Localized and Delocalized States of a Diamine Cation: Resolution of a Controversy

Abstract

Recent Rydberg spectroscopy measurements of a diamine molecule, N,N′-dimethylpiperazine (DMP), indicate the existence of a localized electronic state as well as a delocalized electronic state. This implies that the cation, DMP⁺, can similarly have its positive charge either localized on one of the N atoms or delocalized over both. This interpretation of the experiments has, however, been questioned based on coupled cluster calculations. In this article, results of high-level multireference configuration interaction calculations are presented where a localized state of DMP⁺ is indeed found to be present with an energy barrier separating it from the delocalized state. The energy difference between the two states is in excellent agreement with the experimental estimate. The results presented here, therefore, support the original interpretation of the experiments and illustrate a rare shortcoming of CCSD(T), the “gold standard” of quantum chemistry. These results have implications for the development of density functionals, as most functionals fail to produce the localized state.

A mixed-valence molecule, defined as a molecule containing two or more redox sites in different oxidation states, can exhibit a localized or a delocalized electronic structure. The occurrence and energy difference between the two types of states depend on the detailed molecular structure, such as the distance between redox sites or the chemical bonds connecting them, with either through-space and through-bond mechanisms playing a role.

The radical cation of N,N′-dimethylpiperazine (DMP), see Figure 1, has been identified as an interesting organic mixed-valence molecule for studying lone-pair interactions and delocalization/localization phenomena.¹⁻⁷ Experimental studies (EPR/Raman) on DMP⁺ were originally performed in solution,¹⁻³ where resonance Raman spectra together with calculations indicated the electronic structure of DMP⁺ to be most consistent with a C_2h-symmetric delocalized state (here called DMP-D⁺).^1,2 Until recently, a localized electronic structure of DMP⁺ had not been observed experimentally. Ultrafast time-resolved Rydberg spectroscopy was carried out in the gas phase, where an excitation from the ground state of the DMP molecule to the 3p Rydberg state was used to monitor the picosecond time-scale dynamics from a localized to a delocalized state. By varying the energy of the photon, the energy difference between the two states could be determined as 0.33 eV, in favor of the delocalized state.^4,5 This is a rare case where the energy difference between localized and delocalized electronic states in a molecule has been determined experimentally, and it provides an important test case for theoretical methods where the balance in the electronic structure description of the two types of states can be problematic.

Figure 1.

Structure of (a) neutral N,N′-dimethylpiperazine, DMP (the primary eq-eq conformer), (b) and (e) the localized cation, DMP-L⁺, (c) and (f) the delocalized cation, DMP-D⁺, and (d) the saddle point for the transition between DMP-L⁺ and DMP-D⁺, denoted DMP-SP⁺. The spin densities shown in (b) and (c) correspond to an isosurface level of 0.01 electron/ų. The definition of the dihedral angles D1 and D2 that are used to span a cut through the energy surface are shown in (e) and (f), and their values in DMP-L⁺ and DMP-D⁺ are given, as well as the C–C bond length in the two structures (BHLYP level of theory).

Since the Rydberg state electron is distributed over a large region, the Rydberg excited molecule can be assumed to resemble closely the cation. Density functional theory (DFT) calculations with commonly used density functionals, however, fail to give a localized state of DMP⁺ (the BHLYP functional⁸ being the exception), while calculations with a functional where self-interaction error is explicitly removed give results consistent with the Rydberg state experiments.⁵ These DFT results have subsequently led to a debate in the literature, and the existence of a localized state of DMP⁺ on the potential energy surface (PES) been questioned based on the fact that coupled cluster theory calculations at the CCSD(T) level do not produce a localized state.^6,7 The interpretation of the experimental measurements has thus also been questioned based on the assumption that CCSD(T) calculations, the “gold standard” of quantum chemistry, produce a reliable description of the PES. An important question, therefore, arises as to whether DFT and CCSD(T) calculations are sufficiently accurate to describe DMP⁺ or whether the experimental observations need to be reinterpreted.

Organic mixed-valence cations have actually been found to present considerable challenges to both state-of-the-art wave function theory and DFT approaches.^9,10 As discussed by Kaupp and co-workers, wave function theory approaches based on unrestricted Hartree–Fock (UHF) wave functions have an initial bias toward symmetry breaking and hence charge localization, in addition to spin contamination. Electron correlation is imperative for describing the possible delocalization present in such systems, but unrestricted MP2 calculations have been found to suffer from exaggerated spin contamination. Therefore, a robust dynamic correlation treatment appears necessary, and as these systems by nature have near-degeneracies, it may also be important to describe the static correlation reliably from the start (i.e., via multireference approaches) in addition to the dynamic correlation (though electron correlation is not always clearly separable).

