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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The first book to aid in the understanding of multiconfigurational quantum chemistry, Multiconfigurational Quantum Chemistry demystifies a subject that has historically been considered difficult to learn. Accessible to any reader with a background in quantum mechanics and quantum chemistry, the book contains illustrative examples showing how these methods can be used in various areas of chemistry, such as chemical reactions in ground and excited states, transition metal and other heavy element systems. The authors detail the drawbacks and limitations of DFT and coupled-cluster based methods and offer alternative, wavefunction-based methods more suitable for smaller molecules.
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Veröffentlichungsjahr: 2016
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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Conventions and Units
Chapter 1: Introduction
1.1 References
Chapter 2: Mathematical Background
2.1 Introduction
2.2 Convenient Matrix Algebra
2.3 Many-Electron Basis Functions
2.4 Probability Basics
2.5 Density Functions for Particles
2.6 Wave Functions and Density Functions
2.7 Density Matrices
2.8 References
Chapter 3: Molecular Orbital Theory
3.1 Atomic Orbitals
3.2 Molecular Orbitals
3.3 Further Reading
Chapter 4: Hartree–Fock Theory
4.1 The Hartree–Fock Theory
4.2 Restrictions on The Hartree–Fock Wave Function
4.3 The Roothaan–Hall Equations
4.4 Practical Issues
4.5 Further Reading
4.6 References
Chapter 5: Relativistic Effects
5.1 Relativistic Effects on Chemistry
5.2 Relativistic Quantum Chemistry
5.3 The Douglas–Kroll–Hess Transformation
5.4 Further Reading
5.5 References
Chapter 6: Basis Sets
6.1 General Concepts
6.2 Slater Type Orbitals, STO
6.3 Gaussian Type Orbitals, GTO
6.4 Constructing Basis Sets
6.5 Selection of Basis Sets
8.6 References
Chapter 7: Second quantization and multiconfigurational wave functions
7.1 Second quantization
7.2 Second quantization operators
7.3 Spin and spin-free formalisms
7.4 Further reading
8.5 References
Chapter 8: Electron correlation
8.1 Dynamical and nondynamical correlation
8.2 The interelectron cusp
8.3 Broken bonds. (σ)
2
→(σ
∗
)
2
8.4 Multiple bonds, aromatic rings
8.5 Other correlation issues
8.6 Further reading
8.7 References
Chapter 9: Multiconfigurational SCF Theory
9.1 Multiconfigurational SCF Theory
9.2 Determination of the MCSCF Wave Function
9.3 Complete and Restricted Active Spaces, the CASSCF and RASSCF Methods
9.4 Choosing the Active Space
9.5 References
Chapter 10: The RAS State-Interaction method
10.1 The biorthogonal transformation
10.2 Common one-electron properties
10.3 Wigner–Eckart coefficients for spin–orbit interaction
10.4 Unconventional usage of RASSI
10.5 Further Reading
10.6 References
Chapter 11: The Multireference CI Method
11.1 Single-Reference CI. Nonextensivity
11.2 Multireference CI
11.3 Further Reading
11.4 References
Chapter 12: Multiconfigurational Reference Perturbation Theory
12.1 CASPT2 Theory
12.2 References
Chapter 13: CASPT2/CASSCF Applications
13.1 Orbital Representations
13.2 Specific Applications
References
Summary and Conclusion
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 The and orbitals and associated occupation numbers in the molecule at the equilibrium geometry.
Chapter 3: Molecular Orbital Theory
Figure 3.1 Shells and subshell of atoms.
Figure 3.2 Two hydrogen atoms at large separation.
Figure 3.3 Two hydrogen atoms close to equilibrium.
Figure 3.4 Potential curves of and of . Bond distance and energy in atomic units.
Figure 3.5 MO diagram for molecule.
Figure 3.6 MO diagram for molecule.
Figure 3.7 Radial distribution function for 1s and 2s of lithium atom, atomic units.
Figure 3.8 MO diagram for molecule. (The energy difference between 1s and 2s levels is not shown with the correct scale.)
Figure 3.9 MO diagram for two 2p orbitals.
Figure 3.10 MO diagram for the molecule.
Figure 3.11 MO diagram for the molecule.
Figure 3.12 MO diagram for the HF molecule.
Figure 3.13 MO diagram for the CO molecule.
Chapter 4: Hartree–Fock Theory
Figure 4.1 Occupation numbers for a pair of bonding and antibonding orbitals.
Chapter 6: Basis Sets
Figure 6.1 General versus segmented versus overlapping contraction.
Figure 6.2 Raffenetti contraction.
Figure 6.3 Contraction errors, in Hartrees, for Cl atom using nonrelativistic basis sets with a DKH Hamiltonian to second order.
