In the full optimized reaction space and valence-complete active space self-consistent field (vCAS) methods, a set of active orbitals is defined as the union of the valence orbitals on the atoms, all possible configurations involving the active orbitals are generated, and the orbitals and configuration coefficients are self-consistently optimized. Such wave functions have tremendous flexibility, which makes these methods incredibly powerful but can also lead to inconsistencies in the description of the electronic structure of molecules. In this paper, the problems that can arise in vCAS calculations are illustrated by calculations on the BH and BF molecules. BH is well described by the full vCAS wave function, which accounts for molecular dissociation and 2*s*–2*p* near-degeneracy in the boron atom. The same is not true for the full vCAS wave function for BF. There is mixing of core and active orbitals at short internuclear distances and swapping of core and active orbitals at large internuclear distances. In addition, the virtual 2π orbitals, which were included in the active space to account for the 2*s*–2*p* near degeneracy effect, are used instead to describe radial correlation of the electrons in the F2*p*π-like pairs. Although the above changes lead to lower vCAS energies, they lead to higher vCAS+1+2 energies as well as irregularities and/or discontinuities in the potential energy curves. All of the above problems can be addressed by using the spin-coupled generalized valence bond-inspired vCAS wave function for BF, which includes only a subset of the atomic valence orbitals in the active space.

## I. INTRODUCTION

The full optimized reaction space^{1–3} (FORS) and valence complete active space self-consistent field^{4–6} (vCAS) methods are powerful and popular means of describing the electronic structure of molecules that require more than a single configuration wave function. In both the FORS and vCAS methods, an active space is defined consisting of the union of the valence orbitals of the atoms, all possible configurations using the active space orbitals are constructed, and the orbitals and configuration coefficients are optimized self-consistently. FORS and vCAS wave functions account for all of the non-dynamical correlation in the atoms and molecules, but they also include an unspecified amount of dynamical correlation.^{7} An ensuing FORS+1+2 or vCAS+1+2 calculation is then used to account for the remaining dynamical correlation. Since the orbitals in the FORS+1+2 or vCAS+1+2 wave functions are taken directly from the FORS or vCAS wave function, it is critical that these wave functions and orbitals be appropriate for the FORS+1+2 and vCAS+1+2 calculations. However, this is not guaranteed.

Once the active orbitals for the FORS/vCAS calculations have been selected, the calculations can be run as “black boxes” and are often performed in this way. However, optimization of the orbitals and configuration coefficients may not lead to a consistent description of the molecule as the geometry is varied. We have, in fact, encountered such inconsistencies in a number of our studies with full vCAS and vCAS+1+2 calculations on the ground state of BF being the “poster child” for the various types of problems that can arise. Although, the closely related BH molecule is well described by the FORS/vCAS wave function and presents no problems, substitution of a fluorine atom for the hydrogen atom in BH dramatically complicates the FORS/vCAS and FORS/vCAS+1+2 calculations. In this article, we will illustrate how the FORS/vCAS descriptions of BH and BF differ and how these differences impact the subsequent FORS+1+2/vCAS+1+2 calculations. We shall also show that a spin-coupled generalized valence bond^{8–10} (SCGVB) inspired wave function eliminates these inconsistencies and provides a theoretically consistent description of the electronic structure of both BH and BF.

In Sec. II, we briefly describe the SCGVB and vCAS wave functions for BH and BF. In Sec. III, we present and discuss the results of the SCGVB, vCAS, and vCAS+1+2 calculations on these two molecules. In Sec. IV, we present the conclusions drawn from the studies described in Sec. III. In the remainder of this article, we will use the notation vCAS(*n*_{e}, *n*_{a}) to represent a complete active space self-consistent field wave function for *n*_{e} electrons in *n*_{a} active orbitals. The notation vCAS(*n*_{e}, *n*_{a})+1+2 represents a vCAS(*n*_{e}, *n*_{a}) wave function plus all single and double excitations from the active orbitals plus any doubly occupied valence orbitals.

## II. THEORETICAL AND COMPUTATIONAL METHODS

The traditional SCGVB wave functions for the ground (X^{1}Σ^{+}) states of BH and BF are

In Eq. (1), {*ϕ*_{c1}} refers to the B1*s* core orbital and {*φ*_{a1}, *φ*_{a2}, *φ*_{a3}, *φ*_{a4}} refers to the active valence orbitals—one orbital for each electron, i.e., a SCGVB(4, 4) wave function. In the separated atom limit, these orbitals become the B2*s*_{−}, B2*s*_{+}, B2*p*_{z}, and H1*s* atomic orbitals (see below for the definition of the B2*s*_{−} and B2*s*_{+} lobe orbitals). In Eq. (2), {*ϕ*_{c1}, *ϕ*_{c2}} refers to the F1*s* and B1*s* core orbitals, {*φ*_{a1}, *φ*_{a2}, *φ*_{a3}, *φ*_{a4}} refers to the active valence orbitals that become the B2*s*_{−}, B2*s*_{+}, B2*p*_{z}, and F2*p*_{z} atomic orbitals in the separated atom limit, and {*ϕ*_{v1}, *ϕ*_{v2}, *ϕ*_{v3}} refers to the doubly occupied valence orbitals that become the F2*s*, F2*p*_{x}, and F2*p*_{y} orbitals in the separated atom limit.