In this study, multireference wave function calculations of the PES of the DMP cation are carried out with the aim of resolving the controversy and answering definitively the question of whether both localized and delocalized states exist on the ground-state DMP⁺ energy surface. Using a 78-point cut of the energy surface calculated with multireference configuration interaction (FIC-MRCI+Q), we firmly establish the PES of the DMP cation as containing both a localized state and a delocalized state.

The lowest energy structure of the neutral DMP molecule is a chair conformer of C_2h symmetry with both methyl groups in equatorial positions (Figure 1a), which clearly reveals the presence of lone pairs on each sp³-hybridized nitrogen (atoms N2 and N6 in Figure 1e,f). After removing one electron from a nitrogen lone-pair orbital, a positive hole remains, which results in Jahn–Teller-type distortion and the formation of a distorted structure of C_s symmetry. The geometry of this localized structure, DMP-L⁺, is shown in Figure 1b (a symmetrically equivalent conformer also exists where the hole is localized on the other nitrogen atom). The structure of DMP-L⁺ clearly reflects the different electronic nature of the nitrogen atoms, where the ionized nitrogen site exhibits more sp²-like character and the nonionized site remains sp³-like. This is also seen in the spin density in Figure 1b, showing the unpaired electron localized on only one nitrogen atom. Alternatively, the charge can be delocalized between both nitrogen atoms, resulting in a DMP-D⁺ structure of C_2h symmetry. The delocalized DMP⁺ structure shown in Figure 1c reveals the spin density as delocalized over both nitrogen atoms, and interestingly a contribution from the bridging C–C atoms can be seen, suggesting the involvement of the C–C bonds in lone-pair interactions that result in the delocalized state. The transition from the localized to the delocalized state of DMP⁺ involves geometrical changes such as the bending and rotation of the methyl groups, as well as elongation of C–C bonds. Taking this into account, we find that two dihedral angles, D1 and D2, serve the purpose of being suitable descriptors for characterizing a cut of the PES that connects both delocalized and localized minima. The D1 (D2) angle is defined via the two planes created by C5–N6–C7 (C1–N2–C3) and N6–C7–C8 (N2–C3–C4) atoms (Figure 1e,f).

Previous theoretical studies of DMP⁺ have involved geometry optimizations to find the lowest energy atomic configuration and minimum energy paths at the lower levels of theory as well as single-point calculations at higher levels of theory. However, the absence of the DMP-D⁺ and DMP-L⁺ minima for some electronic structure methods complicates comparisons, introduces an unfortunate dependence on the minimization algorithm employed as well as the initial structure, and ideally requires analytical gradients. In order to conveniently compare different electronic structure methods (with or without available analytical gradients) and to quantify the differences between methods, we instead utilize a 78-point PES cut where the dihedral angles D1 and D2 vary from 70 to 175°. The surfaces are interpolated using a biharmonic spline interpolation provided in Matlab.¹¹ For each surface point, constrained geometry optimizations at the BHLYP/aug-cc-pVDZ level are performed where the D1 and D2 angles are fixed while the energy is minimized with respect to all other atom coordinates. Single-point energy evaluations for all methods are then carried out on the constraint-optimized BHLYP structures (an alternative choice of configurations is discussed in the Supporting Information (SI)).

Figure 2a shows the energy surfaces calculated with the BLYP and B3LYP density functionals. Only a single minimum is found, corresponding to the delocalized state at ∼ ±90°. These results represent the behavior of most common density functionals. Figure 2b (bottom) shows the energy surface obtained with the BHLYP functional, where instead two well-resolved minima are obtained, one corresponding to a localized state, DMP-L⁺, at D1 = −132.4° and D2 = 169.4°, and the other corresponding to a delocalized state, DMP-D⁺, at ±88.7°. These values are obtained from full optimization. A first-order saddle point representing a transition state between the two states (DMP-SP⁺ in Figure 1d) is obtained at D1 = −97.3° and D2 = 151.8°. The energy difference between DMP-D⁺ and DMP-L⁺ is found to be +0.178 eV using fully relaxed structures, with DMP-D⁺ being more stable. An energy barrier of +0.033 eV is found for the transition from DMP-L⁺ to DMP-D⁺. Figure 2b (top) shows the energy surface calculated using the highest level of theory considered in this work, the multireference wave function method, MRCI+Q. A cc-pVDZ basis set was used, and the structures were those obtained from the BHLYP surface. The reference in the MRCI+Q calculation is a CASSCF wave function with a large active space of 11 electrons in 12 orbitals (CAS(11,12)). The fully internally contracted version of MRCI (FIC-MRCI, implemented in ORCA¹²) was used, and the Davidson size-consistency correction¹³ for unlinked quadruples (Q) was applied. The internal contraction avoids bottlenecks associated with the traditional uncontracted MRCI approaches by applying an excitation operator to the whole reference wave function. The CAS(11,12) active space, including σ H₃C–N and C–C orbitals, as well as the corresponding virtual orbitals and the natural orbitals are shown in Figure 3. This active space was deemed large enough to capture the essential orbital interactions in the system.

