for Dietary Supplements and Recommend...
Labelling Requirements and Label Claims for Dietary Supplements and Recommend...
 
Justdial Call Girls In Indirapuram, Ghaziabad, 8800357707 Escorts Service
Justdial Call Girls In Indirapuram, Ghaziabad, 8800357707 Escorts ServiceJustdial Call Girls In Indirapuram, Ghaziabad, 8800357707 Escorts Service
Justdial Call Girls In Indirapuram, Ghaziabad, 8800357707 Escorts Service
 
CELL -Structural and Functional unit of life.pdf
CELL -Structural and Functional unit of life.pdfCELL -Structural and Functional unit of life.pdf
CELL -Structural and Functional unit of life.pdf
 
Pests of mustard_Identification_Management_Dr.UPR.pdf
Pests of mustard_Identification_Management_Dr.UPR.pdfPests of mustard_Identification_Management_Dr.UPR.pdf
Pests of mustard_Identification_Management_Dr.UPR.pdf
 
Chemical Tests; flame test, positive and negative ions test Edexcel Internati...
Chemical Tests; flame test, positive and negative ions test Edexcel Internati...Chemical Tests; flame test, positive and negative ions test Edexcel Internati...
Chemical Tests; flame test, positive and negative ions test Edexcel Internati...
 
TEST BANK For Radiologic Science for Technologists, 12th Edition by Stewart C...
TEST BANK For Radiologic Science for Technologists, 12th Edition by Stewart C...TEST BANK For Radiologic Science for Technologists, 12th Edition by Stewart C...
TEST BANK For Radiologic Science for Technologists, 12th Edition by Stewart C...
 
Site Acceptance Test .
Site Acceptance Test                    .Site Acceptance Test                    .
Site Acceptance Test .
 
Nightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43b
Nightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43bNightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43b
Nightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43b
 
High Profile 🔝 8250077686 📞 Call Girls Service in GTB Nagar🍑
High Profile 🔝 8250077686 📞 Call Girls Service in GTB Nagar🍑High Profile 🔝 8250077686 📞 Call Girls Service in GTB Nagar🍑
High Profile 🔝 8250077686 📞 Call Girls Service in GTB Nagar🍑
 
GBSN - Biochemistry (Unit 1)
GBSN - Biochemistry (Unit 1)GBSN - Biochemistry (Unit 1)
GBSN - Biochemistry (Unit 1)
 
Presentation Vikram Lander by Vedansh Gupta.pptx
Presentation Vikram Lander by Vedansh Gupta.pptxPresentation Vikram Lander by Vedansh Gupta.pptx
Presentation Vikram Lander by Vedansh Gupta.pptx
 
Hire 💕 9907093804 Hooghly Call Girls Service Call Girls Agency
Hire 💕 9907093804 Hooghly Call Girls Service Call Girls AgencyHire 💕 9907093804 Hooghly Call Girls Service Call Girls Agency
Hire 💕 9907093804 Hooghly Call Girls Service Call Girls Agency
 
Formation of low mass protostars and their circumstellar disks
Formation of low mass protostars and their circumstellar disksFormation of low mass protostars and their circumstellar disks
Formation of low mass protostars and their circumstellar disks
 
SCIENCE-4-QUARTER4-WEEK-4-PPT-1 (1).pptx
SCIENCE-4-QUARTER4-WEEK-4-PPT-1 (1).pptxSCIENCE-4-QUARTER4-WEEK-4-PPT-1 (1).pptx
SCIENCE-4-QUARTER4-WEEK-4-PPT-1 (1).pptx
 
GBSN - Microbiology (Unit 2)
GBSN - Microbiology (Unit 2)GBSN - Microbiology (Unit 2)
GBSN - Microbiology (Unit 2)
 

