# Molecular orbitals and active space

During the same minute as asking this question, I also asked this at Quantum Computing SE.

In Qiskit, each qubit corresponds to one spin orbital. For example, the $$\ce{N2}$$ molecule would have 10 molecular orbitals, which correspond to 20 spin orbitals for alpha and beta spins for the sto-3g basis set. In my opinion, this is the case because the for each atom contained in the molecule, we need to account for the orbitals included, which are 1s, 2s, 2p for the nitrogen atom. However, in this case, how could one deal with the molecular orbitals when in the situation like choosing the active space for the molecule?

• +1 especially for a question that attracted such a thorough and detailed answer! However, what does this question have to do with PySCF? And what exactly are you asking? What precisely do you want to know when you ask: "how could one deal with the molecular orbitals when in the situation like choosing the active space for the molecule?" and what does that have to do with STO-3G or Qiskit? Aug 9 at 15:54
• Thanks for the advice. I have already modified the question. Aug 13 at 14:18
• At this point I would recommend you ask a new question. Aug 13 at 14:37

## 1 Answer

As you said above, nitrogen has 2 s orbitals and one p orbital. However, one would typically freeze the chemically inactive 1s orbital, leaving you 5 electrons in 4 orbitals per atom, or 10 electrons in 8 orbitals for N$$_2$$; this is denoted $$(10e,8o)$$.

Going beyond the minimal STO-3G basis, in addition to the occupied molecular orbitals (which are poorly described by STO-3G), you will have a large number of unoccupied molecular orbitals that are necessary to describe electron correlation.

Choosing the active space for complete-active space (CAS) self-consistent field (SCF) calculations is traditionally quite difficult, see e.g. Int. J. Quantum Chem. 111, 3329 (2011). The traditional CAS-SCF approach is equivalent to exact diagonalization of the real two-electron Hamiltonian in the space of electron configurations, i.e. the full configuration interaction (FCI) method. Since the challenge of the FCI solution scales exponentially, in practice the problem size is limited in practice to $$\lesssim(20e,20o)$$. Because of this limitation, it is not possible to include all valence orbitals in the CASSCF calculation already for diatomic molecules; however, the results one gets with various choices for the active orbitals may differ.

There are a number of alternative approaches to the FCI problem, such as the density matrix renormalization group (DMRG) method or various selected CI methods, which are able to push back the scaling wall and which have been found to be immensely useful for chemical applications. As a rule of thumb, these methods can handle strongly correlated electrons in three dimensions up to problem sizes of roughly (50e,50o). Future large-scale quantum computers are hailed as a panacea for chemistry, because they might be able to push back the problem size limit further to hundreds of electrons in hundreds of orbitals, at which point full valence orbital active spaces become feasible.

Because of the importance of the choice of the active orbitals in traditional CAS-SCF calculations, a multitude of ways in which to choose the active orbitals has been suggested. For general 3D molecular systems, a variety of strategies can be chosen. Using the canonical orbitals is not advised, because the unoccupied orbitals are not good estimates of excited states. Instead, common ways to pick the natural orbitals include employing natural orbitals from a lower level of theory; for instance, unrestricted Hartree-Fock natural orbitals (UNO) have been found to yield good orbitals for CAS calculations; MP2 and CISD are other commonly-used alternatives. Another established alternative are the so-called improved virtual orbitals, which are a way to improve the choice for the unoccupied orbitals to include in the active space. It is also possible to define the active space directly in terms of a reference set of gas-phase atomic orbitals, see J. Chem. Theory Comput. 2017, 13, 9, 4063–4078

In diatomic molecules like N$$_2$$, the choice can be done simply by symmetry: as the Hamiltonian does not depend on the angle measured around the bond, the orbitals are periodic with respect to this angle, $$\exp(i m \phi)$$ (see e.g. Int J. Quantum Chem. 119, e25968 (2019)), and the orbitals can be classified into $$\sigma$$ ($$m=0$$), $$\pi$$ ($$m=\pm 1$$), $$\delta$$ ($$m=\pm 2$$), $$\varphi$$ ($$m=\pm 3$$), etc orbitals. In N$$_2$$, the two s orbitals both yield a $$\sigma$$ orbital and the two p orbitals yield 2 $$\pi$$ and 2 $$\sigma$$ orbitals, yielding an active space of 2 $$\pi$$ and 4 $$\sigma$$ orbitals (a $$\sigma$$ orbital fits 2 electrons, while the $$\pi$$ and higher orbitals each fit 4 electrons).

However, the choice of the initial orbitals should matter less and less when the size of the active space is increased. If all the valence orbitals fit into the active space, the orbital optimization in CASSCF should converge quite rapidly even when begun from the canonical orbitals.

• +1 for another very thorough answer! I agree with your rule of thumb about being able to handle strongly correlated electrons up to roughly (50e,50o). We did (54e,54o) in this paper: aip.scitation.org/doi/10.1063/1.5063376, but it was not an easy calculation (and this particular active space was not the worst in terms of strong correlation). Aug 9 at 15:49
• Thanks for the thorough answer! That really helps a lot. I still have one technical question. Take the H2O molecule as an excample, according to the MO diagram here commons.wikimedia.org/wiki/File:H2O-MO-Diagram.svg, I would like to choose the 3a1 & 1b1 orbital to be not in the active space. In this case, is the PySCF pakage arrange the orbitals according to the MO orbitals so that I can just specify a list like [0,1,4,5] to be the active MO space assume that the organization is from down to up in PySCF? Aug 13 at 14:02
• @ironmanaudi You can turn on symmetry in PySCF; some of the MO integrals will be zero. The active space can then be defined directly in terms of the number of orbitals in each symmetry block. This should be illustrated in the PySCF examples. Aug 13 at 16:40