Constructing Molecular π-Orbital Active Spaces for Multireference Calculations of Conjugated Systems

Abstract

Molecules with conjugated π-systems often feature strong electron correlation and therefore require multireference methods for a reliable computational description. A key prerequisite for the successful application of such methods is the choice of a suitable active space. Herein the automated π-orbital space (PiOS) method for selecting active spaces for multireference calculations of conjugated π-systems is presented. This approach allows the construction of small but effective active spaces based on Hückel theory. To demonstrate its performance, π→π* excitations for benzene, octatetraene, and free-base porphin are computed. In addition, this technique can be combined with the automated atomic valence active space method to compute excitations in complex systems with multiple conjugated fragments. This combined approach was used to generate two-dimensional potential energy surfaces for multiple electronic states associated with photoinduced electron-coupled double proton transfer in the blue light-using flavin photoreceptor protein. These types of methods for the automated selection of active space orbitals are important for ensuring consistency and reproducibility of multireference approaches for a wide range of chemical and biological systems.

Graphical Abstract

I. Introduction

Conjugated π systems have attracted considerable attention in the field of chemistry. Organic polymers with conjugated π-bonds often exhibit properties of semiconductors or superconductors^1–4 and therefore serve as valuable components of organic photovoltaic devices^5–6 and organic light-emitting diodes.^7–8 Many biologically important molecular complexes containing conjugated π-bonds, including chlorophyll, hemoglobin, vitamin B12, β-carotene, and cytochromes, are responsible for vital functions in living organisms. The electronic structure characteristics of conjugated π-systems that are responsible for this rich chemistry also complicate the theoretical study of such systems. In particular, these systems often exhibit strong electron correlation, with multiple near-degenerate electronic states and multiconfigurational character. Therefore, a reliable computational description of such systems requires multiconfigurational and/or multireference methods.^9–39 A major challenge in applying such methods in practice is the choice of a suitable active space, which strongly impacts the quality of the results.

The traditional methods of active space selection are accompanied by several problems, including difficulties in identifying the molecular orbitals (MOs) responsible for the principal chemical properties and the necessity of manual selection of the active orbitals. While the manual selection of an active space following general guidelines⁴⁰ is still a feasible task in the case of small molecules or single geometries, it becomes impractical in studies of potential energy surfaces of large complex systems. To ensure a consistent active space (i.e., the same size and same atomic orbital character) along various reaction coordinates and while exploring multidimensional potential energy surfaces, a more systematic, and preferably automated, approach is desirable. This issue is even more amplified by the recent surge in novel electronic structure methods that allow the efficient treatment of large active spaces.^{29, 31, 37, 41–43}

Recently several efforts^44–47 have been directed toward the development of automated active space selection algorithms. These efforts aim to turn the selection procedure into black-box routines, thus avoiding the extensive user input still required for most calculations. In addition, such methods also improve the reproducibility of calculations by other researchers, which is often a challenge. Ref. ⁴⁶ describes the atomic valence active space (AVAS) method, an automated active space selection scheme based on the projection of occupied and virtual molecular orbitals onto the target valence atomic orbitals. This technique is related to other techniques of constructing valence virtual orbital spaces^48–49 but differs in both purpose and mathematical construction. The AVAS method has been shown to be general and flexible for constructing active spaces for many types of systems. In the case of conjugated π-systems, however, the active spaces generated from projecting the MOs onto all valence atomic orbitals of the target atoms will generally be much larger than required because only projection onto the valence atomic orbitals contributing to the π-system is necessary.

To alleviate this problem, herein we describe an automated approach based on Hückel MO theory,^50–52 which is particularly well-suited for conjugated π-systems. This π-orbital space (PiOS) method first builds initial π-orbitals from a single-reference wavefunction, using a projection approach similar to that presented in Ref. ⁴⁶. In this initial π-space, an effective Hamiltonian is diagonalized to obtain a set of energy-ordered π MOs that span the space. Depending on the specific application, the user can either select a small active space consisting of only a few of the highest-occupied molecular orbitals (HOMOs) and lowest-unoccupied molecular orbitals (LUMOs) of this space, or a larger active space consisting of more MOs, up to the entire π-orbital space if this is required. On the basis of classical MO theory, we expect this procedure to allow the selection of the minimal set of the most important π-orbitals. We examined the performance of this proposed method on typical π-conjugated systems, such as benzene, octatetraene, and free-base porphine. We also demonstrated how this approach can be combined with the AVAS method to study the large active site of the blue light-using flavin (BLUF) photoreceptor protein. This approach is thereby shown to enable chemically and biologically relevant calculations with quantitative multi-reference techniques even for multidimensional potential energy surfaces of complex systems, a task that currently would be difficult to execute without such methods.

