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Electronic structure of methyl radical photodissociation

  • Aiswarya M. Parameswaran , Dayou Zhang and Donald G. Truhlar ORCID logo EMAIL logo
Published/Copyright: August 27, 2025
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Pure and Applied Chemistry
From the journal Pure and Applied Chemistry Volume 97 Issue 10

Abstract

The calculation of accurate excitation energies and potential energy surfaces of photochemical reactions is a major challenge of current quantum chemistry, especially when one wants to keep the cost low enough to make detailed dynamical simulations affordable. Methods that might be affordable for large complex molecules can be tested against benchmark results for smaller molecules, where accurate benchmarks can be available. Here we report such testing for the excitation and dissociation of the methyl radical, yielding both singlet and triplet CH2. The emphasis is on multistate pair density functional theory using compressed-multistate (CMS-PDFT) and linearized (L-PDFT) formulations. We also consider the less affordable XMS-CASPT2 method with the same state-averaged-complete-active-space (SA-CASSCF) reference wave functions, which has seven active electrons in 10 active orbitals. The calculations use state averaging over seven states and a model space that spans the seven lowest SA-CASSCF eigenvectors. We study three on-top density functionals: tPBE, tPBE0, and MC23. Vertical excitation energies, adiabatic excitation energies, and dissociation energies, along with cuts through the potential surfaces along the dissociation coordinate, were computed with the (7, 10) active space. XMS-CASPT2 and L-PDFT with the MC23 functional show consistent and reliable performance for excitation energies, closely reproducing benchmark values, and producing smooth, physically reasonable potential energy surfaces essential for nonadiabatic dynamics simulations, but they are less accurate for bond energies. The L-PDFT calculations with the tPBE functional are more accurate for dissociation energies, but less accurate for excitation energies.

Keywords: chemistry; computer simulation; quantum chemistry; quantum mechanics; quantum science and technology; radical reactions; spectroscopy

Introduction

Quantum mechanics is used in many kinds of chemical modeling. The first Nobel Prize for quantum chemistry was awarded in 1954 to Pauling for his work on bonding and ground-state molecular structure. 1 The second Nobel Prize in quantum chemistry was awarded in 1966 to Mulliken “for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method.” His Nobel Prize Lecture, in contrast to Pauling’s, included excited electronic states, the spectroscopy of which was important to his conception of molecular orbitals. 2 Theoretical interpretation of electronically nonadiabatic chemical reactions has been of interest since the first decade of quantum mechanics, 3 and already in the 1930s, models were proposed 4 , 5 , 6 , 7 involving trajectories or wave packets on adiabatic potential energy surfaces, interrupted by “jumps” to other surfaces. Such models have become one standard way to interpret such processes, as in the popular method of trajectory surface hopping. 8 , 9 , 10 , 11 As for ground states though, quantitative progress on excited-state chemistry was at first impeded by lack of quantitative knowledge of potential energy surfaces. The quantum mechanical treatment of excited electronic states requires more complicated considerations and methods than the modeling of ground states.

Progress on energies of electronically excited states was qualitative at first, 12 , 13 , 14 but great progress has now been made and the low-lying excitation energies of many small and medium-sized molecules have now been calculated more accurately than they have been measured experimentally. 15 The next stage of progress is understanding the dynamics of processes involving multiple electronic states. For systems with a small number of atoms, when given the coupled potential energy surfaces, it is now possible to solve the Schrödinger equation for nuclear motion accurately even for electronically nonadiabatic reactions. 16 , 17 , 18 , 19

Dynamics of processes involving multiple electronic states are usually modelled in the adiabatic representation where the electronic energies calculated with fixed nuclei serve as coupled potential energy surfaces governing nonclassical nuclear motion, 20 with the latter treated either semiclassically or quantum mechanically. This often requires globally valid potential energy surfaces, valid even for bond breaking, and many methods of electronic structure that are very useful for energies at molecular equilibrium geometries are inadequate for global potential energy surfaces even for ground states, with even more serious difficulties for global potential energy surfaces of coupled electronic states. The most successful methods available at this time for global potential energy surfaces of coupled electronic states are two-step methods starting with a complete active space self-consistent field (CASSCF) multiconfigurational calculation 21 , 22 to obtain molecular orbitals followed by a second step (a post-CASSCF step) to obtain the electronic energies. The second step is required because, although CASSCF can give qualitatively correct wave functions, it is quantitatively inadequate for energies. Two-step methods beginning with a multiconfigurational reference function are called multireference methods.

