Multi-configurational self-consistent field
Encyclopedia
Multi-configurational self-consistent field (MCSCF) is a method in quantum chemistry
used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory
are not adequate (e.g., for molecular ground states which are quasi-degenerate with low lying excited states or in bond breaking situations). It uses a linear combination of configuration state function
s (CSF) or configuration determinants to approximate the exact electronic wavefunction
of an atom or molecule. In an MCSCF calculation, the set of coefficients of both the CSFs or determinants and the basis functions in the molecular orbitals are varied to obtain the total electronic wavefunction with the lowest possible energy. This method can be considered a combination between configuration interaction
(where the molecular orbitals are not varied but the expansion of the wave function) and Hartree–Fock (where there is only one determinant but the molecular orbitals are varied).
MCSCF wave functions are often used as reference states for Multireference configuration interaction
(MRCI) or multi-reference perturbation theories like complete active space perturbation theory (CASPT2). These methods can deal with extremely complex chemical situations and, if computing power permits, may be used to reliably calculate molecular ground- and excited states if all other methods fail.
where Ni is a normalization constant. The ground state wavefunction for H2 at the equilibrium geometry is dominated by the configuration (φ1)2, which means the molecular orbital φ1 is nearly doubly occupied. The Hartree–Fock model assumes it is doubly occupied, which leads to a total wavefunction of
where Θ2,0 is the singlet (S = 0) spin function for two electrons. The molecular orbitals in this case φ1 are taken as sums of 1s atomic orbitals on both atoms, namely N1(1sA + 1sB). Expanding the above equation into atomic orbitals yields
This Hartree-Fock model gives a reasonable description of H2 around the equilibrium geometry - about 0.735Å for the bond length (compared to a 0.746Å experimental value) and 84 kcal/mol for the bond energy (exp. 109 kcal/mol). This is typical of the HF model, which usually describes closed shell systems around their equilibrium geometry quite well. At large separations, however, the terms describing both electrons located at one atom remain, which corresponds to dissociation to H+ + H−, which has a much larger energy than H + H. Therefore, the persisting presence of ionic terms leads to an unphysical solution in this case.
Consequently, the HF model cannot be used to describe dissociation processes with open shell products. The most straightforward solution to this problem is introducing coefficients in front of the different terms in Ψ1:
which forms the basis for the valence bond
description of chemical bond
s. With the coefficients CIon and CCov varying, the wave function will have the correct form, with CIon=0 for the separated limit and CIon comparable to CCov at equilibrium. Such a description, however, uses non-orthogonal basis functions, which complicates its mathematical structure. Instead, multiconfiguration is achieved by using orthogonal molecular orbitals. After introducing an anti-bonding orbital
the total wave function of H2 can be written as a linear combination of configurations built from bonding and anti-bonding orbitals:
where Φ2 is the electronic configuration (φ2)2. In this multiconfigurational description of the H2 chemical bond, C1 = 1 and C2 = 0 close to equilibrium, and C1 will be comparable to C2 for large separations.
SCF method (CASSCF), where the linear combination of CSF
s includes all that arise from a particular number of electrons in a particular number of orbitals (also
known as full-optimized reaction space (FORS-MCSCF)). For example, one might define CASSCF(11,8) for the molecule, NO, where the 11 valence electrons are distributed between all configurations that can be constructed from 8 molecular orbitals.
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
are not adequate (e.g., for molecular ground states which are quasi-degenerate with low lying excited states or in bond breaking situations). It uses a linear combination of configuration state function
Configuration state function
In quantum chemistry, a configuration state function , is a symmetry-adapted linear combination of Slater determinants. A CSF must not be confused with a configuration.-Definition:...
s (CSF) or configuration determinants to approximate the exact electronic wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
of an atom or molecule. In an MCSCF calculation, the set of coefficients of both the CSFs or determinants and the basis functions in the molecular orbitals are varied to obtain the total electronic wavefunction with the lowest possible energy. This method can be considered a combination between configuration interaction
Configuration interaction
Configuration interaction is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, configuration simply describes the linear combination...
(where the molecular orbitals are not varied but the expansion of the wave function) and Hartree–Fock (where there is only one determinant but the molecular orbitals are varied).
MCSCF wave functions are often used as reference states for Multireference configuration interaction
Multireference configuration interaction
In quantum chemistry, the multireference configuration interaction method consists in a configuration interaction expansion of the eigenstates of the electronic molecular Hamiltonian in a set of Slater determinants which correspond to excitations of the ground state electronic configuration but...