In both equations (1) and (2), the B2*s* lobe orbitals, which would be described by a doubly occupied orbital in the Hartree–Fock (HF) wave function, are described by a pair of non-orthogonal orbitals with the spins of the electrons in the two orbitals singlet coupled as they are in the HF wave function. These two orbitals are given by

with *λ* = 0.427 (the bond axis is the *z*-axis) and the orbitals (*φ*_{B2s}, $\phi B2px$) are the atomic orbitals of the boron atom. The wave function for the (B2*s*_{−} and B2*s*_{+}) pairs in Eqs. (1) and (2) can be alternately written as a linear combination of 2*s*^{2} and 2*p*_{x}^{2} configurations. Thus, this pair of orbitals accounts for the 2*s–*2*p* near-degeneracy effect in the boron atom. However, Eqs. (1) and (2) include only one of the two 2*p*^{2} configurations describing the 2*s*–2*p* near-degeneracy effect—it is missing the 2*p*_{y}^{2} configuration. Furthermore, the wave functions in Eqs. (1) and (2) do not have pure ^{1}Σ^{+} symmetry. Both of these problems can be addressed by using a ^{1}Σ^{+}-projected SCGVB wave function, e.g., for the BH molecule at *R* = ∞, this wave function is

The SCGVB wave function in Eq. (4) describes four electrons in five orbitals—the *φ*_{B2s}, $\phi B2px$, $\phi B2py$, $\phi B2pz$, and *φ*_{H1s} orbitals (note that the $\phi B2px$ and $\phi B2py$ orbitals need not be equivalent to the $\phi B2pz$ orbital). We refer to this wave function as a SCGVB(4,5) wave function. This wave function can be used for all *R* values by optimizing the orbitals and spin coupling coefficients in $\Theta 0,04$ in the first or second line of Eq. (4) or (*c*_{1}, *c*_{2}) in the bottom line of Eq. (4). The SCGVB(4,5) wave function for BF has the same general form as the SCGVB(4,5) wave function for BH, and the CASVB module^{11–14} in Molpro^{15,16} has been used to compute the SCGVB(4,5) wave functions for both molecules. In this paper, we will only report the results for the SCGVB(4,5) wave functions for the BH and BF molecules.

The active space for full vCAS calculations on the ground state (X^{1}Σ^{+}) of BH consists of the (B2*s*, B2*p*) orbitals and the H1*s* orbital. In C_{2v} symmetry, this means that there are three orbitals of σ(a_{1}) symmetry and one orbital each of π_{x}(b_{1}) and π_{y}(b_{2}) symmetry in the active space, i.e., a vCAS(4,5) wave function with four electrons in five orbitals. The configurations in the vCAS(4,5) wave function include the configurations required to describe the dissociation of BH into atoms as well as all of the configurations needed to describe the 2*s*–2*p* near-degeneracy effect in the boron atom.^{17} There are additional configurations in the vCAS(4,5) wave function whose coefficients approach zero as *R* → ∞ but are otherwise non-zero. These configurations describe a small subset of the dynamical correlation effects in BH.

For the ground state (X^{1}Σ^{+}) of BF, the full valence active space includes the (B2*s*, B2*p*) and (F2*s*, F2*p*) orbitals, resulting in an active space consisting of four orbitals of σ(a_{1}) symmetry and two each of π_{x}(b_{1}) and π_{y}(b_{2}) symmetry, i.e., a vCAS(10,8) wave function. This valence active space can also describe the dissociation of the BF molecule into atoms as well as the full boron atom 2*s*–2*p* near-degeneracy effect. The use of “can” in the last sentence was purposeful—the vCAS wave function will seek to describe those correlation effects that minimize the energy; for further details, see the discussion in Sec. III. Just as in BH, some of the configurations included in the vCAS(10,8) wave function can describe dynamical correlation effects.

An alternate set of active orbitals can be constructed for the BF molecule inspired by the SCGVB(4,5) wave function. In the SCGVB(4,5) wave function, the F2*s*, F2*p*_{x}, and F2*p*_{y} orbitals are doubly occupied and, therefore, are excluded from the active space. This set of active orbitals describes the dissociation of the BF molecule into a HF description of the fluorine atom plus the full 2*s*–2*p* near-degeneracy effect in the boron atom. This wave function is a vCAS(4,5) wave function,^{18} which has the same basic structure as the vCAS(4,5) wave function for BH. This wave function would be expected to primarily describe non-dynamical correlation effects, accounting for less dynamical correlation energy than is possible in the vCAS(10,8) wave function.

The vCAS+1+2 calculations included all single and double excitations from the vCAS wave function and account for dynamical correlation effects. The results of coupled cluster calculations including singles and doubles^{19} plus perturbative triples^{20} [CCSD(T)] were included in the study for comparison.