Chapter 8: Electron correlation
Figure 8.1 The two-electron wave function of Hooke's ‘atom’ (see text).
Figure 8.2 The two-electron density function for the valence electrons of the Be atom (see text).
Figure 8.3 The two-electron wave function of near the equilibrium distance, from HF (a) and Full CI (b).
Figure 8.4 The same, but with the molecule stretched to about three times the equilibrium distance.
Figure 8.5 Cr (a): CASPT2 natural orbitals; (b): Occupation numbers, from CASPT2.
Figure 8.6 Potential curves of CO computed by SCF (dotted), CASSCF (dashed), and CASPT2 (solid).
Chapter 9: Multiconfigurational SCF Theory
Figure 9.1 Potential curves for showing the erratic behavior of the RHF curve (solid line) as a function of the internuclear distance. For comparison, the MCSCF curve, which dissociates correctly, is also given (dashed line).
Figure 9.2 The competing valence structures for the ozone molecule.
Figure 9.3 The potential energies for the reactant () and product () electronic configurations along the reaction coordinate.
Figure 9.4 The valence orbitals of the water molecule plus the and extra valence orbitals. The top row orbitals are the bonding orbitals and the second row contains the corresponding correlating orbitals.
Figure 9.5 Illustration of the active orbitals for a 3-in-2 CAS, extended to a RAS of SD type.
Figure 9.6 The standard active space selection for second-row atoms. For the initial set of elements, Li-C, the 2s and 2p near degeneracy has to be considered. However, for N and onward the 2s orbital can be left inactive.
Figure 9.7 The standard active space selection for transition metals. For the first-row transition metals we have 4s, 3d, and 4p as active orbitals. For elements with more than five d-electrons, a second complete -shell might be required. This effect decreases for higher row transition metals.
Figure 9.8 The standard active space selection for lanthanides. For the lanthanides, we have 6s, 4f, 5d, and 6p as active orbitals. The 5d-shell normally only is required in the worst case.
Figure 9.9 The standard active space selection for actinides. For the actinides, we have 7s, 5f, 6d, and 7p as active orbitals.
Chapter 12: Multiconfigurational Reference Perturbation Theory
Figure 12.1 The Full CI Hamiltonian matrix structured using the CASPT2 excitation operators. Gray areas: Nonzero matrix elements. (Not drawn to scale).
Figure 12.2 The CASPT2 Fock matrix, used to define . Gray areas: Nonzero matrix elements.
Chapter 13: CASPT2/CASSCF Applications
Figure 13.1 The occupied (bottom row) and virtual (top row) canonical SCF molecular orbitals of benzene, from a calculation with no explicit symmetry constraints. In a calculation with a minimal basis (ANO-RCC). This set of orbitals constitute the HOMO-4, HOMO-1, HOMO, LUMO, LUMO+1, and LUMO+2 SCF orbitals of the molecular system. Note how these orbitals are delocalized over the whole conjugated system.
Figure 13.2 The energetically six lowest occupied canonical SCF orbitals of benzene, from a calculation with no explicit symmetry constraints and a minimal basis (ANO-RCC). Here we note that the canonical SCF molecular orbitals are not only delocalized but also that they are a mixture of CC and CH bonds. This makes these orbitals rather useless as starting orbitals for a CASSCF in which, for example, one would like to study the hydrogen abstraction reaction.
Figure 13.3 The energetically six lowest virtual canonical SCF orbitals of benzene, from a calculation with no explicit symmetry constraints in conjunction with a basis set of triple quality (ANO-RCC). The first observation is that these orbitals are not found among the three lowest virtual orbitals. The LUMO and the following virtual orbitals can best be described as garbage orbitals to fulfill the orthogonality conditions of the canonical virtual orbitals, for all what concerns us they completely lack any character that will help in the selection of starting orbitals (compare with the upper row in Figure 13.2). The second observation is that these orbitals contain a heavy mixture of other types of antibonding orbitals, again rendering these useless for the purpose of generating CASSCF starting orbitals.
Figure 13.4 The trans-butadiene molecule. The carbon atoms of this conjugated system are hybridized. The remaining carbon unhybridized 2p orbitals form a set of orbitals perpendicular to the plane of the molecular system. This set of orbitals constitute the orbital subspace of the conjugated orbitals.
Figure 13.5 The orthonormal atomic 2p carbon orbitals of trans-butadiene, which are the building blocks of the conjugated systems. These orbitals have been generated from an initial set of atomic orbitals, which through the PAO procedure have been made orthonormal to each other.
Figure 13.6 The converged natural orbitals of the CASSCF active space designed from the orbitals of the conjugated system, that is, the (bottom row) and (top row) orbitals. The values below each set of orbitals represent the partial natural occupation number.
Figure 13.7 The localized Pipek–Mezey molecular orbitals of the conjugated system, the bonding (b) and antibonding (a) orbitals.