Introduction to Electron Correlation

  • 2. Hartree-Fock (SCF) methods What it does well Geometries What it doesn’t do well Energies No electron correlation The Fock operator assumes an average electron-electron repulsion Each orbital energy is independent of another Average interaction is optimized through the SCF procedure
  • 3. Perturbation Theory Corrections to the Hartree-Fock energy and wavefunction Requires that the corrections be small HF wavefunction and energy must be pretty good Møller-Plesset perturbation (MP2, MP3, etc.) Perturbations are not convergent Coupled Cluster CCSD, CCSD(t), CCSDT, CCSDTQ, etc. Perturbations are guaranteed to converge
  • 4. Correlated electron motion He atom ˆH = − 1 2 2 1 − Z r1Z − 1 2 2 2 − Z r2Z + 1 r1,2 This is not the Hartree-Fock Hamiltonian There is no analytic solution We must use hyrdrogen atom solutions as a guide The Hamiltonian diverges at r1,2 = 0 r1,2 = r1 − r2 r1Z = r1 − RZ r2Z = r2 − RZ and at the nuclei ri = 0. The kinetic energy must also diverge to achieve a finite sum.
  • 5. A determinant is a wavefunction Ψ(x1, x2, . . . , xn) = 1 √ n! χ1(x1) χ2(x1) · · · χn(x1) χ1(x2) χ2(x2) · · · χn(x2) ... ... · · · ... χ1(xn) χ2(xn) · · · χn(xn) Rows are electron positions and spins Columns are spin orbitals Ψ(x1, x2, . . . ) = −Ψ(x2, x1, . . . ) Exchange either two rows or two columns: antisymmetric
  • 6. Probability density Ψ(x1, x2) = χ1(x1)χ2(x2) P(r1, r2) = dω1dω2 |Ψ|2 dr1dr2 The two-electron density Probability of finding electron 1 at r1 in a volume dr1 while simultaneously finding electron 2 at r2 in a volume dr2.
  • 7. The helium atom: ground state singlet wavefunction Hartree-Fock single determinant for the 11 S state Minimal 1-electron basis: 2 1s restricted spin orbitals Ψ(x1, x2) = 1 √ 2 ψ1s(r1)α(ω1) ψ1s(r1)β(ω1) ψ1s(r2)α(ω2) ψ1s(r2)β(ω2) = 2− 1 2 (ψ1s(r2)β(ω2)ψ1s(r1)α(ω1) −ψ1s(r2)α(ω2)ψ1s(r1)β(ω1)) |Ψ|2 = 1 2 . . . = ψ1s(r1) 2 ψ1s(r2) 2 The spatial orbital is identical for α and β spin Spatial motion in uncorrelated
  • 8. The helium atom: triplet wavefunction Hartree-Fock single determinant for the 23 S state Minimal 1-electron basis: 1s and 2s spin orbitals Ψ(x1, x2) = 1 √ 2 ψ1s(r1)α(ω1) ψ2s(r1)α(ω1) ψ1s(r2)α(ω2) ψ2s(r2)α(ω2) = 2− 1 2 (ψ2s(r2)α(ω2)ψ1s(r1)α(ω1) −ψ1s(r2)α(ω2)ψ2s(r1)α(ω1)) |Ψ|2 = 1 2 . . . = ψ1s(r1) 2 ψ2s(r2) 2 + ψ1s(r2) 2 ψ2s(r1) 2 Parallel spin electrons have Fermi correlation
  • 9. Cusp conditions Figure: Hylleraas Hamiltonian for Helium with relative coordinates.(Figure from Fred Manby) electron-nucleas cusp limri →0 ∂Ψ ∂ri ave = −ZΨ(ri = 0) Exists in Slater orbitals, but approximated in Gaussian orbitals More angular momentum means a sharper cusp electron-electron cusp limr1,2→0 ∂Ψ ∂r1,2 ave = 1 2 Ψ(r1,2 = 0) Leads to a depletion in in the two-electron density at r1,2
  • 10. Hylleraas wavefunction Figure: Hartree-Fock wavefunction and Hylleraas wavefunction. (Figure from Fred Manby) One slater determinant is not enough To help electrons “avoid” each other, we can distribute them among more orbitals using many determinants
  • 11. Helium determinants: double zeta basis Singlet Helium ψ2s is a “virtual” or unoccupied spin orbital Ψ2 is a “doubly excited” determinant CI Singles and Double (CISD) Why no singles? Ψ0 = ψ1s ¯ψ1s Ψ2 = ψ2s ¯ψ2s Ψ(x1, x2) = K1 ψ1s ¯ψ1s + K2 ψ2s ¯ψ2s P(r1, r2) = K1ψ1s(r1)ψ1s(r2) + K2ψ2s(r1)ψ2s(r2) 2 lim r1→r2 P(r1, r2) = K1 ψ1s(r1) 2 + K2 ψ2s(r1) 2 2
  • 12. Configuration Interaction Solving for the determinant coefficients Ψ = K0Ψ0 + K2Ψ2 H = Ψ0|H|Ψ0 Ψ0|H|Ψ2 Ψ2|H|Ψ0 Ψ2|H|Ψ2 Ψ2|H|Ψ0 = 1¯1 2¯2 − 1¯1 ¯22 = K12 H K0 K2 = ECISD K0 K2
  • 13. Helium convergence ERHF = −2.85516 a.u. Table: FCI/cc-pVQZ Spatial Orbital CI coefficient 1s 0.995974 2s -0.039200 2p -0.028429 3d -0.005164 3f -0.001472 Table: Helium Full CI total energy J. Chem. Phys. 127 , 224104 (2007) Basis Set Energy (a.u.) cc-pVDZ -2.8875948 cc-pVTZ -2.9002322 cc-pVQZ -2.9024109 cc-pV5Z -2.9031519 Exact Energy -2.9037225 The difference between Full CI and Hartree-Fock energies is called the correlation energy