All of the SCGVB(4,5), vCAS, and CCSD(T) calculations were performed with the Molpro suite of computational chemistry programs^{15,16} (version 2010.1). The SCGVB(4,5) calculations used the CASVB module^{11–14} in Molpro. Augmented correlation consistent basis sets of quadruple zeta quality (aVQZ) were used for the hydrogen, boron, and fluorine atoms.^{21,22}

## III. RESULTS AND DISCUSSION

The total energies (*E*_{e}), equilibrium bond lengths (*R*_{e}), dissociation energies (*D*_{e}), and fundamental frequencies (ω_{e}) for the SCGVB, vCAS, vCAS+1+2, and CCSD(T) calculations on the ground state (X^{1}Σ^{+}) of BH are listed in Table I. The results of the corresponding calculations on the ground state (X^{1}Σ^{+}) of BF are listed in Table II. Also included in the tables are the results of recent computational studies of BH^{23,24} and BF^{25,26} along with the available experimental data.^{27–29}

Reference . | Calculation . | E_{e}
. | R_{e}
. | D_{e}
. | ω_{e}
. |
---|---|---|---|---|---|

Present | SCGVB(4,5) | −25.187 009 | 1.2490 | 77.42 | 2287 |

Present | vCAS(4,5) | −25.187 421 | 1.2495 | 77.68 | 2280 |

Present | vCAS(4,5)+1+2 | −25.235 358 | 1.2337 | 84.22 | 2355 |

Present | CCSD(T) | −25.235 343 | 1.2334 | 84.42 | 2357 |

Gagliardi et al.^{23} | Full CI | −25.235 654 | 1.2338 | 84.23 | 2354 |

Miliordos and Mavridis^{24} | CASSCF | −25.287 578 | 1.2301 | 84.78 | 2358 |

Miliordos and Mavridis^{24} | CCSD(T) | −25.257 203 | 1.2296 | 84.78 | 2361 |

Expt.^{27,28} | ⋯ | 1.2322 | 84.9–85.1 | 2367 |

Reference . | Calculation . | E_{e}
. | R_{e}
. | D_{e}
. | ω_{e}
. |
---|---|---|---|---|---|

Present | SCGVB(4,5) | −25.187 009 | 1.2490 | 77.42 | 2287 |

Present | vCAS(4,5) | −25.187 421 | 1.2495 | 77.68 | 2280 |

Present | vCAS(4,5)+1+2 | −25.235 358 | 1.2337 | 84.22 | 2355 |

Present | CCSD(T) | −25.235 343 | 1.2334 | 84.42 | 2357 |

Gagliardi et al.^{23} | Full CI | −25.235 654 | 1.2338 | 84.23 | 2354 |

Miliordos and Mavridis^{24} | CASSCF | −25.287 578 | 1.2301 | 84.78 | 2358 |

Miliordos and Mavridis^{24} | CCSD(T) | −25.257 203 | 1.2296 | 84.78 | 2361 |

Expt.^{27,28} | ⋯ | 1.2322 | 84.9–85.1 | 2367 |

Reference . | Calculation . | E_{e}
. | R_{e}
. | D_{e}
. | ω_{e}
. |
---|---|---|---|---|---|

Present | SCGVB(4,5) | −124.217 983 | 1.2661 | 153.80 | 1401 |

Present^{a,b} | vCAS(4,5) | −124.218 072 | 1.2663 | 153.85 | 1399 |

Present^{b,c} | vCAS(10,8) | −124.271 392 | 1.2647 | 181.63 | 1420 |

Present^{a,b} | vCAS(10,8)^{*} | −124.271 051 | 1.2651 | 187.10 | 1414 |

Present^{a,b} | vCAS(4,5)+1+2 | −124.519 866 | 1.2654 | 177.45 | 1406 |

Present^{b,c} | vCAS(10,8)+1+2 | −124.516 288 | 1.2705 | 194.36 | 1386 |

Present^{a,b} | vCAS(10,8)^{*}+1+2 | −124.524 177 | 1.2654 | 180.16 | 1405 |

Present | CCSD(T) | −124.541 670 | 1.2688 | 180.77 | 1391 |

Feller and Sordo^{25} | CCSD(T) | −124.538 564 | 1.2680 | 180.4 | 1398 |

Feller and Sordo^{25} | CCSDT | −124.538 710 | 1.2675 | 180.1 | ⋯ |

Magoulas et al.^{26} | CAS+1+2 | −124.537 21 | 1.2635 | 184.25 | 1409 |

Magoulas et al.^{26} | CCSD(T) | ⋯ | 1.2668 | 181.54 | ⋯ |

Expt.^{29} | ⋯ | 1.2627 | 182.7 | 1394 |

Reference . | Calculation . | E_{e}
. | R_{e}
. | D_{e}
. | ω_{e}
. |
---|---|---|---|---|---|