Figure 13.8 The localized Pipek–Mezey (a) and (b) orbitals of the terminal CH bond.
Figure 13.9 The converged natural orbitals of the CASSCF active space to study the hydrogen abstraction process of a terminal hydrogen in the trans-butadiene molecule. The lower row represents the bonding orbitals and above each of these orbitals, the corresponding correlating orbital is depicted. The number below each orbital represents the natural occupation number. Note that for each pair of correlating orbitals the occupation numbers add up to 2.00.
Figure 13.10 The (a) and (b) orbitals of the central CC bond of trans-butadiene.
Figure 13.11 The converged natural orbitals of the CASSCF active space for the study of the fragmentation of the trans-butadiene molecule into two vinyl radicals. The number below each orbital represents the natural occupation number.
Figure 13.12 The converged natural orbitals of the CASSCF active space for the study of the fragmentation of the trans-butadiene molecule into two vinyl radicals in a triple quality basis set. The number below each orbital represents the natural occupation number.
Figure 13.13 The calicheamicin molecule discovered in 1987 in bacteria living in chalky rocks.
Figure 13.14 The Bergman cyclization reaction, the transformation of ()-hex-3-1,5-diyne into the
para
-benzyne molecule.
Figure 13.15 The starting atomic orbitals of the active space of ()-3-ene-1,5-diyne. The top two rows display the out-of-plane orbitals and orbitals, respectively; the first two orbital sets on the third and fourth row display the in-plane and orbitals. The last orbital set, on the same rows, displays the and orbitals of the – bond.
Figure 13.16 The starting atomic orbitals of the active space of
para
-benzyne. The top two rows display the out-of-plane and orbitals, respectively; the first two orbital sets on rows three and four display the and orbitals. The last orbital set, on the same rows, displays the two radical orbitals.
Figure 13.17 The natural orbitals of the active space and the natural orbital occupation numbers for ()-hex-3-ene-1,5-diyne treated with 12 electrons in 12 orbitals CASSCF. The top two rows display the out-of-plane and orbitals and occupation numbers, respectively, the first two sets of orbitals on rows three and four display similarly the in-plane space. Finally, the last orbital set, on the same rows, displays the and orbitals of the – bond.
Figure 13.18 The natural orbitals of the active space and the natural orbital occupation numbers for
para
-benzyne treated with 12 electrons in 12 orbitals CASSCF. The top row displays the out-of-plane and orbitals and occupation numbers, respectively. The first two orbital sets on rows three and four display the manifold. Finally, the last orbital set, on the same rows, displays the radical orbitals.
Figure 13.19 The water valence, 3s Rydberg, 3p Rydberg, and 3d Rydberg orbitals, from a full valence CASSCF calculation of O cation using a minimal valence and 2s2p2d molecular Rydberg basis set. The isosurface of the valence and Rydberg orbitals are 0.040 and 0.017 au, respectively.
Figure 13.20 The converged active orbitals from a 6 root SA-CASSCF calculation of 8 electrons in 10 orbitals in conjunction with an uncontracted molecular Rydberg basis and a triple plus polarization ANO-L basis. The number below the orbitals indicates the SA-CASSCF natural orbital occupation number. Isosurface values for the valence and Rydberg orbitals are 0.040 and 0.017 au, respectively.
Figure 13.21 The -nitroaniline system, an important precursor in the synthesis of dyes.
Figure 13.22 The starting active orbitals of -nitroaniline system using a ANO-L minimal valence basis set.
Figure 13.23 The optimized single-root active orbitals of -nitroaniline system using a ANO-L minimal valence basis set. Note that the leftmost orbitals on the lower row correspond to oxygen 2s orbitals and not as expected a pair of in-plane lone-pair orbitals.
Figure 13.24 The optimized three root SA-CASSCF active orbitals and associated natural orbital occupation numbers of -nitroaniline system using a ANO-L minimal valence basis set.
Figure 13.25 The 16-in-16 SA-RASSCF 16 root RAS1, RAS2, and RAS3 natural active orbitals and associated occupation numbers of p-nitroaniline in conjunction with an ANO-L VDZP basis set. Isosurface at 0.02 au.
Figure 13.26 The experimental vapor spectra of
para
-nitroaniline by Millefiori [18] and the computed RASPT2 values of the transitions in the 197–350 nm region. The intensity scale is arbitrary. The experimental curve is a digitalization from the 1977 publication. The solid gray lines are the three most intense RASPT2 transitions normalized to have the most intense transition, at 293 nm, to normalize to the same height as the most intense experimental transition. The triangle symbols indicate transitions which, for all practical purposes, can be considered dipole-forbidden transitions.
Figure 13.27 The molecular structure of 1,2-dioxetane at (a) the ground state equilibrium structure and (b) the O–O bond-breaking transition state.