There are two main types of methods used to calculate electronic energies, namely wave function theory 23 , 24 and density functional theory, 25 although hybrid methods 26 , 27 , 28 , 29 are also widely used. Density functional theory is very popular and has been transformatively successful in the form proposed by Kohn and Sham, 30 based on a single Slater determinant; the success is due in part to the low cost/accuracy ratio for density functional calculations as compared to accurate wave function calculations. However, conventional approximations to the density functionals for single Slater determinants can represent most bond breaking processes only by breaking spin symmetry, and optimized Slater determinants also often break symmetry for other problems involving nearly degenerate electronic states. (This is also a known problem for Hartree–Fock methods, but CASSCF methods do not usually have this problem and can correctly describe static correlation without breaking spin symmetry). While symmetry breaking is often not an impediment to calculating accurate energies near equilibrium geometries for ground states, it can pose serious problems for bond breaking in ground states and for modeling coupled excited states globally. In such cases, the artificial mixing of singlet and triplet states introduced by symmetry breaking can lead to ambiguities that undermine the reliability of the modeling. This is one reason for preferring post-CASSCF methods (like XMS-CASPT2, 31 QD-NEVPT2, 32 and the L-PDFT method introduced below) for nonadiabatic simulations of wide-amplitude motion and closely coupled states.

Obtaining post-CASSCF energies by wave function methods can be expensive in terms of computer time and memory. While CASSCF calculations are often practical for moderately sized systems, post-CASSCF calculations by perturbation theory (e.g., CASPT2, 33 XMS-CASPT2, NEVPT2, 34 or QD-NEVPT2), configuration interaction (e.g., MR-CISD 35 ), or coupled cluster theory (e.g., MR-CC 36 ) can become computationally prohibitive as the size and complexity of the active space and the size of the molecule increase, even when CASSCF is affordable. One attempt to circumvent this is multiconfiguration pair-density functional theory (MC-PDFT) in which one starts with a multiconfiguration reference wave function (for example, a CASSCF wave function) that provides the correct spin symmetry and a qualitatively correct treatment of bond-breaking processes and near degeneracies, then uses this reference wave function to compute the classical electronic energy and uses a pair-density functional (called the on-top functional) to compute the nonclassical electronic energy (For this purpose, the classical energy is defined as the sum of the reference kinetic energy and reference classical Coulomb energy, and the nonclassical energy is defined as the remainder. In Kohn–Sham theory, the nonclassical energy is usually called exchange–correlation energy). Such calculations are often more affordable than the post-CASSCF wave-function methods mentioned in the second sentence of this paragraph. Further cost savings for both wave function theory and pair-density functional theory can be achieved by using other kinds of multiconfigurational reference wave functions (e.g., RASSCF, 37 , 38 RASCI, 39 GASSCF, 40 , 41 selected CI, 42 , 43 or DMRG 38 , 44 , 45 , 46 ) which can be more affordable than CASSCF for large systems. We refer the reader to the literature for further discussion of these alternative reference wave functions.

For correct modeling of nonadiabatic dynamics, it is important for the electronic structure method to give the correct topology of the manifold of electronic states. For nonlinear molecules in the absence of spin–orbit coupling, the correct topology is that adiabatic potential energy surfaces have conical intersections along an (F – 2)-dimensional seam of cuspidal ridges, where F is the number of internal degrees of freedom equal to 3N – 6, where N is the number of atoms. 47 One such method that gives the correct topology of potential energy surfaces is the compressed-state multistate pair-density functional theory 48 (CMS-PDFT). CMS-PDFT is a multi-state extension of MC-PDFT 49 , 50 , 51 specifically designed to describe systems with strongly coupled electronic states where the accurate description of conical intersections and avoided crossings become essential for modeling. Another multi-state extension of MC-PDFT is the more recent linearized pair-density functional theory 52 , 53 , 54 (L-PDFT). The present paper compares CMS-PDFT and L-PDFT predictions to CASPT2 predictions for the excited electronic states of methyl radical. (Multi-state methods are methods in which the final step is the diagonalization of an effective Hamiltonian in a model space of the electronic states of interest; this final step is the key to obtaining the correct topology of the interacting surfaces).

The methyl radical (CH3) plays a pivotal role in numerous fundamental and applied areas of chemistry, including hydrocarbon combustion, 55 atmospheric processes, 56 and interstellar reactions. 57 In photochemistry, it serves as an important prototype for investigating the photodissociation dynamics of larger open-shell hydrocarbons. Since Herzberg’s seminal work in 1961, 58 , 59 it has been established that the methyl radical adopts a planar geometry in its ground electronic state, rendering many single-photon electronic transitions to excited states dipole-forbidden. However, the advent of advanced multiphoton spectroscopic techniques, such as resonance-enhanced multiphoton ionization (REMPI), has opened new avenues for probing the excited states of the methyl radical 60 that were previously inaccessible.

Despite significant experimental advances, 61 , 62 , 63 , 64 , 65 , 66 theoretical investigations into the photodissociation dynamics of the methyl radical remain limited. 65 , 66 , 67 , 68 , 69 , 70 Many open questions persist – particularly regarding the dissociation mechanisms and the nature of the electronic states involved. While several experimental studies have reported accurate lifetimes for the excited states, 65 , 66 the corresponding theoretical simulations often fall short of reproducing the full dynamics, typically offering only qualitative agreement. A major limitation of these theoretical efforts (except for the treatment of Zanchet et al., which includes the H–C–H bending mode as well as a C–H stretch) is the use of one-dimensional potential energy surfaces (PESs) in trajectory-based simulations; 69 one-dimensional treatments are inadequate for capturing essential features such as conical intersections and nonadiabatic effects that require higher-dimensional representations. They are especially limiting in the context of this study, where multidimensional dynamics are critical for accurately describing the formation of the H2 photoproduct.