Present | SCGVB(4,5) | −124.217 983 | 1.2661 | 153.80 | 1401 |

Present^{a,b} | vCAS(4,5) | −124.218 072 | 1.2663 | 153.85 | 1399 |

Present^{b,c} | vCAS(10,8) | −124.271 392 | 1.2647 | 181.63 | 1420 |

Present^{a,b} | vCAS(10,8)^{*} | −124.271 051 | 1.2651 | 187.10 | 1414 |

Present^{a,b} | vCAS(4,5)+1+2 | −124.519 866 | 1.2654 | 177.45 | 1406 |

Present^{b,c} | vCAS(10,8)+1+2 | −124.516 288 | 1.2705 | 194.36 | 1386 |

Present^{a,b} | vCAS(10,8)^{*}+1+2 | −124.524 177 | 1.2654 | 180.16 | 1405 |

Present | CCSD(T) | −124.541 670 | 1.2688 | 180.77 | 1391 |

Feller and Sordo^{25} | CCSD(T) | −124.538 564 | 1.2680 | 180.4 | 1398 |

Feller and Sordo^{25} | CCSDT | −124.538 710 | 1.2675 | 180.1 | ⋯ |

Magoulas et al.^{26} | CAS+1+2 | −124.537 21 | 1.2635 | 184.25 | 1409 |

Magoulas et al.^{26} | CCSD(T) | ⋯ | 1.2668 | 181.54 | ⋯ |

Expt.^{29} | ⋯ | 1.2627 | 182.7 | 1394 |

^{a}

In the separated atom limit, the 2π orbitals correlate the B2*s* pair. In this case, the 2π orbitals are B2*p*_{x,y} orbitals.

^{b}

The dissociation energies were computed using the energies for the wave functions at Δ*R* = 10 Å.

^{c}

In the separated atom limit, the 2π orbitals correlate the F2*p*_{x,y} pairs. In this case, the 2π orbitals are F3*p*_{x,y} orbitals.

### A. The BH molecule

The configurations in the SCGVB(4,5) wave function of BH are a significant subset of the configurations in the vCAS(4,5) wave function. Thus, it is not surprising that the SCGVB(4,5) calculations predict a bond length (*R*_{e}) that is just 0.0005 Å shorter, a dissociation energy (*D*_{e}) that is just 0.26 kcal/mol less, and a fundamental frequency just 7 cm^{−1} more than the spectroscopic constants predicted by the vCAS(4,5) calculations; see Table I. In fact, the energies of the SCGVB(4,5) and vCAS(4,5) wave functions at their respective *R*_{e}’s differ by only −0.42 millihartrees (m*E*_{h}).

The vCAS(4,5) wave function predicts a *R*_{e} that is 0.0158 Å longer, a *D*_{e} that is 6.54 kcal/mol smaller, and an ω_{e} that is 75 cm^{−1} smaller than the spectroscopic constants predicted by the vCAS(4,5)+1+2 calculations. These differences are significant and are a result of the dynamical correlation included in the vCAS(4,5)+1+2 wave function. Furthermore, the vCAS(4,5)+1+2 calculations predict spectroscopic constants that are in good agreement with the present CCSD(T) calculations—Δ*R*_{e} = 0.0003 Å, Δ*D*_{e} = −0.20 kcal/mol, and Δω_{e} = −2 cm^{−1}—as well as with prior calculations and experimental measurements.

The potential energy curves for BH from the SCGVB(4,5), vCAS(4,5), vCAS(4,5)+1+2, and CCSD(T) calculations are plotted in Fig. 1. The curves from the SCGVB(4,5) and vCAS(4,5) calculations are almost indistinguishable in the plot. However, there is some variation in the curves: at *R* = 0.7337 Å, the difference is −0.56 m*E*_{h} with the difference reaching a maximum of −0.94 m*E*_{h} at *R* = 2.1337 Å before decreasing to zero at *R* = ∞. The potential energy curves for the vCAS(4,5)+1+2 and CCSD(T) calculations are also nearly indistinguishable in Fig. 1, deviating only slightly for R > 2.7 Å where the CCSDS(T) calculations begin to break down because they are based on the RHF wave function. In summary, the SCGVB(4,5) and vCAS(4,5) provide excellent zero-order descriptions of the BH molecule, yielding semi-quantitative predictions of the spectroscopic constants, while the vCAS(4,5)+1+2 wave function provides nearly quantitative descriptions of the BH molecule for all *R*.

A detailed examination of the orbitals and configuration coefficients in the vCAS wave function shows that the configurations corresponding to the 2*s*–2*p* near-degeneracy effect in the boron atom are important at all *R*; see Fig. 2(a). If the configurations in the vCAS(4,5) wave function are classified as (n_{2σ}n_{3σ}n_{4σ} n_{1πx}n_{1πy}), where n_{i} is the occupation of the active valence orbital *i*, the behaviors of the coefficients of the configurations with occupancies of (200 2 0), (020 2 0), and (002 2 0) (and the three corresponding 1π_{y} configurations where n_{1πy} = 2) in the region *R* = 1.8 Å–2.4 Å reflect the fact that the B2*s*-like σ orbital and the BH σ bond orbital smoothly interchange in this region. This leads to an “avoided crossing” in the plot of the occupation numbers of the 2σ and 3σ orbitals; see Fig. 2(b). At large *R* values, 2σ is the B2*s*-like orbital and 3σ is the BH bonding orbital, while the opposite is true for smaller separations including *R*_{e}. For intermediate *R*, as the two orbitals switch character, both orbitals become delocalized over the atoms. The orbitals then re-localize after the interchange. The 4σ orbital, on the other hand, is always the antibonding orbital. Despite this orbital interchange, the SCGVB(4,5) and vCAS(4,5) wave functions, both of which include the 2σ and 3σ orbitals in a symmetrical fashion, fully account for the boron 2*s*–2*p* near-degeneracy effect at all *R* values. The vCAS(4,5) orbitals for BH at *R*_{e}, at an intermediate *R* value where the orbitals are undergoing the interchange, and at a large *R* value are plotted in Fig. 3.