Figure 13.28 The state-specific SA-CASSCF natural orbitals and the associated occupation numbers for the ground state (top two rows) and the first excited state (bottom two rows) at the ground state equilibrium structure of 1,2-dioxetane. Isosurface at 0.04 au.
Figure 13.29 The state-specific SA-CASSCF natural orbitals and the associated occupation numbers for the ground state (top two rows) and the first excited state (bottom two rows) at the transition state of the O–O bond breaking in 1,2-dioxetane. Isosurface at 0.04 au.
Figure 13.30 The molecular structure of uracil, 2,4-dioxopyrimidine, one of the five DNA/RNA bases.
Figure 13.31 The SA-CASSCF natural orbitals and the associated occupation numbers for the uracil molecule with a valence triple zeta plus polarization basis set. Top two rows depict the eight orbitals and the lower row the oxygen lone-pairs. Isosurface at 0.04 au.
Figure 13.32 Cartoon showing the excited state dynamics in uracil after UV absorption.
Figure 13.33 Structure of the pyridine–cyanonaphthalene complex at equilibrium distance 1.45 Å and at 2.8 Å.
Figure 13.34 Selection of active space for pyridine–cyanonaphthalene complex.
Figure 13.35 The chloroiron corrole molecule.
Figure 13.36 The final orbitals and associated occupation numbers for the state of chloroiron corrole as computed with a 14in13 CASSCF.
Figure 13.37 Relative energies (eV), computed in CASPT2 level for lowest singlet, triplet and quintet states in chloroiron corrole. The symbols and indicate up- and down-spin-orbitals. The amd orbitals are corrole HOMOs.
Figure 13.38 The 10 active orbitals in the RAS1 space of chloroiron corrole.
Figure 13.39 The active orbitals of a CASSCF calculation for the molecule, using 12 active electrons in 12 orbitals. (a) The natural occupation numbers; (b) Orbital pictures.
Figure 13.40 The state-specific singlet SA-CASSCF//ANO-VTZP natural orbitals and associated occupation number of Br, at the CASPT2//ANO-VDZP equilibrium structure, for the first doubly degenerate excited states (top panel) and the ground state (lower panel).
Figure 13.41 The state-specific singlet SA-CASSCF//ANO-VTZP natural orbitals and associated occupation number of Br, at a –Br bond distance of 4.00 Å, for the first doubly degenerate excited states (top panel) and the ground state (lower panel).
Figure 13.42 The lowest CASPT2 potential energy curves of Br along the reaction coordinate corresponding to constrained optimizations with fixed –Br bond distances.
Figure 13.43 The active orbitals of a CASSCF calculation for the molecular ion, using 5 active electrons in 11 orbitals. Top row: strongly occupied orbitals. Bottom left: most important correlating orbitals. Bottom right: electron density isosurface at level 0.28 au.
Figure 13.44 The lowest energy levels of at 2.43 Å bond length, computed with (left) and without (right) spin-orbit interaction. Energies are relative to the ground state.
Figure 13.45 The CASPT2 natural orbitals of CUC (linear) for the ground state, with occupation numbers.
Figure 13.46 The CASPT2 natural orbitals of UC for the ground state (, , ), with (averaged) occupation numbers.
Figure 13.47 The lowest electronic energy levels of UC at Å, computed with spin-orbit interaction included (a) and not included (b). Energies are relative to the ground state. There are 72 spin-orbit states in the figure.
Figure 13.48 The potential functions of the lowest few spin-orbit states of UC.
List of Tables
Chapter 6: Basis Sets
Table 6.1 Hartree–Fock Structures and Energies for with Various Basis Sets
Table 6.2 SCF Energies for the Ni Atoms with Respect to Different Contraction Schemes
Table 6.3 SDCI Energies for Ni Atom with Respect to Different Contractions Sizes for an ANO Basis with Bias Toward the State
Table 6.4 SCF Energies for the , and Levels
Table 6.5 SDCI Energies for the , and Levels
Table 6.6 SCF Energies for Ni Atom Using a Nonrelativistic Contraction with a Douglas–Kroll–Hess Hamiltonian to Second Order
Table 6.7 SDCI Energies for Ni Atom Using a Nonrelativistic Contraction with a Douglas–Kroll–Hess Hamiltonian to Second Order
Chapter 12: Multiconfigurational Reference Perturbation Theory
Table 12.1 The Excitation Types Used in CASPT2
Table 12.2 Comparison Between CASPT2 and MP2
Multiconfigurational Quantum Chemistry
By
Prof. Björn O. Roos
Roland Lindh
Per Åke Malmqvist
Valera Veryazov
Per-Olof Widmark
Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Names: Roos, B. O. (Björn O.) 1937-2010, author. | Lindh, Roland, 1958- author. | Malmqvist, Per Åke, 1952- author. | Veryazov, Valera, 1963- author. | Widmark, Per-Olof, 1956- author.