II. Theory

This section explains how to construct π-orbitals from a single-reference wavefunction obtained from self-consistent field (SCF) calculations using linear algebraic transformations and Hückel theory. We first explain how to determine the plane that acts as the effective nodal surface of the π-system by considering the positions of the atoms comprising the π-system. Then we describe how to determine the number of π-electrons and the number of π-orbitals for a given system. Finally, we show how to obtain the molecular π-orbitals for an effective active space.

A. Determining spatial orientation of the π-system

Molecular systems with conjugated π-bonds typically have a planar structure. Herein M denotes the set of main group atoms A that contribute to the π-system within a molecule or complex. If these atoms lie in the xy plane at z ≈ 0, then the corresponding π-system is spanned by their valence p_z orbitals. However, in an arbitrary molecular system, the molecule can be oriented differently. In this case, we need to first identify the spatial direction $n\in {ℝ}^3$ aligned with the direction of the local $p_z^{\prime }$ orbitals of M.

If the molecule were exactly planar, this vector n would be the normal of the plane in which all atoms comprising the π-system lie. If the molecule is not exactly planar, an approximate n must be determined. For this purpose, let {R_A;A ∈ M} denote the three-dimensional column vectors denoting the positions of the atoms A of M contributing to the π-system. We first determine the “center of local π-system coordinates” (i.e., its centroid, which is an analogue of the center of mass but without mass weighting), represented by a 3-dimensional column vector as:

$$R_C:=\frac{1}{\left|M\right|}\sum _{A\in M}{R}_A\mathrm{..}$$ (1)

Here | M | denotes the number of atoms comprising the π-system. Then the system’s 3×3 inertial tensor matrix T around R_c is formed as

$$T:=\sum _{A\in M}\left(R_A-R_C\right){\left(R_A-R_C\right)}^{†}.$$ (2)

Diagonalization of this positive definite matrix T. yields the principal axes for the local π-system coordinates. The (normalized) eigenvector with the smallest eigenvalue is selected as the direction n. Note that this eigenvalue will be exactly zero if the molecule is exactly flat and is typically very close to zero for approximately planar structures.

B. Generating spatially oriented atomic orbitals

Once the direction n is determined, the local molecule-oriented $p_{z,A}^{\prime }$ orbital of each atom A is then obtained as a linear combination of the three valence p_x,A, p_y,A., p_z,A atomic orbitals (AOs) of the atom A in the global Cartesian coordinate system as:

$$p_{z,A}^{\prime }:=n_x{p}_{x,A}+n_y{p}_{z,A}+n_z{p}_{z,A}(A\in M)$$ (3)

In general, basis functions of an arbitrary computational basis set have more variational degrees of freedom than atomic orbitals and do not necessarily correspond to the s-, p-, d-type atomic orbitals in a chemically intuitive sense. Therefore, a second auxiliary AO basis should be employed. A minimal basis set of tabulated free-atom AOs, denoted MINAO,⁵³ or subsets of atomic natural orbital basis sets, such as ANO-RCC,^54–55 ano-pVnZ,⁵⁶ or ANO-VT-XZ,⁵⁷ can be used for this purpose.

To simplify notation, we will assume that the set of locally oriented, $p_z^{\prime }$ orbitals is collected in a single matrix O with the dimension of N_MINAO × N_π-AO, where N_MINAO. denotes the number of basis functions in a suitable minimal AO basis set of the entire molecule or complex (see also Sec. D), and N_π-AO denotes the number of valence π-orbitals on M. Typically, each main group atom in M will contribute exactly one valence $p_z^{\prime }$ orbital to the π system, such that | M | = N_π–AO. Each column of matrix O then represents one of the functions in Eq. (3) on a given main group atom. Specifically, it contains the coefficients n_x, n_y, n_z in rows corresponding to atom A’s valence p_x, p_y, p_z. AOs, respectively, and 0 for all other rows.