### B. The BF molecule

For the BF molecule, the situation is quite different; see Table II. However, again, the spectroscopic constants from the SCGVB(4,5) and vCAS(4,5) calculations are quite similar, Δ*R*_{e} = −0.0002 Å, Δ*D*_{e} = −0.05 kcal/mol, and Δω_{e} = +2 cm^{−1}. In addition, Δ*E*_{e} = −0.09 m*E*_{h} at their respective *R*_{e}’s. The potential energy curves from the SCGVB(4,5) and vCAS(4,5) curves are plotted in Fig. 4. The two curves are almost indistinguishable, with the difference in the curves varying from 0.06 m*E*_{h} at *R* = 0.8705 Å and reaching a maximum of 1.30 m*E*_{h} at *R* = 1.8705 Å before decreasing to zero at *R* = ∞. In summary, the SCGVB(4,5) wave function clearly captures the essential features of the vCAS(4,5) wave function for BF.

Comparing the predictions from the vCAS(4,5) wave function and the vCAS(10,8) wave function, we see that the predicted *R*_{e} and ω_{e} are very similar, Δ*R*_{e} = +0.0016 Å and Δω_{e} = −21 cm^{−1}, but *D*_{e}, which is calculated as the difference between the energies at *R*_{e} and Δ*R* = 10 Å (Δ*R* = *R* − *R*_{e} and *R*_{e} = 1.2705 Å), is much larger for the vCAS(10,8) wave function than that for the vCAS(4,5) wave function: Δ*D*_{e} = −27.78 kcal/mol. In fact, the dissociation energy calculated with the vCAS(10,8) wave function is slightly larger than that from the present CCSD(T) calculations, a surprising result. For the vCAS+1+2 calculations, we also see reasonable agreement for (*R*_{e}, ω_{e}) between the vCAS(4,5)+1+2 and vCAS(10,8)+1+2 results: Δ*R*_{e} = −0.0051 Å and Δω_{e} = +20 cm^{−1}. However, the calculated vCAS(10,8)+1+2 *D*_{e}, 194.36 kcal/mol, is more than 10 kcal/mol larger than the experimental value.

The unusual results from the vCAS(10,8) and vCAS(10,8)+1+2 calculations on BF led us to carefully examine the details of the calculations. We found a number of problems with the vCAS(10,8) calculations, problems that also impacted the vCAS(10,8)+1+2 calculations. The first set of problems involve mixing or swapping of the core and active orbitals, while the second problem involves a change in the nature of the 2π orbitals, the orbitals that nominally arise from the B2*p*_{x,y} atomic orbitals in the vCAS(10,8) wave function.

At short

*R*values, the core F1*s*orbital mixes with the active F2*s*-like orbital in the vCAS(10,8) wave function with the amount of mixing increasing as*R*decreases. Although this mixing lowers the energy of the vCAS(10,8) wave function unexpectedly, it increases the energy of the vCAS(10,8)+1+2 wave function, i.e., the resulting active orbital is not optimal for the vCAS(10,8)+1+2 calculations. This is the reason that the energy of the vCAS(10,8)+1+2 wave function is 3.6 m*E*_{h}*above*the energy of the vCAS(4,5)+1+2 wave function at their respective*R*_{e}’s; see Table II.The nature of the orbitals in the vCAS(10,8) wave function depend on the direction of the scan: from

*R*_{e}to Δ*R*= 10 Å or from Δ*R*= 10 Å to*R*_{e}(Δ*R*=*R*−*R*_{e},*R*_{e}= 1.2705 Å).Starting at

*R*_{e}and increasing*R*, the 2π orbitals in the vCAS(10,8) wave function describe radial correlation of the two F2*p*_{x,y}-like pairs with the 2π orbitals resembling F3*p*_{x,y}-like orbitals. In addition, the core F1*s*-like orbital and active B2*s*-like orbital swap places at Δ*R*≈ 3.9 Å. These changes again result in lower vCAS(10,8) energies unexpectedly, but at Δ*R*= 10 Å, the vCAS(10,8)+1+2 energy is 30.5 m*E*_{h}*higher*than the vCAS(4,5)+1+2 energy because of the change in the active orbitals (B2*s*replaced by F1*s*); see Table III.A vCAS(10,8) calculation at Δ

*R*= 10 Å results in 2π orbitals that describe correlation of the B2*s*-like pair (2*s*–2*p*near-degeneracy effect). This remains the case as*R*is decreased until Δ*R*≈ 1.65 Å. At this point, the 2π orbitals switch to being F3*p*_{x,y}-like orbitals that describe radial correlation of the F2*p*_{x,y}-like pairs. In addition to this change, the core B1*s*and active F2*s*orbitals swap places at Δ*R*= 10 Å and remain this way until Δ*R*≈ 2.0 Å. At Δ*R*= 10 Å, the energy of this vCAS(10,8)+1+2 wave function is 40.4 m*E*_{h}*higher*than the vCAS(10,8)+1+2 wave function that uses the 2π orbitals to correlate the F2*p*_{x,y}pairs; see Table III.

