Combinatorics within Configuration Interaction

[Morten]

Consider six electrons allocated in three orbitals in a closed shell restrictred HF ground state determinant. Now, consider three excited states: How many CI configurations can result from this system when only two electrons are excited per configuration? Is it true that there are 9 * 9 + 36 = 117 configurations, which would be as follows:

Same orbitals --> Same orbtials: 9, Same orbitals --> Different orbitals: 9, Different orbitals --> Same orbitals: 9, --> Different orbitals --> Different orbitals: 9. This is 36 possibilites. I assume that the 81 possibilities arise from spin combinations. If this is true (if not, where do they then origin from), which of the four cases would have different spin combinations?

 

[eljose79]

I would like to clarify and provide some additional information regarding the question posed in the forum post.

Firstly, when discussing excited states in a closed shell restricted HF ground state determinant, it is important to note that these states are typically referred to as virtual orbitals, as they are not occupied by electrons in the ground state. Therefore, the terminology used in the post may be a bit misleading.

Additionally, when considering excited states, it is important to specify which type of excitation is being referred to. In this case, it is assumed that the excitation refers to single electron excitations, meaning that only one electron is excited from one orbital to another, resulting in a singly excited state.

Now, to address the question of how many CI configurations can result from this system when only two electrons are excited per configuration, the answer is not as straightforward as the post suggests. Firstly, it is important to note that CI (Configuration Interaction) is a method used to approximate the wavefunction of a molecule by including multiple electronic configurations, rather than just the ground state determinant. Therefore, the number of CI configurations that can result from this system depends on the number of electronic configurations that are included in the CI calculation.

Secondly, the number of possible electronic configurations that can be included in a CI calculation is not solely dependent on the number of electrons that are excited per configuration. It also depends on the number of orbitals and the spin multiplicity of the system. For example, in the case of a closed shell restricted HF ground state determinant with six electrons, there are 15 possible electronic configurations that can be included in a CI calculation (assuming all singly excited states). This is calculated by taking the number of possible ways to distribute two electrons among six orbitals (15), multiplied by the number of possible spin combinations (2 for each configuration).

Therefore, the total number of CI configurations that can result from this system when only two electrons are excited per configuration is not 117, but rather 15. This number may increase if higher excitations (such as double or triple excitations) are included in the calculation.

In summary, the number of possible CI configurations that can result from this system depends on the number of electronic configurations that are included in the calculation, which in turn depends on the number of orbitals and the spin multiplicity of the system. The number of electronic configurations is not solely determined by the number of electrons that are excited per configuration.