(MRCI) or multi-reference perturbation theories like complete active space perturbation theory (CASPT2). These methods can deal with extremely complex chemical situations and, if computing power permits, may be used to reliably calculate molecular ground- and excited states if all other methods fail.
Introduction
For the simplest single bond, found in the H2 molecule, molecular orbitals can always be written in terms of two functions χiA and χiB (which are atomic orbitals with small corrections) located at the two nuclei,where Ni is a normalization constant. The ground state wavefunction for H2 at the equilibrium geometry is dominated by the configuration (φ1)2, which means the molecular orbital φ1 is nearly doubly occupied. The Hartree–Fock model assumes it is doubly occupied, which leads to a total wavefunction of
where Θ2,0 is the singlet (S = 0) spin function for two electrons. The molecular orbitals in this case φ1 are taken as sums of 1s atomic orbitals on both atoms, namely N1(1sA + 1sB). Expanding the above equation into atomic orbitals yields
This Hartree-Fock model gives a reasonable description of H2 around the equilibrium geometry - about 0.735Å for the bond length (compared to a 0.746Å experimental value) and 84 kcal/mol for the bond energy (exp. 109 kcal/mol). This is typical of the HF model, which usually describes closed shell systems around their equilibrium geometry quite well. At large separations, however, the terms describing both electrons located at one atom remain, which corresponds to dissociation to H+ + H−, which has a much larger energy than H + H. Therefore, the persisting presence of ionic terms leads to an unphysical solution in this case.
Consequently, the HF model cannot be used to describe dissociation processes with open shell products. The most straightforward solution to this problem is introducing coefficients in front of the different terms in Ψ1:
which forms the basis for the valence bond
Valence bond theory
In chemistry, valence bond theory is one of two basic theories, along with molecular orbital theory, that were developed to use the methods of quantum mechanics to explain chemical bonding. It focuses on how the atomic orbitals of the dissociated atoms combine to give individual chemical bonds...
description of chemical bond
Chemical bond
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...
s. With the coefficients CIon and CCov varying, the wave function will have the correct form, with CIon=0 for the separated limit and CIon comparable to CCov at equilibrium. Such a description, however, uses non-orthogonal basis functions, which complicates its mathematical structure. Instead, multiconfiguration is achieved by using orthogonal molecular orbitals. After introducing an anti-bonding orbital
the total wave function of H2 can be written as a linear combination of configurations built from bonding and anti-bonding orbitals:
where Φ2 is the electronic configuration (φ2)2. In this multiconfigurational description of the H2 chemical bond, C1 = 1 and C2 = 0 close to equilibrium, and C1 will be comparable to C2 for large separations.
Complete active space SCF
A particularly important MCSCF approach is the complete active spaceComplete active space
In quantum chemistry, a complete active space is a type of classification of molecular orbitals. Spatial orbitals are classified as belonging to three classes:* core, always hold two electrons* active, partially occupied orbitals...
SCF method (CASSCF), where the linear combination of CSF
Configuration state function
In quantum chemistry, a configuration state function , is a symmetry-adapted linear combination of Slater determinants. A CSF must not be confused with a configuration.-Definition:...
s includes all that arise from a particular number of electrons in a particular number of orbitals (also
known as full-optimized reaction space (FORS-MCSCF)). For example, one might define CASSCF(11,8) for the molecule, NO, where the 11 valence electrons are distributed between all configurations that can be constructed from 8 molecular orbitals.
Restricted active space SCF
Since the number of CSFs quickly increases with the number of active orbitals, along with the computational cost, it may be desirable to use a smaller set of CSFs. One way to make this selection is to restrict the number of electrons in certain subspaces, done in the restricted active space SCF method (RASSCF). One could, for instance, allow only single and double excitations from some strongly-occupied subset of active orbitals, or restrict the number of electrons to at most 2 in another subset of active orbitals.See also
- Charlotte Froese FischerCharlotte Froese FischerAcad. Prof. Dr. Charlotte Froese Fischer PhD is a Canadian-American applied mathematician and computer scientist who gained world recognition for the development and implementation of the Multi-configurational Hartree-Fock approach to atomic structure calculations and for her theoretical...
- Douglas HartreeDouglas HartreeDouglas Rayner Hartree PhD, FRS was an English mathematician and physicist most famous for the development of numerical analysis and its application to the Hartree-Fock equations of atomic physics and the construction of the meccano differential analyser.-Early life:Douglas Hartree was born in...
- Hartree–Fock method
- Quantum chemistry computer programsQuantum chemistry computer programsQuantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock and some post-Hartree–Fock methods. They may also include density functional theory , molecular mechanics or semi-empirical quantum...