. | E_{vCAS(4,5)}
. | E_{vCAS(4,5)+1+2}
. | E_{vCAS(10,8)}
. | E_{vCAS(10,8)+1+2}
. |
---|---|---|---|---|

F2p^{a} | ⋯ | ⋯ | −123.981 947 | −124.206 553 |

B2s^{b} | −123.972 896 | −124.237 083 | −123.972 912 | −124.166 156 |

B + F^{c} | −123.972 663 | −124.243 488 | −123.972 663 | −124.243 488 |

ΔE(F2p)^{a} | ⋯ | ⋯ | 9.284 | −36.935 |

ΔE(B2s)^{b} | 0.233 | −6.405 | 0.249 | −77.332 |

. | E_{vCAS(4,5)}
. | E_{vCAS(4,5)+1+2}
. | E_{vCAS(10,8)}
. | E_{vCAS(10,8)+1+2}
. |
---|---|---|---|---|

F2p^{a} | ⋯ | ⋯ | −123.981 947 | −124.206 553 |

B2s^{b} | −123.972 896 | −124.237 083 | −123.972 912 | −124.166 156 |

B + F^{c} | −123.972 663 | −124.243 488 | −123.972 663 | −124.243 488 |

ΔE(F2p)^{a} | ⋯ | ⋯ | 9.284 | −36.935 |

ΔE(B2s)^{b} | 0.233 | −6.405 | 0.249 | −77.332 |

^{a}

In the separated atom limit, the 2π orbitals correlate the F2*p*_{x,y} pairs; the 2π orbitals are F3*p*_{x,y} orbitals.

^{b}

In the separated atom limit, the 2π orbitals correlate the B2*s* pair; the 2π orbitals are B2*p*_{x,y} orbitals.

^{c}

The zero-order wave functions for the boron and fluorine atoms were the vCAS(3,4) and HF wave functions, respectively.

As can be seen, the problems in the vCAS(10,8) calculations on BF are wide ranging, involving both core-active orbital mixing and/or swapping. This is further complicated by changes in the basic nature of the 2π orbitals as a function of *R*. These problems lead to much higher vCAS(10,8)+1+2 asymptotic energies, discontinuities, and other irregularities in the vCAS(10,8) and vCAS(10,8)+1+2 potential energy curves for BF.

We explored a number of options for addressing the above problems in the vCAS(10,8) calculations. Our initial focus was on restricting the valence active space. In this regard, we sought vCAS(*n*_{e}, *n*_{a}) wave functions with *n*_{e} ≤ 10 and *n*_{a} ≤ 8 that satisfied the following two criteria:

The vCAS(

*n*_{e},*n*_{a}) wave function provides a theoretically consistent description of the electronic structure of BF for all*R*, without any mixing or swapping of the core and active orbitals.The vCAS(

*n*_{e},*n*_{a}) wave function uses the 2π orbitals to correlate the B2*s*pair in the separated atom limit since these orbitals give the lowest vCAS+1+2 energy for this limit.

We found only one vCAS(*n*_{e}, *n*_{a}) wave function that satisfied both of these criteria—the SCGVB(4,5)-inspired vCAS(4,5) wave function. Some of the other vCAS wave functions used the 2π orbitals to correlate the F2*p*-like pairs at all *R* values. Other vCAS wave functions used the 2π orbitals to correlate the F2*p*-like pairs at short *R* values and then discontinuously switched to using a different set of 2π orbitals to correlate the B2*s*-like pair at large *R* values. Many of these wave functions also exhibited core-active orbital mixing and/or orbital swapping. In summary, only the vCAS(4,5) and vCAS(4,5)+1+2 wave functions were found to provide a theoretically consistent methodology for describing the electronic structure of both the BH and BF molecules, exhibiting none of problems associated with the vCAS(10,8) and vCAS(10,8)+1+2 wave functions for BF. The potential energy curves from the vCAS(4,5) and vCAS(4,5)+1+2 calculations for BF are shown in Fig. 4, along with the potential energy curve from the present CCSD(T) calculations.

