A qualitative sketch of multi-reference wavefunction methods

Complete active space perturbation theory (CASPTn) is a multireference electron correlation method for computational investigation of molecular systems, especially for those with heavy atoms such as

transition metals In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that ca ...

lanthanides The lanthanide () or lanthanoid () series of chemical elements comprises the 15 metallic chemical elements with atomic numbers 57–71, from lanthanum through lutetium. These elements, along with the chemically similar elements scandium and ytt ...

, and

actinides The actinide () or actinoid () series encompasses the 15 metallic chemical elements with atomic numbers from 89 to 103, actinium through lawrencium. The actinide series derives its name from the first element in the series, actinium. The inform ...

. It can be used, for instance, to describe electronic states of a system, when single reference methods and

density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...

cannot be used, and for heavy atom systems for which quasi-relativistic approaches are not appropriate. Although perturbation methods such as CASPTn are successful in describing the molecular systems, they still need a Hartree-Fock wavefunction to provide a valid starting point. The perturbation theories cannot reach convergence if the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are degenerate. Therefore, the CASPTn method is usually used in conjunction with the

Multi-configurational self-consistent field 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 mole ...

method (MCSCF) to avoid near-degeneracy correlation effects.

History

In the early 1960s, the

perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...

in quantum chemical applications was introduced. Since, there has been a wide spread of uses of the theory through software such as

Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...

. The perturbation theory correlation method is used routinely by the non-specialists. This is because it can easily achieve the property of size extensivity comparing to other correlation methods. During the starting point of the uses of perturbation theory, the applications using the method were based on nondegenerate many-body perturbation theory (MBPT). MBPT is a reasonable method for atomic and molecular system which a single non-degenerate

Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...

can represent zeroth-order electronic description. Therefore, MBPT method would exclude atomic and molecular states, especially excited states, which cannot be represented in zeroth order as single Slater determinants. Moreover, the perturbation expansion would converges very slowly or not at all if the state is degenerate or near degenerate. Such degenerate states are often the case of atomic and molecular valence states. To counter the restrictions, there was an attempt to implement second-order perturbation theory in conjunction with complete active space self-consistent field (CASSCF) wave functions. At the time, it was rather difficult to compute three- and four-particle density matrices which are needed for matrix elements involving internal and semi-internal excitations. The results was rather disappointing with little or no improvement from usual CASSCF results. Another attempt was made in 1990, where the full interacting space was included in the first-order wave function while zeroth-order Hamiltonian was constructed from a Fock-type one-electron operator. For cases which has no active orbitals, the Fock-type one-electron operator that reduces to the Møller–Plesset-Plesset Hartree-Fock (HF) operator. A diagonal Fock operator was also used to make a computer implementation simple and effective.

References

{{Reflist Electronic structure methods