Complete Active Space Self Consistent Field

The Complete Active Space Self Consistent Field Method describes a method for solving the time-independent Schrödinger equation (or its relativistic generalizations), which is used in quantum chemistry . It represents a logical extension of the Configuration Interaction (CI) and belongs to the group Multi-configurational self-consitent field methods (MCSCF).

Basics

In the configuration interaction, the orbitals of a converged Hartree-Fock calculation are typically used to construct the excited configuration state functions (CSF) and remain unchanged during the optimization of the CI coefficients. However, the RF orbitals are optimized for a single determinant wave function and therefore not necessarily optimal for a multi ‐ determinant wave function. With an MCSCF, not only the coefficients of the CSFs, but also the orbitals are varied. Since this method is much more complex than CI, the CI development must be kept small. In contrast to CI, where all excitations of a certain degree are taken into account, with MCSCF the selection of the CSFs to be used is selected at the level of the orbitals. For the CASSCF, an active orbital space (Complete Active Space) is selected. This usually consists of the orbitals that change most during the reaction under consideration, i.e. generally the frontier orbitals and the nearer occupied and unoccupied orbitals. A full CI is now carried out in this orbital space, ie all possible suggestions that can be constructed with these orbitals are used for the CI development. The CASSCF wave function usually describes only a relatively small part of the correlation energy. If, on the other hand, you consider relative energies, which is what you usually do in chemistry (e.g. thermodynamics, energy barriers, etc.), this procedure makes perfect sense, because the lower orbitals hardly change during a reaction and as long as they are equally bad for products and starting materials an inaccurate description of these orbitals does not falsify the result. The orbitals that change the most and thus provide the greatest relative energy contribution to a reaction are well described by the CASSCF, which is necessary to obtain reasonable energies.

Meaning for multi-reference systems

The CASSCF has a great advantage that comes into play with so-called multi-reference (MR) systems. Multi-reference means that the wave function can no longer be described qualitatively correctly by a single Slater determinant , as in the Hartree-Fock theory. In many cases, one-determinant method delivers qualitatively correct results. In some cases, with homolytic bond cleavage or with briadical systems, qualitatively incorrect results are obtained with a determinant. In these cases, the system is said to have a multi-reference character. In such cases, the RF determinant does not provide a sufficient description and consequently its orbitals are not necessarily meaningful. The CASSCF orbitals are more suitable here, as they provide a qualitatively correct description with the correct active space. Just as in post-HF methods you start with an HF determinant (“reference”) and add higher stimuli to it (e.g. through configuration interaction or disturbance theory ), you can also start with a CASSCF as a reference and to this Add suggestions. This then leads to the highly precise MR methods such as B. MRCI and MRMP.

Process of a CASSCF invoice

The orbital expansion coefficients and the expansion coefficients of the CI development are optimized in the calculation. This is generally carried out in the so-called Super-CI procedure, which is based on the Brillouin-Levy-Berthier theorem (BLB):

${\displaystyle \langle {0}|{\hat {H}}({\hat {E}}_{pq}-{\hat {E}}_{qp})|0\rangle {}=0}$

The spin-mediated ladder operators are defined as

${\displaystyle {\hat {E}}_{pq}=\sum _{\sigma }{\hat {a}}_{p\sigma }^{\dagger }{\hat {a}}_{q\sigma }}$

Here stands for the spin and the indices and relate to all orbital of the active space. The BLB theorem thus represents a condition for the variational optimization of the state (analogous to the Brillouin theorem in the CI formalism). For this purpose, the so-called Super-CI wave function is defined which contains a sum of all single excitations relative to : ${\displaystyle \sigma }$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle |0\rangle {}}$ ${\displaystyle |SCI\rangle {}}$ ${\displaystyle |0\rangle {}}$

${\displaystyle |SCI\rangle {}=|0\rangle {}+\sum _{p<q}x_{pq}({\hat {E}}_{pq}-{\hat {E}}_{qp})|0\rangle {}=|0\rangle {}+\sum _{p<q}x_{pq}|p\rightarrow q\rangle {}.}$

Where are the expansion coefficients. An MCSCF invoice then generally proceeds as follows: ${\displaystyle x_{pq}}$

• The normal CI secular equations for the MCSCF wave function are solved with the start orbitals (e.g. from a Hartree-Fock calculation) to obtain the CI expansion coefficients.
• The super CI wave function is constructed and the corresponding secular equation is solved to obtain. ${\displaystyle x_{pq}}$
• The new set of orbitals is calculated according to the equation . ${\displaystyle \phi _{p}^{(1)}=\phi _{p}^{(0)}+\sum _{p\neq q}x_{pq}\phi _{q}^{(0)}}$

These steps are repeated until convergence is reached, so that the orbitals no longer need to be corrected.

Individual evidence

1. ↑ Björn O. Roos, Peter R. Taylor, Per EM Sigbahn: A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach . In: Chemical Physics . tape 48 , no. 2 , May 1980, ISSN  0301-0104 , p. 157-173 , doi : 10.1016 / 0301-0104 (80) 80045-0 .
2. ↑ ^{a b} Jørgensen, Poul; Olsen, Jeppe .: Molecular electronic-structure theory . Wiley, Chichester 2000, ISBN 0-471-96755-6 .
3. ^ Björn O. Roos: The complete active space SCF method in a fock-matrix-based super-CI formulation . In: International Journal of Quantum Chemistry . tape 18 , S14, June 19, 2009, pp. 175-189 , doi : 10.1002 / qua.560180822 .
4. ^ Bernard Levy, Gaston Berthier: Generalized brillouin theorem for multiconfigurational SCF theories . In: International Journal of Quantum Chemistry . tape 2 , no. 2 , March 1968, ISSN  0020-7608 , p. 307-319 , doi : 10.1002 / qua.560020210 .
5. ↑ Christof Walter: Excitonic States and Optoelectronic Properties of Organic Semiconductors - A Quantum-Chemical Study Focusing on Merocyanines and Perylene-Based Dyes Including the Influence of the Environment . 2015, urn : nbn: de: bvb: 20-opus-123494 (English, uni-wuerzburg.de ).