C. Determining the number of occupied and virtual π MOs

For a given molecule, the number of electrons N_π,e its π-system is supposed to contain must be determined. For example, a neutral benzene molecule, as well as an anionic Cp⁻ ligand, has six π electrons. The PiOS method finds the number of π-electrons automatically based on atomic connectivity using the following algorithm. First, the method determines the bond order for each atom of a given π-system, assuming two atoms i and j with covalent radii R_i and R_j., respectively, (see Ref. ⁵⁸, for example) are bonded if r_ij < 1.3(R_i + R_j). Based on this information, the scheme identifies the number of electrons each atom in M normally contributes to the π-system. For example, each sp² carbon atom in M typically contributes one π-electron regardless of connectivity, while each N or P atom in the aromatic system M will typically contribute one or two π-electrons if it is connected by two or three σ-bonds, respectively, to the rest of the system. If necessary, for individual atoms, the number of contributed π-electrons can be fixed manually based on user input, and the total number of π-electrons, as obtained via the sum of atomic contributions, can be adjusted up for anionic or down for cationic systems. This procedure allows for a simple and yet flexible way of fixing the number of π-electrons. Note that the validity and accuracy of these user choices can be easily established based on the projection eigenvalues discussed below.

Based on N_π,e., the total number of π-electrons in M, and N_π-AO, the number of locally-oriented $p_z^{\prime }$ orbitals in the π-system of M, the number of the full molecular system’s occupied and virtual MOs representing occupied and virtual π spatial MOs on M can be determined. Specifically, these numbers are

$$N_{\pi ,\text{occ}}=\frac{N_{\pi ,e}}{2}$$ (4)

$$N_{\pi ,\text{vir }}=N_{\pi -\text{AO}}-N_{\pi ,\text{occ}}.$$ (5)

D. Representation of the full molecule’s core and valence atomic orbitals

To describe the core and valence atomic orbitals of the molecular system, we use Intrinsic Atomic Orbitals (IAOs),⁵³ rather than using the MINAO directly. The IAOs represent polarized AOs and describe the occupied orbitals of the molecular SCF wavefunction | Φ〉 exactly, i.e., they correspond to chemical atomic orbitals. While the basis functions of an arbitrarily chosen computational basis set do not correspond directly to the AOs, the MINAO or ANO basis sets account for the molecular environment. Thus, it is advantageous to obtain the IAOs for a given molecular system and use them to describe the AOs. [Note that for cases in which users wish to create a “double-shell” π-space,^59–60 they would need to use an ANO basis set from which to draw reference AOs, rather than the MINAO or IAOs described here, and then include both valence shell n and (n+1) p_z AOs in the PiOS construction. This procedure is fully analogous to that used to include such effects in the AVAS method.⁴⁶]

Let B₁ denote the main computational basis set (e.g., cc-pVTZ) and B₂. denote the auxiliary basis set (e.g., MINAO). S₁. and S₂ represent the overlap matrices for the basis functions μ ∈ B₁ and ν ∈ B₂, respectively. S₁₂ denotes the overlap matrix between functions of the two bases with elements [S₁₂]_μν = 〈μ | ν〉. Herein we employ the updated IAO construction approach,⁶¹ in which the | B₁ |× | B₂ | coefficient matrix R of the IAOs in the B₁ basis set is given as:

$$C_{\text{occ}}^{\prime }=S_2{}^{-1}{S}_{21}{C}_{\text{occ}},$$ (6)

$$S^{\prime }=C_{\text{occ}}^{\prime }{}^{†}{S}_2{C}_{\text{occ}}^{\prime }.,$$ (7)

$$R=S_1^{-1}{S}_{12}+C_{\text{occ}}{C}_{\text{occ}}^{†}{S}_{12}-S_1^{-1}{S}_{12}{C}_{\text{occ}}^{\prime }{S}^{\prime -1}{C}_{\text{occ}}^{\prime }{}^{†}{S}_2,$$ (8)

where C_occ denotes the occupied MO matrix of the molecular system. Note that the IAOs are not orthogonalized.

E. Isolation of the MOs corresponding to the π-system

Let | Φ〉 denote a closed-shell Slater determinant with N_occ occupied MOs describing the electronic structure of the system at the SCF level (i.e., Hartree-Fock or Kohn-Sham formulation of density functional theory). Its occupied and virtual MOs are expressed as:

$$|i\rangle =\sum _{\mu \in B_1}|\mu \rangle C_i^{\mu },$$ (9)

$$|a\rangle =\sum _{\mu \in B_1}|\mu \rangle {\overline{C}}_a^{\mu }.,$$ (10)

where μ are basis functions from the employed computational basis set B₁, and $C_i^{\mu }={\left[C_{\text{occ}}\right]}_{\mu i}$ and ${\overline{C}}_a^{\mu }={\left[C_{\text{vir}}\right]}_{\mu a}$ are the coefficients of the basis function μ in the expansion of the occupied orbital i and virtual orbital a MOs, respectively. C_occ and C_vir denote the | B₁ | × N_occ occupied and | B₁ | × N_vir virtual sub-matrices of the | B₁ | × | B₁ | SCF orbital matrix C. Note that N_occ + N_vir = | B₁ | because each orbital is either occupied or virtual.