Title: Multiconfigurational quantum chemistry / Bjorn Olof Roos, Roland Lindh, Per Åke Malmqvist, Valera Veryazov and Per-Olof Widmark.
Description: Hoboken, New Jersey : John Wiley & Sons, 2016. | Includes bibliographical references and index.
Identifiers: LCCN 2016010465 (print) | LCCN 2016015079 (ebook) | ISBN 9780470633465 (cloth) | ISBN 9781119277873 (pdf) | ISBN 9781119277880 (epub)
Subjects: LCSH: Quantum chemistry-Textbooks.
Classification: LCC QD462 .R66 2016 (print) | LCC QD462 (ebook) | DDC 541/.28-dc23
LC record available at http://lccn.loc.gov/2016010465
Cover image courtesy of Dr.Valera Veryazov
Dedication
In memory of and dedicated to Björn O. Roos 1937–2010
The work on this book was started in 2009 by Professor Björn O. Roos. He was in charge of the planning and wrote significant parts before passing away on February 22, 2010. Despite being marked by the deteriorating impact of his condition, Björn spent most of his limited awaken time working on this project. Inspired by Björn's enthusiasm and dedication to multiconfigurational wave function theory, we decided to complete the work, as outlined by him, as a testament and a tribute for his contributions in this field.
Thanks Björn!
Preface
The intention of this book is to introduce the reader into the multiconfigurational approaches in quantum chemistry. These methods are more difficult to learn to use and there does not exist any textbook in the field that takes the students from the simple Hartree–Fock method to the advanced multireference methods such as multireference configuration interaction (MRCI), or the complete active space self-consistent field (CASSCF) method. The intention is to describe these and other wave function-based methods such that the treatment can be followed by any student with basic knowledge in quantum mechanics and quantum chemistry. Using many illustrative examples, we shall show how these methods can be applied in various areas of chemistry, such as chemical reactions in ground and excited states, transition metal, and other heavy element systems. These methods are based on a well-defined wave function with exact spin and symmetry and are therefore well suited for detailed analysis of various bonding situations. A simple example is the oxygen molecule, which has a ground state. Already this label tells us much about the wave function and the electronic structure. It is a triplet state (), it is symmetric around the molecular axis (), it is a gerade function, and it is antisymmetric with respect to a mirror plane through the molecular axis. None of these properties are well defined in some methods widely used today. It becomes even worse for the first excited state, , which cannot be properly described with single configurational methods due to its multiconfigurational character. This failure can have severe consequences in studies of oxygen-containing biological systems. It is true that these wave function-based methods cannot yet be applied to as large systems as can, for example, density functional theory (DFT), but the method development is fast and increases the possibilities for every year.
Computational quantum chemistry is today dominated by the density functional theory and to some extent coupled-cluster-based method. These methods are simple to use and DFT can be applied to larger molecules. They have, however, several drawbacks and failures for crucial areas of applications, such as complex electronic structures, excited states and photochemistry, and heavy element chemistry. Many students learn about the method and how to use it but have often little knowledge about the more advanced wave function-based methods that should preferably be used in such applications.
Conventions and Units
In this book, we use the conventional systems of units of quantum chemistry, which are the Hartree-based atomic units (au), a set of rational units derived from setting the reduced Planck constant , the electron mass , the elementary charge , and the Coulomb constant ( times the vacuum permittivity) , see Tables 1 and 2. The resulting formulae then appear to be dimensionless, and to avoid confusion they are sometimes written in full, that is,
for the kinetic energy term for an electron. Similarly, the electrostatic interaction energy between two electrons can be written with or without the explicit constants:
The first form can be used with any (rational) units. The Bohr, or Bohr radius, and the Hartree, are then used as derived units for length and energy, with symbols and , respectively. The speed of light is numerically equal to , the reciprocal Sommerfeld fine-structure constant, in atomic units.
Throughout the book, we follow the conventions in Table 3 except where otherwise stated. For example, will be used as the symbol for a wave function, whereas will be used for configuration state functions. The Hartree–Fock determinant might then in one circumstance be denoted by if it is the wave function at hand or perhaps as if it is part of an MCSCF expansion.
Table 1 One Atomic Unit in Terms of SI Units
Quantity
Symbol
Value
Action
J s
Mass
kg
Charge
C
Coulomb constant
F/m (exact)
Length
m
Energy
J
Electric dipole moment
C m
Time
s
Temperature
K
Note: Most of the values above have been taken from web pages of the National Institute of Standards and Technology: http://physics.nist.gov/cuu/.