Figure 2.

Potential energy surfaces calculated at the (a) B3LYP (upper) and BLYP (lower), (b) FIC-MRCI+Q(11,12) (upper) and BHLYP (lower), and (c) CCSD(T) (upper) and CCSD (lower) levels of theory. The aug-cc-pVDZ basis set is used, except in the MRCI calculation where cc-pVDZ is used. The BHLYP functional with 50% exact exchange produces a minimum on the energy surface (unlike BLYP and B3LYP) corresponding to a localized state, 0.18 eV higher in energy than the delocalized state. The structures are relaxed at the BHLYP level, subject to the constraints on the two dihedral angles defining the energy surface, and the other calculations are carried out for those structures. The high-level FIC-MRCI+Q calculations give a localized state that is 0.34 eV higher in energy than the delocalized state, in close agreement with the experimental estimate of 0.33 eV. The CCSD(T) calculation fails to give a localized state even though it is present in the CCSD calculation.

Figure 3.

CAS(11,12) active space for (a) DMP-D⁺ and (b) DMP-L⁺. Natural orbitals calculated at the CASSCF/aug-cc-pVDZ level of theory on the BHLYP/aug-cc-pVDZ structures are shown with their natural occupation numbers. The active space contains two N lone pairs, two σ H₃C–N and two σ C–C bonding orbitals, and the corresponding virtual orbitals. As seen, the orbitals associated with the N lone pairs have acquired considerable C–C bond character in both DMP-D⁺ and DMP-L⁺ states, and the C–C σ* orbitals have significant fractional occupation (0.02–0.03), suggesting the importance of the C–C bonds in a through-bond delocalization mechanism. It was found necessary to include virtual N 3p atom-like orbitals for a balanced active space.

The PESs of BHLYP and FIC-MRCI+Q are overall remarkably similar, exhibiting resolved minima for both DMP-D⁺ and DMP-L⁺ for similar coordinate values as well as a clearly defined energy barrier region. The magnitude of the energy difference between the delocalized and localized regions is the main difference, with BHLYP overstabilizing the localized state.

Energy surfaces were also calculated at the coupled cluster level of theory using either a singles–doubles expansion (CCSD) or the singles–doubles and perturbative triples expansion (CCSD(T)) as shown in Figure 2c. Importantly, for each surface point, a stability analysis (calculation of the electronic Hessian) was performed for the UHF SCF solution, and in many cases instabilities were found; new stable solutions were subsequently generated. In fact, it was found that all stable UHF SCF solutions had a localized electronic structure, and no delocalized minimum could be found at the HF level. The CCSD surface (shown in Figure 2c, lower) is overall comparable to BHLYP surface in terms of relative energy, while the position of the DMP-L⁺ minimum is closer to the MRCI+Q minimum. The two minima are directly visible on the surface and are consistent with the results of previous CCSD geometry optimizations.^5,6 The energy surface shows, furthermore, the presence of a low-energy energy barrier. Spin population analysis of the unrelaxed CCSD density for the D1 = −90°, D2 = 90° point (Mulliken nitrogen spin populations of 0.38 and 0.30, respectively) confirms the electronic structure as mostly delocalized with CCSD, despite being expanded from a localized UHF reference wave function. Despite the minor symmetry breaking present, the CCSD wave function appears to describe the electronic structure and energy surface of DMP⁺ qualitatively correctly. The energy difference between DMP-D⁺ and DMP-L⁺ is calculated to be +0.22 eV, in agreement with previous results (0.23 eV).⁵ However, remarkably, when perturbative triples excitations are added (i.e., the CCSD(T) method), the PES changes substantially. The CCSD(T) surface (Figure 2c, upper) has only one minimum, analogous to the results of most DFT methods. The localized state is missing.