Let S_AO,π denote the overlap matrix of M’s selected and locally-oriented $p_z^{\prime }$ orbitals, given as

$$S_{\text{AO},\pi }={\left(RO\right)}^{†}{S}_1(RO)$$ (11)

where O denotes the selection-and-orientation matrix from Sec. IIB and R denotes the | B₁ | × | B₂ | matrix of IAO coefficients from Sec. IID. With this, we can compute the N_occ × N_occ overlap matrix of the occupied orbitals with the locally-oriented $p_z^{\prime }$ orbitals (i.e., the overlap matrix of the occupied orbitals projected onto the π-space) as:

$$X:=\left({(RO)}^{†}{S}_1{C}_{\text{occ}}\right)$$ (12)

$$S_{\text{occ},\pi }=X^{†}{\left(S_{\text{AO},\pi }\right)}^{-1}X$$ (13)

Diagonalizing S_occ,π. yields a set of eigenvalues σ_i; i = 1,…,N_occ and an N_occ × N_occ (orthogonal) matrix of eigenvectors U, which define a rotation of the occupied orbitals. The unitarily transformed occupied orbital matrix

$$C_{\text{occ}}^{\prime }:=C_{\text{occ}}U$$ (14)

contains as the ith column the coefficients of a unitarily transformed occupied orbital with an overlap σ_i with the molecule M’s π-subspace:

$$\left|i\rangle \mapsto \right|i^{\prime }\rangle =\sum _k|k\rangle {\left[U\right]}_{ki}$$ (15)

The MOs with σ_i = 0 have no overlap with the π-system, and the MOs with σ_i ≠ 0 have an overlap with the π-system and thus should be selected into the π-active space. In total we need to select the subset of the N_π,occ columns corresponding to the largest eigenvalues σ_i for the occupied MO part of the π-space. As a result, we obtain a | B₁ | × N_π,occ matrix C_occ,π.

Once determined, the occupied π-orbitals are semi-canonicalized by computing

$$f_{\text{occ},\pi }:=C_{\text{occ},\pi }^{†}f{C}_{\text{occ},\pi },$$ (16)

where f is the full molecular system’s Fock matrix, diagonalizing f_occ,π, and using the resulting unitary transformation (given by the eigenvector matrix) to transform C_occ,π again. This procedure provides a set of energy-ordered occupied orbitals representing the occupied part of the π-system of M.

The entire procedure in this section is then repeated with the virtual orbitals to yield the virtual part of the π-system of M. The resulting occupied and virtual orbitals together span the same space as the $p_z^{\prime }$ orbitals from Sec. IIB.

F. Inactive orbitals

This subsection describes how the inactive orbitals are generated. In the simplest case of a molecular complex with one π-system, the inactive occupied orbitals are formed by the MOs from Eqs. (14) and (15) with σ_i = 0, and the inactive virtual orbitals are obtained in a similar manner from the subspace of virtual orbitals. In the case of a molecular complex with multiple π-systems, the active orbitals are obtained for each of the π-systems as described in Sec. IIE. After forming the entire active space, the inactive occupied and inactive virtual orbitals are obtained by constructing the subspace of the full system of occupied and virtual orbitals, respectively, that are orthogonal to the active orbitals.

For the case with multiple π-systems, let C_act denote the active orbital matrix containing the coefficients of the occupied and virtual active orbitals as the columns. First, the active orbitals should be orthogonalized, as the active orbitals obtained from different π-systems are not necessarily orthogonal:

$${\tilde{C}}_{\text{act}}=C_{\text{act}}{S}_{\text{act}}^{-1/2},$$ (17)

where $S_{\text{act}}=C_{\text{act}}^{†}{S}_1{C}_{\text{act}}$ is the overlap matrix of the active orbitals. Then the matrix representation of the projector to the active orbitals in the basis of occupied orbitals is computed as

$$P_{\text{act},\text{occ}}=C_{\text{occ}}^{†}{S}_1{\tilde{C}}_{\text{act}}{\tilde{C}}_{\text{act}}^{†}{S}_1{C}_{\text{occ}}$$ (18)