  • 14. Helium cusp Figure: Helium wavefunction cusps for cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pV5Z basis sets from Martin Sch¨utz Slater determinant expansions have even powers of r1,2 (Helgaker)
  • 15. Configuration Interaction Exact when all possible determinants used with a complete basis set Full CI Basis sets HF has a factorizable two-electron density and requires fewer basis functions to converge Correlation methods seek to improve the two-electron density and require much more functions to converge Orbitals are frozen at the HF solution
  • 16. More than two electrons Excited determinants and matrix elements (Slater Rules) Single excitations are zero by Brilloun’s theorem Determinants differening by one spin orbital Ψ|H|Ψp m = m ˆh p + N n ( mn|pn − mn|np ) Determinants differing by two spin orbitals Ψ|H|Ψpq mn = mn|pq − mn|qp determinants differing by more than two spin orbitals contributes 0 H =   Ψ0 H Ψ0 0 Ψ0 H Ψ2 0 0 · · · Ψ1 H Ψ1 Ψ1 H Ψ2 Ψ1 H Ψ3 0 · · · Ψ2 H Ψ2 Ψ2 H Ψ3 Ψ2 H Ψ4 · · · Ψ3 H Ψ3 Ψ3 H Ψ4 · · · Ψ4 H Ψ4 · · · ...  
  • 17. Expansions Slow convergence with excitation Scaling problem Full CI grows factorially Many billion determinant Full CI is possible, but only small molecules Truncation CI expansions CISD: CI Singles and Doubles CISDT: With triple excitations CISDTQ: With quadruple excitations QCISD is closely related to coupled-cluster
  • 18. What is still missing? Bond breaking Including transition states Diradicals Degenerate orbital pictures
  • 19. H2 dissociation UHF Not smooth due to spin contamination RHF MP2 likely to fail
  • 20. H2 dissociation determinants A single determinant has a spurious J integral at long range ψσ = φA + φB ψσ∗ = φA − φB Eeq = 2hσσ + Jσσ E∞ = 2hσσ + Jσσ = 2hσ∗σ∗ + Jσ∗σ∗
  • 22. Separate correlation Dynamic correlation Short range cusp conditions Perturbation theories are very efficient Requires large basis sets Static correlation Degeneracies in orbitals or determinants “chemical intuition” Full CI does both
  • 23. Diradicals What is the Lewis structure of ozone?
  • 24. Ozone What are the molecular orbitals? Each oxygen supplies 1s, 2s and 3 2p orbitals Hartree-Fock will only fill the first 12 orbitals The HOMO energy is -0.4810 a.u. The LUMO energy is -0.0299 a.u.
  • 25. Ozone
  • 26. Ozone Distribute 4 electrons in the 3 π orbitals Full CI = CISD 0.929 -0.224 -0.224 -0.166 -0.066 0.066 Determinant weights for Hartree-Fock orbitals
  • 27. Ozone Multi-configurational Self Consistent Field (MCSCF) Improve the energy by minimizing the orbitals New definition of Fock matrix Fully Optimized Reaction Space (FORS) Complete Active Space Self Consistent Field (CAS) Full CI within a “chosen” orbital space MCSCF does not imply CASSCF
  • 28. Ozone Determinant weights for CASSCF optimized orbitals 0.864 -0.240 -0.240 -0.295 -0.159 0.159
  • 29. What about the orbitals? Hartree-Fock orbitals have little physical meaning DFT orbitals even less Orbital energies may be helpful to pick active space Determinants define electron configurations Natural orbitals have populations Diagonalize the density matrix U are orbitals and n are electron occupations Ψ = k Ak Ψk P = Ψ∗ Ψ PU = nU
  • 30. Ozone Electron populations in natural orbitals for the active space 1.993 1.598 0.409
  • 31. Dynamic correlation CASSCF has no dynamic correlation Similar to Hartree-Fock Multi-reference CI (MR-CISD) Very accurate for excitation energies Still not size-consistent Alternatives to MR-CI expansions, perturbation theory MRPT (MRMP), CASPT, MCQDPT MRCC: Multiple types Good for potential energy surfaces
  • 32. Pople Diagram Figure: From Chem. Soc. Rev., 2012,41, 6259-6293