Table 2 Constants and Conversion Factors
Quantity
Symbol
Value
The fine structure constant
Vacuum speed of light
Chapter 1Introduction
How do we define multiconfigurational (MC) methods? It is simple. In Hartree–Fock (HF) theory and density functional theory (DFT), we describe the wave function with a single Slater determinant. Multiconfigurational wave functions, on the other hand, are constructed as a linear combination of several determinants, or configuration state functions (CSFs)—each CSF is a spin-adapted linear combination of determinants. The MC wave functions also go by the name Configuration Interaction (CI) wave function. A simple example illustrates the situation. The molecule (centers denoted A and B) equilibrium is well described by a single determinant with a doubly occupied orbital:
where is the symmetric combination of the atomic hydrogen orbitals (; the antisymmetric combination is denoted as ). However, if we let the distance between the two atoms increase, the situation becomes more complex. The true wave function for two separated atoms is
which translates to the electronic structure of the homolytic dissociation products of two radical hydrogens. Two configurations, and , are now needed to describe the electronic structure. It is not difficult to understand that at intermediate distances the wave function will vary from Eq. 1.1 to Eq. 1.2, a situation that we can describe with the following wave function:
where and , the so-called CI-coefficients or expansion coefficients, are determined variationally. The two orbitals, and , are shown in Figure 1.1, which also gives the occupation numbers (computed as and ) at a geometry close to equilibrium. In general, Eq. 1.3 facilitates the description of the electronic structure during any bond dissociation, be it homolytic, ionic, or a combination of the two, by adjusting the variational parameters and accordingly.
Figure 1.1 The and orbitals and associated occupation numbers in the molecule at the equilibrium geometry.
This little example describes the essence of multiconfigurational quantum chemistry. By introducing several CSFs in the expansion of the wave function, we can describe the electronic structure for a more general situation than those where the wave function is dominated by a single determinant. Optimizing the orbitals and the expansion coefficients, simultaneously, defines the approach and results in a wave function that is qualitatively correct for the problem we are studying (e.g., the dissociation of a chemical bond as the example above illustrates). It remains to describe the effect of dynamic electron correlation, which is not more included in this approach than it is in the HF method.
The MC approach is almost as old as quantum chemistry itself. Maybe one could consider the Heitler–London wave function [1] as the first multiconfigurational wave function because it can be written in the form given by Eq. 1.2. However, the first multiconfigurational (MC) SCF calculation was probably performed by Hartree and coworkers [2]. They realized that for the state of the oxygen atom, there where two possible configurations, and , and constructed the two configurational wave function:
The atomic orbitals were determined (numerically) together with the two expansion coefficients. Similar MCSCF calculations on atoms and negative ions were simultaneously performed in Kaunas, Lithuania, by Jucys [3]. The possibility was actually suggested already in 1934 in the book by Frenkel [4]. Further progress was only possible with the advent of the computer. Wahl and Das developed the Optimized Valence Configuration (OVC) Approach, which was applied to diatomic and some triatomic molecules [5, 6].
An important methodological step forward was the formulation of the Extended Brillouin's (Brillouin, Levy, Berthier) theorem by Levy and Berthier [7]. This theorem states that for any CI wave function, which is stationary with respect to orbital rotations, we have
where is an operator (see Eq. 9.32) that gives a wave function where the orbitals and have been interchanged by a rotation. The theorem is an extension to the multiconfigurational regime of the Brillouin theorem, which gives the corresponding condition for an optimized HF wave function. A forerunner to the BLB theorem can actually be found already in Löwdin's 1955 article [8, 9].
The early MCSCF calculations were tedious and often difficult to converge. The methods used were based on an extension of the HF theory formulated for open shells by Roothaan [10]. An important paradigm change came with the Super-CI method, which was directly based on the BLB theorem [11]. One of the first modern formulations of the MCSCF optimization problem was given by Hinze [12]. He also introduced what may be called an approximate second-order (Newton–Raphson) procedure based on the partitioning: , where is the unitary transformation matrix for the orbitals and is an anti-Hermitian matrix. This was later to become . The full exponential formulation of the orbital and CI optimization problem was given by Dalgaard and Jørgensen [13]. Variations in orbitals and CI coefficients were described through unitary rotations expressed as the exponential of anti-Hermitian matrices. They formulated a full second-order optimization procedure (Newton–Raphson, NR), which has since then become the standard. Other methods (e.g., the Super-CI method) can be considered as approximations to the NR approach.
One of the problems that the early applications of the MCSCF method faced was the construction of the wave function. It was necessary to keep it short in order to make the calculations feasible. Thus, one had to decide beforehand which where the most important CSFs to include in the CI expansion. Even if this is quite simple in a molecule like , it quickly becomes ambiguous for larger systems. However, the development of more efficient techniques to solve large CI problems made another approach possible. Instead of having to choose individual CSFs, one could choose only the orbitals that were involved and then make a full CI expansion in this (small) orbital space. In 1976, Ruedenberg introduced the orbital reaction space in which a complete CI expansion was used (in principle). All orbitals were optimized—the Fully Optimized Reaction Space—FORS [14].