Diagonalizing the P_act,occ matrix yields a set of eigenvalues λ_i; i = 1,…,N_occ and an N_occ × N_occ orthogonal matrix of eigenvectors U_act,occ, which define a rotation of the occupied orbitals:

$$P_{\text{act},\text{occ}}{U}_{\text{act},\text{occ}}=U_{\text{act},\text{occ}}\text{diag}\left({\lambda }_1,{\lambda }_2,\dots \right).$$ (19)

Now the columns of the unitarily transformed occupied orbital matrix

$${\overline{C}}_{\text{occ}}=C_{\text{occ}}{U}_{\text{act},\text{occ}}$$ (20)

that correspond to eigenvalues λ_i = 0 (i.e., that have no overlap with the active space) define the new inactive occupied (core) orbitals. The inactive virtual orbitals are obtained through the same set of transformations performed on the virtual orbitals. As for the active orbitals, in practice the inactive occupied and virtual orbitals are semi-canonicalized after construction to improve the convergence behavior of subsequent calculations using a similar procedure as in Eq. 16.

G. Differences between AVAS and PiOS methods

The spirit of the PiOS method is similar to the AVAS method, but there are several key differences with important practical consequences. First, as the PiOS method is targeted at π-systems, we employ only locally-oriented $p_z^{\prime }$ orbitals as target orbitals for the projector, instead of all valence atomic orbitals as used in the AVAS method. This difference leads to a smaller initial active space that does not include the electronic degrees of freedom associated with the σ-system, which is typically unreactive, from the outset. For conjugated systems in which σ bonds break along a reaction coordinate of interest, the PiOS and AVAS methods can be combined to describe both cases.

Second, the AVAS scheme does not contain any kind of canonicalization or semi-canonicalization of the orbitals generated by the projection onto the target AO space. This additional procedure allows the selection of the most important orbitals based on energy criteria. In the general systems for which AVAS is designed, the single-particle orbital energies computable from Hartree-Fock or Kohn-Sham wave functions are often not meaningful for identifying what needs to be included in an active space to describe chemistry (e.g., bond dissociations), and therefore AVAS employs only overlap criteria. In the PiOS scheme, however, we obtain an essentially complete characterization of the molecular π-system at the Hückel MO level because all occupied and unoccupied π-orbitals of the π-system are constructed. Thus, semi-canonicalizing this full set of MOs in the PiOS method generates a meaningful energy ordering of the π-orbitals. This procedure allows a user to select, for example, only π-HOMOs and π-LUMOs, instead of all orbitals on the target fragment as in the AVAS method. In contrast to the general case treated by the AVAS method, the energy ordering of orbitals for π-systems is often highly relevant for characterizing the π-system’s electronic states and transitions (e.g., in π-systems, the lowest excited state is typically characterized by the HOMO→LUMO excitation).

Finally, the PiOS scheme can be applied to multiple fragments (e.g., on different monomers of a complex), by building a projector for each π-space, whereas the AVAS method can be applied only once with a single projector combining all AOs for a given system. Thus, the PiOS method can be used to describe phenomena such as singlet splitting or exciton recombination, which can often be expressed in terms of HOMOs and LUMOs of π-systems associated with different monomers.

III. Computational Details

The only user-defined input parameter in the PiOS method is the set of main-group atoms contributing to the π-system. In addition, the user can choose between the options of selecting the entire π-orbital space or selecting the number of π-orbitals in the active space from the generated occupied and unoccupied MO subspaces. For some systems, not all π-orbitals need to be included in the active space to capture most of the essential chemistry, particularly the aspects related to the lowest lying electronic states and their transitions. In this case, the user can decide to choose only the HOMO and LUMO π-orbitals, or some additional low-lying virtual or high-lying occupied orbitals, as the chemistry demands. In any case, these orbitals can be directly identified with the method described above.

In addition to comparing to high-level reference calculations and experimental data, we also employed an internal criterion to assess the quality of the obtained active spaces. Specifically, we computed the overlap matrix between the initial guess and the optimized final active space orbitals and then computed its singular value decomposition (SVD). An overlap of the optimized active space with the initial active space equal to 1.0 indicates a complete coincidence. If all singular values are close to 1.0, the active space remains mostly unchanged during the optimization. A singular value close to 0.0 for a given initial active orbital indicates that this orbital had to be completely replaced by a different orbital. This combination has been successfully used before to validate other active space construction methods.⁴⁶

All complete active space self-consistent field (CASSCF) and strongly-contracted N-electron valence state perturbation theory (NEVPT2)^{21, 62} calculations presented in this paper were carried out using the PYSCF package.⁶³ All geometries, computed electronic states, and other technical details are provided in the Supporting Information.