The last option that we explored was to combine the vCAS(4,5) and vCAS(10,8) approaches to determine if the “correct” behavior of the vCAS(4,5) wave function would address the problems in the vCAS(10,8) calculations. At each value of *R*, we computed the vCAS(4,5) wave function and then used this wave function as the initial guess for the vCAS(10,8) calculations, freezing the B1*s* and F1*s* orbitals to be those from the vCAS(4,5) wave function. We refer to the wave functions from these calculations as the vCAS(10,8)^{*} and vCAS(10,8)^{*}+1+2 wave functions. This approach did, indeed, solve many of the above problems, giving an energy that was 4.3 m*E*_{h} below that of the vCAS(4,5)+1+2 wave function at *R*_{e} and leading to values for the spectroscopic constants for BF—*R*_{e} = 1.2654 Å, *D*_{e} = 180.16 kcal/mol, and ω_{e} = 1405 cm^{−1}—that are in excellent agreement with the present CCSD(T) calculations; see Table II. However, this wave function used the 2π orbitals to describe the correlation of the two F2*p*-like pairs at short *R* values, and then, in the region Δ*R* ≈ 1.4 Å, the 2π orbitals switched to correlating the B2*s*-like pair. This inconsistent description of BF as a function of *R* resulted in a discontinuity in the vCAS(10,8)^{*} potential energy curve; see Fig. 4. Fortunately, a discontinuity is not present in the curve from the vCAS(10,8)^{*}+1+2 calculations, although the behavior of the curve at large *R* values is unusual.

The valence orbitals from the vCAS(4,5) and vCAS(10,8)^{*} wave functions for BF at *R*_{e} and *R*_{e} + 4.0 Å (essentially the separated atom limit) are plotted in Fig. 5. Of particular interest are the 1π and 2π orbitals. For the vCAS(4,5) wave function, these orbitals correspond to the F2*p*_{x,y}-like (1π) and B2*p*_{x,y}-like (2π) orbitals at all *R* values. However, for the vCAS(10,8)^{*} wave function, these orbitals correspond to the F2*p*_{x,y} (1π) and B2*p*_{x,y} (2π) orbitals in the separated atom limit and F2*p*_{x,y}-like (1π) and F3*p*_{x,y}-like (2π) orbitals at *R*_{e}.

As a check on the performance of the vCAS and vCAS+1+2 wave functions at large *R* values, the last row in Table III lists the sum of the vCAS(3,4) and HF energies (−24.563 454 h, −99.409 209 h) as well as the vCAS(3,4)+1+2 and HF+1+2 energies (−24.601 200 h, −99.642 289 h) of the boron and fluorine atoms. The vCAS(4,5) and vCAS(10,8) energies for BF at Δ*R* = 10 Å are slightly below the sum of the atomic energies because the vCAS(3,4) calculations on the boron atom yield orbitals that are pure symmetry functions (*s*, *p*), whereas the vCAS(4,5) and vCAS(10,8) calculations yield orbitals that have only C_{2v} symmetry, e.g., the 2*p*_{x,y} orbitals are slightly different than the 2*p*_{z} orbitals. This additional flexibility in the vCAS BF calculations at large *R* values yields a slightly lower energy (−0.2 *mE*_{h}). The differences in the vCAS+1+2 calculations at Δ*R* = 10 Å are a result of multiple factors. The difference in the vCAS(4,5)+1+2 energy and the corresponding sum of the atomic energies is a result of the fact that the vCAS+1+2 energy is not size consistent. As a result, the sum of the atomic energies differs by −6.4 m*E*_{h} from that of the vCAS(4,5)+1+2 wave function at Δ*R* = 10 Å. The errors in the vCAS(10,8)+1+2 energies are much larger, −36.9 m*E*_{h} (F2*p*) and −77.3 m*E*_{h} (B2*s*), because of the interchange of core and active orbitals as well as size inconsistency in energies from these wave functions.

The problems in the full vCAS(10,8) and vCAS(10,8)+1+2 descriptions for BF are a direct result of the extraordinary variational freedom implicit in the vCAS wave function. Optimization of both the orbitals and the configuration coefficients in the full configuration space associated with the active orbitals enables the vCAS wave function to describe non-dynamical and dynamical correlation effects in a potentially unpredictable way. Although the full vCAS wave functions for both BH and BF are able to describe the 2*s*–2*p* near-degeneracy effect in the boron atom, only for BH is this found to be the case at all internuclear separations. In BF, radial correlation of the electrons in the doubly occupied F2*p*-like orbitals, a dynamical correlation effect, is energetically more favorable at intermediate to short internuclear distances, and the 2π orbitals are used to describe this effect, not the B2*s*–2*p* near-degeneracy effect. In addition, because of orbital mixing and core-active orbital swapping, the active orbitals from the full vCAS calculations are often not optimal for the vCAS+1+2 calculations, leading to an unreasonably large *D*_{e}’s.

The scope of the vCAS and vCAS+1+2 calculations could, of course, be increased by including additional orbitals in the active space. Although this extension is beyond the scope of the current work, the above results suggest that it will still be critical to monitor the nature of the orbitals as a function of *R* to ensure that a consistent description of the BF molecule is obtained for all values of *R*.