An important prerequisite for such an approach was the possibility to solve large CI expansions. A first step was taken with the introduction of the Direct CI method in 1972 [15]. This method solved the problem of performing large-scale SDCI calculations with a closed-shell reference wave function. It was not useful for MCSCF, where a more general approach is needed that allows an arbitrary number of open shells and all possible spin-couplings. The generalization of the direct CI method to such cases was made by Paldus and Shavitt through the Graphical Unitary Group Approach (GUGA). Two papers by Shavitt explained how to compute CI coupling coefficients using GUGA [16, 17]. Shavitt's approach was directly applicable to full CI calculations. It formed the basis for the development of the Complete Active Space (CAS) SCF method, which has become the standard for performing MCSCF calculations [18, 19].
However, an MCSCF calculation only solves part of the problem—it can formulate a qualitatively correct wave function by the inclusion of the so-called static electron correlation. This determines the larger part of the wave function. For a quantitative correct picture, we need also to include dynamic electron correlation and its contribution to the total electronic energy. We devote a substantial part of the book to describe different methods that can be used. In particular, we concentrate on second-order perturbation theory with a CASSCF reference function (CASPT2). This method has proven to be accurate in many applications also for large molecules where other methods, such as MRCI or coupled cluster, cannot be used. The combination CASSCF/CASPT2 is the main computational tool to be discussed and illustrated in several applications.
This book mainly discusses the multiconfigurational approach in quantum chemistry; it includes discussions about the modern computational methods such as Hartree–Fock theory, perturbation theory, and various configuration interaction methods. Here, the main emphasis is not on technical details but the aim is to describe the methods, such that critical comparisons between the various approaches can be made. It also includes sections about the mathematical tools that are used and many different types of applications. For the applications presented in the last chapter of this book, the emphasis is on the practical problems associated with using the CASSCF/CASPT2 methods. It is hoped that the reader after finishing the book will have arrived at a deeper understanding of the CASSCF/CASPT2 approaches and will be able to use them with a critical mind.
1.1 References
[1] Heitler W, London F. Wechselwirkung neutraler Atome und homopolare Bindung nach der Quantenmechanik.
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[2] Hartree DR, Hartree W, Swirles B. Self-consistent field, including exchange and superposition of configurations, with some results for oxygen.
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[3] Jucys A. Self-consistent field with exchange for carbon.
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[4] Frenkel J.
Wave Mechanics, Advanced General Theory
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[5] Das G, Wahl AC. Extended Hartree-Fock wavefunctions: optimized valence configurations for
and
, optimized double configurations for
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[6] Wahl AC, Das G. The multiconfiguration self-consistent field method. In: Schaefer HF III, editor.
Methods of Electronic Structure Theory
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[7] Levy B, Berthier G. Generalized Brillouin theorem for multiconfigurational SCF theories.
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[8] Löwdin PO. Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction.
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[9] Roos BO. Perspective on “Quantum theory of many-particle systems I, II, and III” by Löwdin PO, [Phys Rev 1995;97:1474–1520].
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:228–230.
[10] Roothaan CCJ. Self-consistent field theory for open shells of electronic systems.
Rev Mod Phys
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[11] Grein F, Chang TC. Multiconfiguration wavefunctions obtained by application of the generalized Brillouin theorem.
Chem Phys Lett
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[12] Hinze J. MC-SCF. I. The multi-configuration self-consistent-field method.
J Chem Phys
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:6424–6432.
[13] Dalgaard E, Jørgensen P. Optimization of orbitals for multiconfigurational reference states.
J Chem Phys
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69
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[14] Ruedenberg K, Sundberg KR. MCSCF studies of chemical reactions. I. Natural reaction orbitals and localized reaction orbitals. In: eds Calais JL, Goscinski O, Linderberg J, Öhrn Y, editors.
Quantum Science; Methods and Structure
. New York: Plenum Press; 1976. p. 505.
[15] Roos BO. A new method for large-scale CI calculations.
Chem Phys Lett
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[16] Shavitt I. Graph theoretical concepts for the unitary group approach to the many-electron correlation problem.
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Chapter 2Mathematical Background
2.1 Introduction
From a basic point of view, orbitals are not the orbitals of some electron system, but they are a convenient set of one-electron basis functions. They may, or may not, solve some differential equations.
The ones that are most used in contemporary Quantum Chemistry are described in more technical detail further on. Here we just mention a few basic properties, and some mathematical facts and notations that come in handy. Later in the book it also describes the methods whereby the wave functions, which are detailed descriptions of the quantum states, can be approximated.
