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,

1

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:

2

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.

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/.

Quantity	Symbol	Value
The fine structure constant
Vacuum speed of light		au
Avogadro's number
Energy		J
		kcal/mol
		eV
Electric dipole moment		C m
		Debye

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/.

Symbol	Meaning
	Wave function
	Configuration state function
	Orbital (spatial part)
	Spin orbital
	Spin wave function
	Basis function
	Electronic density
	Occupation number of (spin)orbital

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:

1.1

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

1.2

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:

1.3

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.

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:

1.4

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

1.5

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.