## IV. CONCLUSIONS

Although it is tempting to use the complete active space self-consistent field (CAS) method as a “black box”—select a set of active orbitals and run the calculation—the full valence-CAS (vCAS) calculations on BF presented here serve as a cautionary tale. Selecting the active orbitals as the union of a set of atomic orbitals does not imply that these orbitals maintain their basic identities for the calculations. In BH, the 1π orbitals kept their basic identities as boron 2*p*-like orbitals in the full vCAS calculations, describing the 2*s*–2*p* near-degeneracy effect in the boron atom. In the full vCAS wave function of BF, on the other hand, the corresponding 2π orbitals are F3*p*-like orbitals because the correlation of the two fluorine 2*p*π pairs is energetically more important than the correlation of the B2*s* pair (2*s*–2*p* near-degeneracy effect) at *R*_{e}. The full vCAS calculations are further complicated by the fact that there is mixing between the core F1*s* and active F2*s* orbitals at short *R*, swapping of the core B1*s* and active F2*s* orbitals and between the core F1*s* and active B2*s* orbitals at large *R*, with changes in the nature of the orbitals depending on how *R* is scanned (from *R*_{e} → ∞ or ∞ → *R*_{e}). Although these changes lower the full vCAS energy, they do not necessarily lead to the lowest vCAS+1+2 energy.

Although a larger active space will always provide a better vCAS energy than a smaller active space of which it is a superset, it does not follow that the vCAS+1+2 energies will do the same. This is the case for BF where the vCAS(4,5) wave function gave a better vCAS(4,5)+1+2 energy than the vCAS(10,8)+1+2 calculation. This rather unsettling finding is a consequence of the unusual behavior of the vCAS(10,8) wave function where core-active orbital mixing and swapping cause the active orbitals to be suboptimal for the vCAS(10,8)+1+2 wave function. We found only one computationally sound means of addressing the problems in the vCAS(10,8) and vCAS(10,8)+1+2 calculations on BF—the use of the SCGVB(4,5)-inspired vCAS(4,5) wave function for the vCAS+1+2 calculations. This approach provides very good predictions of (*R*_{e}, *D*_{e}, and ω_{e}) in comparison to the results from the present CCSD(T) calculations: Δ*R*_{e} = −0.0034 Å, Δ*D*_{e} = −3.32 kcal/mol, and Δω_{e} = +15 cm^{−1}. Furthermore, the potential energy curve exhibits no discontinuities or other troublesome features. In comparison, the differences in these same quantities for the BH molecule are Δ*R*_{e} = 0.0003 Å, Δ*D*_{e} = −0.20 kcal/mol, and Δω_{e} = −2 cm^{−1}.

We found that the use of the F1*s* and B1*s* orbitals from the vCAS(4,5) wave function in the vCAS(10,8) calculations, yielding the CAS(10,8)^{*} wave function, eliminates the problems with the mixing and swapping of the core and active orbitals. Furthermore, the spectroscopic constants predicted by vCAS(10,8)^{*}+1+2 calculations are in excellent agreement with those from the present CCSD(T) calculations, Δ*R*_{e} = −0.0034 Å, Δ*D*_{e} = −0.61 kcal/mol, and Δω_{e} = 14 cm^{−1}. Unfortunately, at large *R*, there was a discontinuous change in the nature of the 2π orbitals in the vCAS(10,8)^{*} calculations, from F3*p*_{x,y}-like orbitals at *R* ≲ 2.7 Å to B2*p*_{x,y}-like orbitals at *R* ≳ 2.7 Å, resulting in a discontinuity on the vCAS(10,8)^{*} potential energy curve. Although the curve from the corresponding vCAS(10.8)^{*}+1+2 calculations did not exhibit such a discontinuity, it still exhibited strange behavior as it approached the asymptote at large *R*.

We have found similar problems with full vCAS calculations on other molecules, including molecules that contain other atoms in the latter part of the first row. Although the multiconfiguration vCAS method is a powerful means of describing the electronic structure of molecules that are not well described by a single configuration method, the tremendous flexibility of the vCAS method is a mixed blessing. As shown here, it is critical to pay close attention to both the orbitals and configuration coefficients in the vCAS calculations to ensure that the calculations provide a theoretically consistent description of the electronic structure of molecules for all geometries of interest. Although this check was relatively easy to perform for BH and BF, it will be far more challenging, yet undoubtedly just as important, for vCAS calculations on larger molecules. The problems with the full vCAS wave function of BF can be avoided by using the SCGVB(4,5)-inspired vCAS(4,5) and vCAS(4,5)+1+2 wave functions, and a similar approach could be used for other molecules.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ACKNOWLEDGMENTS

This research was supported, in part, by the Center for Scalable Predictive Methods for Excitations and Correlated phenomena (SPEC), which is funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, as part of the Computational Chemical Sciences Program at Pacific Northwest National Laboratory. We wish to thank Professor Kirk A. Peterson for his reading of and comments on this manuscript and Professor David L. Cooper for his advice on the use of the CASVB module in Molpro for the SCGVB(4,5) calculations.

## REFERENCES

It is, of course, possible to define an active space that also correlates the doubly occupied F2*s*, F2*p*_{x}, and F2*p*_{y} orbitals. However, the resulting vCAS calculations rapidly become intractable for larger molecules.