IV. Results and Discussions

A. Benzene

We begin by considering the electronic structure of benzene, one of the simplest conjugated systems that is studied for illustrative purposes. We carried out an initial restricted Hartree-Fock (RHF) calculation using the aug-cc-pVTZ⁶⁴ basis set for the singlet ground state at the geometry optimized with the UB3LYP/cc-pVTZ basis set in Gaussian 09.⁶⁵ The PiOS method generated the π-orbital space with six orbitals: three bonding and three antibonding type orbitals, as expected (Fig. 1). The singlet π→π* excitations were computed using the state-averaged CASSCF method (state-averaged over 7 states, the ground state and 6 excited states with two pairs of degenerate states) and the NEVPT2 method with this (6e, 6o) active space. Table 1 provides the singlet excited states obtained in this work and from multireference calculations with the manually chosen (6e, 6o) active space, as available from the literature. As shown from these data, the computed π→π* excitations are in good agreement with available computed and experimental values,^66–69 and small differences can be explained by the use of different geometries, basis sets, and methodologies (see Table 1). Specifically, in Ref. ⁶⁶ each electronic state was obtained by a separate CASSCF optimization, while in this work all states were obtained in a state-averaged manner; thus, the CASSCF reference wavefunctions for the dynamic correlation treatment were different. Additionally, different methods were used for the dynamic correlation correction (i.e., NEVPT2 is based on the bielectronic Dyall Hamiltonian and CASPT2 is based on the one-electron Fock-like zero-order Hamiltonian), explaining the more significant discrepancies observed for PT-corrected energies than for the CASSCF energies.

Figure 1.

Six π orbitals generated for benzene by PiOS before CASSCF optimization.

Table 1.

Excitation Energies (eV) of the Singlet States for Benzene with Six Active π-orbitals

State	CASSCF	CASSCF/NEVPT2	CASSCFa	CASPT2a	CASSCFb	MRMPb	Exp.
11 B2u	4.87	5.34	4.97	4.58	4.82	4.71	4.90c
11 B1u	7.81	6.07	7.85	5.90	7.91	5.83	6.20c
11 E1u	9.21	6.91	9.29	6.54	9.29	6.33	6.95c
21 E2g	8.11	8.54	8.11	7.65	8.01	7.74	7.80d

^aState-specific (6e,6o) CASSCF and CASPT2 calculations with the ANO basis set and the C [4s3p2d]/H [3s2p] contraction scheme from Ref. ⁶⁶.

^bMRMP calculations with state-averaged (6e,6o) CASSCF with the cc-pVTZ and cc-pVDZ basis sets for carbon and hydrogen, respectively, from Ref. ⁶⁷.

^cExperimental values from Ref. ⁶⁸.

^dExperimental value from Ref. ⁶⁹.

The SVD eigenvalues for the overlap between the CASSCF-optimized active space and the initial guess active space were 0.9708, 0.9709, 0.9875, 0.9998, 0.9998, and 1.0. This analysis indicates that the π-orbitals generated by the algorithm require only minimal optimization by the CASSCF procedure, as they are near-optimal from the outset, and they can be used in multireference methods without orbital optimization, such as CASCI.

B. Octatetraene

The next example is the (E,E)-1,3,5,7-octatetraene molecule, where the geometry was obtained from Ref. ⁷⁰. As in the previous example, we started with an RHF calculation for the singlet ground state with the aug-cc-pVTZ⁶⁴ basis set. The algorithm generated four π-orbitals from the occupied orbital subspace and another four π-orbitals from the unoccupied orbital subspace (Fig. 2). This (8e,8o) active space was used to obtain the five lowest singlet states (including the ground state) in the state-averaged CASSCF and subsequent NEVPT2 calculations. The singlet π→π* excitations obtained for octatetraene with this active space were compared with the computations of the electronic states performed with manual selection of the analogous active space (e.g., canonical orbitals from the RHF calculation) of the same size^70–71 (Table 2). The states computed at the CASSCF/NEVPT2 level are identical to those computed at this level with the same size, but manually chosen, active space using the same basis set and geometry as used in Ref. ⁷⁰. The SVD eigenvalues for the overlap between the active spaces before and after orbital optimization by CASSCF ranged from 0.9876 to 0.9999. This analysis implies that the algorithm provides a very accurate initial π-orbital active space for the CASSCF calculations of the octatetraene molecule as well.

Figure 2.

Eight π orbitals generated for octatetraene by PiOS before CASSCF optimization.

Table 2.