This chapter is also concerned with the practical methods to represent the many-electron wave functions and operators that enter the equations of quantum chemistry, specifically for bound molecular states.
It ends with some of the tools used to get properties and statistics out from multiconfigurational wave functions. They all turn out to be, essentially, “matrix elements,” computed from linear combinations of a basic kind of such matrix elements: the density matrices.
2.2 Convenient Matrix Algebra
There are numerous cases where linear or multilinear relations are used. Formulas may be written and handled in a very compact form, as in the case of orbitals being built from simpler basis functions (or other orbitals). The one-particle basis functions are arranged in a row vector, formally a matrix, and the coefficients for their linear combinations, often called MO coefficients, in an matrix , so that the orbitals , can be written very concisely as a matrix product:
that is,.
As an example, the orbital optimization procedure in a Quantum Chemistry program is frequently carried out by matrix operations such as, for example, thematrix exponential function, as shown in Chapter 9. The same approach can be extended also to handle many-particle wave functions.
Mathematically, the Schrödinger equation is usually studied as a partial differential equation, while computational work is done using basis function expansions in one form or another. The model assumption is that the wave functions lie in a Hilbert space, which contains the square-integrable functions, and also the limits of any convergent sequence of such functions: molecular orbitals, for example, would have to be normalizable, with a norm that is related to the usual scalar product:
This space of orbitals, together with the norm and scalar product, is called , a separable Hilbert space, which means that it can be represented by an infinite orthonormal basis set. Such a basis should be ordered, and calculations carried out using the first basis functions would be arbitrarily good approximations to the exact result if is large enough. There are some extra considerations, dependent on the purpose: for solving differential equations, not only the wave functions but also their derivatives must be representable in the basis, and so a smaller Hilbert space can be used. For quantum chemistry, this can be regarded as requiring that the expectation value of the kinetic energy operator should be finite, and the wave function should then lie in a subspace of , where also is finite (a so-called Sobolev space). While this is naturally fulfilled for most kinds of bases, it is not always so, for example, for finite element functions, wavelets, and in complete generality, issues such as completeness, convergence rate, and accuracy can be complicated.
Operators tend to be positions, partial derivatives, or functions of these. State vectors are usually wave functions with position variables, with spin represented by additional indices such as or . Examples are as follows:
We note that there are vector operators, which act by producing a vector with elements that are wave functions: in the natural way. We also note that operators are defined by their effect when acting on a function. One thing to look up for is using, for example, polar coordinates, a partial derivative or a operator may act on a vector expressed with the basis vectors . These are not constant vectors, and their derivatives yield extra terms, for example, for angular momentum operators. We also note that order matters—operators are usually not commutative, as seen for and . For any two operators and , one defines the commutator and the anticommutator :
The so-called Dirac notation, or bra-ket notation, is common and very useful. It is simply explained by starting with a vector space scalar product, which can be, for example,
where and are some vectors, and is some linear operator in that space. This can be an infinite-dimensional Hilbert space, like the Sobolev spaces, but this notation can be used for any general vector space. Dirac notation implies that another vertical bar symbol is introduced, and the syntax is then that this is a triple of the following constituents:
A vector, written
, called a “ket vector”
An operator, as before written as
A linear functional, written
, with the property that when “acting” on a ket vector, it produces a scalar value, usually complex.
The linear functional is an element of a linear vector space, formally the “dual space” of the ket space. It is called a “bra” vector or a “bra functional.” For a Hilbert space, its dual is also a Hilbert space, isomorphic with the ket space, and for the usual function spaces, they can be simply identified without causing any problems. The actual functions, used in integrals, can be used both as ket and bra vectors just by complex conjugation.
This is not entirely true for all spaces, or when “Dirac distributions” are used in the integrals. However, we usually feel free to use Dirac distributions as if they were functions, usually arising from the “resolution of the identity,” which starts with the well-known formula
which is true for any finite vector space with an orthonormal basis . It is also true for any so-called “separable” infinite Hilbert space, which is simply those in which there are infinite orthonormal bases . This is essentially all spaces that we have reason to use in Quantum Chemistry! There is just a couple of caveats: one must remember that the scalar product, and the norm, are then written in terms of integrals, which do not distinguish between any functions that differ only in isolated points. Function values in isolated points are not “useable,” and Dirac distributions do not formally have any place in the formalism. However, this particular problem disappears with the simple stratagem of regarding expressions involving Dirac distributions as constructs that imply the use of a “mollifier.” In this context it is just a parametrized function, which has the property of being nonnegative, bounded, zero for , and having the integral 1 if integrated from any negative to any positive value, and with the evaluation rule that the limit is to be taken finally. This allows us to define a unit operator as
and translate it to functions as
The multivariate extensions are obvious.
This also allows us to represent operators, by writing