The results presented here (see Table 1) obtained at the MRCI+Q level of theory with a large CASSCF(11,12) reference wave function clearly establish the presence of both a localized and a delocalized minimum on the PES of the DMP cation. Furthermore, the energy difference calculated using the aug-cc-pVDZ basis set is 0.336 eV, in excellent agreement with the measured energy difference between delocalized and localized Rydberg energy states.⁵ While the MRCI+Q calculations depend on the theory level used for structural optimization, we note that the use of alternative structures at the CCSD or DMRG-CASSCF(19,20) level (to be reported on later) result in small changes to the energy difference (less than 0.02 eV). The barrier from DMP-L⁺ to DMP-D⁺ (using the BHLYP-located minima and saddle point geometries) is calculated to be 0.05 eV. The multireference wave function energy surface hence reinforces the original interpretation of the experimental measurements that a localized state exists, separated by an energy barrier from the delocalized state. This still hinges on the assumption that the Rydberg states resemble the states of the molecular cation.

Table 1. Single-Point BHLYP, MP2, CCSD, and FIC-MRCI+Q(11,12) Energies (eV) of DMP-D⁺ and DMP-SP⁺ Relative to DMP-L⁺, Calculated Using the Relaxed BHLYP/aug-cc-pVDZ Structures.

method	ΔE	ESP – Eloc
BHLYP/aug-cc-pVDZ	0.178	0.033
MP2/aug-cc-pVDZ	0.153	0.081
CCSD/aug-cc-pVDZ	0.221	0.012
FIC-MRCI+Q(11,12)/aug-cc-pVDZ	0.336	0.050
experiment5	0.33

The cc-pVDZ basis set was used for the calculations of the energy surface, while a larger aug-cc-pVDZs basis set was used for the calculations of the local minima. We estimate that increasing the basis set further would change the relative energy by less than ∼0.02 eV, based on explicitly correlated CCSD-F12 calculations (CCSD-F12/cc-pVDZ-F12 results give an energy difference of 0.238 eV) as well as large-basis local-correlation CCSD calculations (see the SI).

In a recent comment, Ali et al.⁶ presented CCSD and CCSD(T) calculations of DMP⁺ and compared them with DFT results. They argued that CCSD(T) gives accurate results and that the absence of a localized state in most DFT calculations does not represent a shortcoming of those functionals. From the high-level MRCI+Q results presented here, it is clear that the CCSD(T) calculations are in fact in error, possibly in part due to its single-reference nature. The lack of a stable delocalized solution at the reference UHF level (as shown by stability analysis) and the contrasting behavior of CCSD and CCSD(T) suggest this. In order to understand whether the behavior of CCSD(T) stems from the flawed UHF reference function or alternatively the perturbative triples correction, we performed CCSD(T) calculations using alternative reference wave functions, as detailed in the SI. CCSD(T) calculations using Brueckner orbitals,^14,15 quasi-restricted orbitals,¹⁶ and UKS-DFT orbitals did not, however, lead to an improved CCSD(T) energy surface; see SI for a discussion. We also performed orbital-optimized coupled cluster theory (OO-CCD(T)) calculations, where the orbitals are variationally optimized at the CCSD level (instead of the HF level), thus effectively removing any effect of the HF reference. While this encouragingly gave a relaxed OO-CCD density that showed complete delocalization (without symmetry-breaking) according to spin population analysis of the D1 = −90°, D2 = 90° point (see Table S2 in the SI), the OO–CCD(T) surface is still missing the DMP-L⁺ minimum, as shown in Figure S1 in the SI. These results suggest that the HF reference wave function is not the culprit in the CCSD(T) calculation, but rather the wave function expansion itself, perhaps due to an imbalance in the static and dynamic correlation of the wave function when the perturbative triples correction is included. It would be interesting to see whether these problems are resolved at the (very expensive) CCSDT (full triples)¹⁷ or CCSDT(Q)¹⁸ levels (full triples and perturbative quadruples) or via alternative triples approximations.¹⁹⁻²⁴

In summary, we have presented a combined single-reference and multireference wave function theory investigation of the DMP cation and have shown that the molecule represents an unusual challenge to standard quantum chemistry methods, whether at the DFT level or the wave function theory level. Surprisingly, the CCSD(T) level of theory gives a qualitatively incorrect energy surface, failing to describe the localized state of the DMP cation that is unquestionably present at the MRCI+Q level of theory and is inferred from Rydberg spectroscopy measurements. The DMP cation, despite its apparent simplicity, is a truly challenging case for correlated wave function theory, and the presented results should make the system useful as a benchmark system for the study of electronic state localization and guide the development of more robust and affordable correlated wave function methods as well as density functionals.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.0c03651.

• Computational details, basis set convergence, Mulliken spin populations of wave functions, and Cartesian coordinates of optimized structures (PDF)

• A compressed directory with XYZ files for the entire 78-point BHLYP-constraint-optimized surface (ZIP)

The authors declare no competing financial interest.