Lowest Excitation Singlet States (eV) for Octatetraene with Eight Active π-orbitals

State	CASSCF	CASSCF/NEVPT2	CASSCFa	CASSCF/NEVPT2a	CASSCFb	CASSCF/NEVPT2b
$1^1{B}_{\mu }^{+}$	6.65	4.02	6.65	4.02	6.65	4.02
$2^1{A}_g^{-}$	4.79	4.81	4.79	4.81	4.68	4.75
$1^1{B}_u^{-}$	6.00	6.07	6.00	6.07	5.86	5.99
$3^1{A}_g^{-}$	6.74	6.82	6.74	6.82	6.59	6.73

^a(8e,8o) CASSCF (state-averaged over 5 states) and NEVPT2 calculations with aug-ccVTZ basis set from Ref. ⁷⁰. These values are identical to those produced in the present work.

^b(8e,8o) CASSCF (state-averaged over 8 states) and NEVPT2 calculations with def2-TZVP basis set from Ref. ⁷¹.

C. Free-base porphine

Free-base porphine is an example of an extended π-conjugated system that has 24 valence π- and π*-orbitals with 26 electrons. The geometry was obtained from Ref. ⁷². The RHF calculation for the singlet ground state, as well as the subsequent CASSCF and NEVPT2 calculations, were performed using the cc-pVDZ⁶⁴ basis set. The PiOS algorithm built 13 π-orbitals from the occupied orbital subspace and another 11 π-orbitals from the unoccupied orbital subspace to yield a (26e, 24o) active space. Figure 3 shows all π-orbitals generated by the algorithm. This active space could be used to compute comprehensive electronic absorption spectra with modern methods supporting large active spaces, such as density matrix renormalization group (DMRG) or restricted active space self-consistent field (RASSCF) methods. Here, however, we set the algorithm to select only two HOMOs and two LUMOs to compute the Soret or B-band and the Q-bands of the porphine. All generated π-orbitals are sorted within their occupied and unoccupied subspaces based on the MO energies obtained through the Fock matrix and coefficients of the basis functions in the expansion of the generated π-orbital to compute the Soret or B-band and the Q-bands of the porphine.

Figure 3.

24 π orbitals generated for porphine by PiOS before CASSCF optimization.

The ground state and four excited states, corresponding to the Soret bands (Table 3), were computed using state-averaged CASSCF over five states with equal weights, followed by NEVPT2. The SVD eigenvalues for the overlap between the initial and optimized active spaces are 0.9835, 0.9835, 0.999, and 1.0. The computed CASSCF/NEVPT2 states are in good agreement with available theoretical and experimental values. The ability to easily select only the relevant highest-lying occupied and lowest-lying unoccupied orbitals even in complex and extended π-systems is one of the major advantages of the PiOS method, as it will often allow a massive reduction in computational cost via a decrease in the active space size.

Table 3.

Lowest Excitation Singlet States (eV) of Free-Base Porphine

State	CASSCF	CASSCF/NEVPT2	CASPT2a	MRCIb	Exp.c
11 B3u	3.42	1.96	1.70	1.90	1.98–2.02 (Qx)
11 B2u	3.64	2.37	2.26	2.36	2.33–2.42 (Qy)
21 B2u	5.42	3.06	2.91	3.04	3.13–3.33 (B)
21 B3u	5.43	3.09	3.04	3.06	3.13–3.33 (B)

^a(4e,4o) CASPT2 calculations with ANO-L basis set and the C,N [3s2p]/H[2s] contraction scheme from Ref ⁷³.

^bCalculations with def2-SV(P) basis set from Ref. ⁷².

^cExperimental data from Ref^74–76.

D. Two-dimensional potential energy surfaces for the active site of the BLUF photoreceptor protein

Now we consider a more complex system, namely the blue light-using flavin (BLUF) photoreceptor protein, which transmits a long-range signal to control cellular processes upon activation by light. In particular, we focus on the active site of the Slr1694 BLUF photoreceptor, which controls phototaxis in the cyanobacterium Synechocystis sp. PCC 6903. The active site consists of flavin adenine mononucleotide (FMN), conserved Gln and Tyr residues in the flavin binding pocket, and a semiconserved Trp. Upon photoexcitation of the flavin to its locally-excited (LE) state, the anionic semiquinone radical (FMN^•−) is formed after charge transfer (CT) from Tyr8 to the flavin (FMN), and the neutral flavin semiquinone radical (FMNH^•) is formed after the subsequent proton transfer from Tyr8 to the flavin via Gln50 (Figure 4). These two intermediates have been observed experimentally.^77–83 However, many questions remain open, such as the nature of the double proton transfer reaction, which is presumed to occur in the CT state based on the experimental data but could occur either sequentially or concertedly. We used our PiOS approach in combination with the AVAS algorithm to compute two-dimensional potential energy surfaces along the two proton transfer coordinates for four different electronic states. Here we demonstrate such computations for a single conformation of the active site. The full study of this Slr1694 BLUF photoreceptor is beyond the scope of this work and will be presented in a separate paper.⁸⁴ Herein we describe the core procedure and demonstrate how the active space selection method makes such calculations computationally tractable.

