Published: December 29, 2011
r2011 American Chemical Society 4 dx.doi.org/10.1021/cr200168z
|
Chem. Rev. 2012, 112, 4–74
REVIEW
pubs.acs.org/CR
Explicitly Correlated Electrons in Molecules
Christof H € attig,
†
Wim Klopper,*
,‡
Andreas K€ ohn,
§
and David P. Tew
||
†
Lehrstuhl f€ ur Theoretische Chemie, Ruhr-Universit € at Bochum, D-44780 Bochum, Germany
‡
Abteilung f€ ur Theoretische Chemie, Institut f€ ur Physikalische Chemie, Karlsruher Institut f€ ur Technologie, KIT-Campus S€ ud,
Postfach 6980, D-76049 Karlsruhe, Germany
§
Institut f€ ur Physikalische Chemie, Johannes Gutenberg-Universit € at Mainz, D-55099 Mainz, Germany
)
School of Chemistry, University of Bristol, Bristol BS8 1TS, United Kingdom
CONTENTS
1. Introduction 5
2. Correlation in Electronic Wave Functions 5
2.1. Statistical Correlation 5
2.1.1. Exchange-Correlation Hole in Density
Functional Theory 6
2.1.2. Exchange and Coulomb Holes in Wave
Function Theory 6
2.1.3. Radial, Angular, and LeftRight
Correlation 6
2.2. Fermi Correlation: Shell Structure 6
2.3. Fermi Correlation: Exchange 7
2.3.1. Case Study: He Atom 7
2.4. Coulomb Correlation 9
2.4.1. Static and Dynamic Correlation 9
2.4.2. Case Study: Molecular Hydrogen 9
3. Cusp Conditions and Energy Convergence 11
3.1. Regularity Conditions 11
3.2. Nuclear Cusp Conditions 12
3.3. Electron Coalescence Conditions 12
3.3.1. Singlet Coalescence 13
3.3.2. Triplet Coalescence 13
3.3.3. Unnatural Parity Singlet Coalescence 14
3.4. Three-Particle Coalescence Conditions 14
3.5. Second-Order Coalescence Conditions 14
3.6. Coalescence Conditions and Approximate
Wave Functions 14
3.7. Convergence Properties of CI Wave
Functions 15
3.7.1. One-Electron Convergence 15
3.7.2. Two-Electron Convergence 16
3.7.3. Partial Wave Expansion 16
3.7.4. Principal Expansion 17
3.7.5. Extrapolation 17
3.7.6. Explicit Correlation 17
4. n-Electron Expansions 17
4.1. Hylleraas-Type Wave Functions 19
4.1.1. He Atom 19
4.2. Gaussians 24
4.2.1. ECGs: Exponentionally Correlated
Gaussians 24
4.2.2. GTGs: Gaussian-Type Geminals 25
4.2.3. GGn Methods: GTGs Combined with
Orbital Expansions 26
4.3. Transcorrelated Methods 28
4.4. Quantum Monte Carlo Methods 29
4.5. R12 Methods 30
5. Strategies for Avoiding n-Electron Integrals 33
5.1. Restricting the Wave Function Parameter Space 33
5.2. Nonvariational Approaches 34
5.3. Weak Orthogonality 34
5.4. Stochastic Numerical Integration 34
5.5. Resolution of the Identity 34
5.6. Numerical Quadrature 34
5.7. Two-Electron Integrals 34
6. General F12 Theory 35
6.1. General Ansatz 35
6.2. Auxiliary Basis Sets 36
6.3. Many-Electron Integrals and Standard
Approximations 37
6.4. MP2-F12 Theory 39
6.5. CCSD-F12 Theory 41
6.6. Geminal Basis Functions 42
6.6.1. Correlation Factor 42
6.6.2. Multiple Geminals 43
6.6.3. Open-Shell Aspects 44
6.6.4. Extension of the Generating Orbital Space 44
6.7. CABS Singles 45
6.8. Explicitly Correlated Triples 46
6.9. Multireference Methods 48
6.9.1. Overview 49
6.9.2. Details of the Theories 49
7. Computationally Efficient Formulations 51
7.1. Perturbational Analysis 52
Special Issue: 2012 Quantum Chemistry
Received: May 13, 2011