Figure 4.

The active site of the Slr1694 BLUF photoreceptor protein at the geometry with two protons positioned near their donor atoms. The two proton transfer reactions are indicated by green arrows, and the two proton transfer coordinates correspond to the axes connecting the heavy atoms in each of the two hydrogen bonds.

For a fixed conformation of the active site, we generated a two-dimensional grid by varying the positions of the two transferring protons along the lines connecting their respective donor and acceptor atoms (Figure 4) with a 0.1 Å step size. Our goal was to compute the two-dimensional potential energy surfaces for the ground state (GS) and the relevant excited states, namely the LE state within the flavin and the CT_Tyr and CT_Trp states associated with electron transfer from Tyr8 or Trp91 to the flavin. To obtain the active orbitals (both bonding and antibonding types) associated with the two transferring protons, we used the AVAS algorithm, and to obtain the four π-conjugated fragments associated with FMN, Tyr8, Gln50, and Trp91, we used the PiOS method. This application utilizes the advantage of the PiOS method over the AVAS method for treating multiple π-systems within a molecular complex separately (i.e., selecting the HOMO and LUMO from each π-system individually) and subsequently combining them. In this case, the active space obtained in this manner is also combined with additional orbitals obtained with the AVAS method for modeling the bond formation and bond breaking associated with the two transferring hydrogen atoms.

We performed preliminary tests on a few geometries corresponding to different proton positions to determine the active space that is a reasonable size but also suitable for the description of all electronic states of interest, namely the GS, LE state, and two charge transfer states, CT_Tyr and CT_Trp. The (18e, 15o) active space used for this system consists of two HOMOs and one LUMO for each of the FMN, Tyr8, and Trp91 (Figure 4), one HOMO and one LUMO for Gln50, and a pair of orbitals for each transferring proton (Figure 5). To obtain the complete two­dimensional potential energy surfaces, we computed six electronic states in a state-averaged manner with CASSCF, followed by the NEVPT2 calculations. All of these calculations were performed with the 6–31++G**^85–87 basis set.

Figure 5.

(18e,15o) active space generated for the active site of the Slr1694 BLUF photoreceptor protein at the geometry with two protons positioned near their donor atoms.

Figure 6 shows the computed potential energy surfaces as a function of the two proton transfer coordinates. Based on these potential energy surfaces and the analogous surfaces generated for other active site conformations, we discovered that double proton transfer can occur in the CT_Tyr state but not in the other states studied, corroborating the experimental data suggesting that electron transfer occurs prior to the double proton transfer reaction. In addition, the calculations suggest that the double proton transfer reaction is sequential, with proton transfer from Tyr8 to Gln50 occurring prior to proton transfer from Gln50 to the FMN. These conclusions are discussed more extensively elsewhere.⁸⁴ Using the automated approach allowed us to compute excited electronic states at the CASSCF/NEVPT2 level with a (18e,15o) active space for more than 500 geometries with different protein active site conformations. This number of high-level calculations using a consistent active space for all geometries studied would not currently be computationally practical with a manual selection approach.

Figure 6.

Two-dimensional potential energy surfaces as a function of the two proton transfer coordinates in the GS, LE state, CT_Trp state, and CT_Tyr state. These potential energy surfaces were computed with the CASSCF(18e,15o)+NEVPT2 method in the gas phase.

V. Conclusions

The automated π-orbital space (PiOS) method for selecting active spaces for multireference calculations of conjugated π-systems was presented. This approach is based on linear algebraic transformations of a single-reference wavefunction and Hückel theory for systems with conjugated π-bonds. This algorithm can be used alone to study π→π* excitations in conjugated systems such as aromatic hydrocarbons, polyenes, and porphyrins or, alternatively, can be combined with other active space selection approaches to study excitations in complex systems with multiple conjugated fragments. This work represents a continuation of efforts to advance the automated selection of active space orbitals to render multireference calculations of complex systems computationally practical. Such efforts are important for ensuring consistency, reproducibility, and broad applicability of multireference approaches for chemical and biological systems.