Photodissociation from the 3s and 3pz Rydberg states of CH3 has been the focus of both experimental and theoretical efforts. Early experimental work includes photofragment translational spectroscopy by North et al., 61 resonance-enhanced Raman spectroscopy by Westre et al., 62 , 63 and the H-atom Rydberg tagging by Wu et al. 64 While insightful, these methods were susceptible to systematic errors due to Doppler and pressure broadening. On the theoretical front, Yu et al. 70 performed one of the first ab initio investigations using restricted Hartree–Fock (RHF, a single-determinant method) and configuration interaction with single and double excitations (CISD), aiming to assess the preferred dissociation pathways of CH3 in its excited states. Their analysis of the B ˜ state of CH3 showed a clear preference for C–H bond rupture leading to CH2(1A1) + H, supported by favorable energetics, tunneling arguments, and orbital correlation considerations. However, they noted that the CH + H2 channel may be activated under high-energy photolysis, though the associated mechanisms remain theoretically complex and experimentally unresolved.

Recently, the work of Balerdi et al. 65 using the femtosecond three-color experiment along with velocity map imaging detection and high-level ab initio calculations proposed that the fastest predissociation mechanism from CH3 to CH2 is from the 3pz 2 A 2 ″ state of the methyl radical to the valence 2B1 state eventually dissociating to CH2+H with the CH2 in its 1B1 excited state. However, a more recent velocity map and slice ion imaging study 66 predicted that the predissociation process from the 3pz Rydberg state from the previous study was followed by two conical intersections later on, letting the system relax to lower electronic states of CH2.

The most recent theoretical investigation by Rodríguez-Fernández and co-workers 69 employed the time-dependent semiclassical forward scattering (TFS) approach on one-dimensional C–H dissociation curves to simulate dynamics from the Rydberg states. While they were able to reproduce the experimental dissociation lifetimes for the 3s state to a reasonable degree, the model failed to quantitatively capture the dynamics from the 3pz state. The authors attributed this discrepancy to the reduced dimensionality of the PES. Given the known role of conical intersections in the nonadiabatic dynamics of CH3, it is clear that a fully dimensional treatment – incorporating all six vibrational degrees of freedom – is essential for accurately describing the photodissociation process. The present article is a first step toward that goal by examining the accuracies of methods capable of treating conical intersections with the correct topology.

Computational details

All multi-state methods considered in this paper begin with a state-averaged complete active space self-consistent field (SA-CASSCF) wave function in which the number of states averaged is the same as the number of states included in the model space. All XMS-CASPT2 calculations in this study were carried out using the Molpro software package. 71 All calculations were performed using an active space consisting of seven electrons and 10 orbitals. The 10 orbitals were carefully chosen, being the 2s and 2p orbitals of carbon, three bonding orbitals of the three C–H bonds in methyl radical, and the three corresponding antibonding orbitals. The ground-state geometry of the methyl radical was optimized by SA(7)-CASSCF(7,10). This is the only geometry optimization done in the study. To calculate the adiabatic PESs, we used SA(7)-CASSCF(7,10) theory followed by XMS(7)-CASPT2(7,10). The 1s orbital of carbon is kept doubly occupied in all SA-CASSCF calculations and frozen in CASPT2 calculations. All CMS-PDFT calculations were carried out in OpenMolcas 72 using the tPBE on-top pair density functional. We ensured that all SA(7)-CASSCF(7,10) energies are identical in OpenMolcas and Molpro for each state of each geometry. All calculations in OpenMolcas and Molpro are performed without using spatial symmetry.

To calculate the L-PDFT energies, we first reproduce the SA(7)-CASSCF(7,10) reference wave function in PySCF 73 by converting converged OpenMolcas CASSCF molecular orbital coefficients into PySCF format, followed by a CASCI calculation. The configuration state function basis (instead of the default Slater determinant basis) is used during the CASCI calculation. The CASCI calculation should give the same energy (within negligible numerical noise) as the CASSCF calculation because the final step of the CASSCF calculation is a CASCI step, and we checked that the results are indeed the same. Once we reproduced the CASSCF wave function in PySCF, we evaluated the L-PDFT energies using three on-top pair density functionals, namely, L-MC23, 74 L-tPBE, 49 and L-tPBE0. 29 (In general, the use of two programs is not necessary, but in the present case it was done because we had originally found the active space orbitals in OpenMolcas, and we wanted to use those active space orbitals in PySCF to avoid repeating the search for a smoothly varying and physically realistic set of active space orbitals).

The modified G3 semidiffuse (MG3S) basis set 75 was used for all calculations performed for the present study. Note that for systems containing only C and H, this basis set is equivalent to the 6-311+G(2df, 2p) basis set.

We also compute and compare the dissociation energies, D e , for the homolytic bond dissociation reaction of methyl radical, CH3 → CH2 + H. Here, D e refers to the bond dissociation energy on the Born–Oppenheimer potential energy surface, corresponding to the depth of the potential energy well at the equilibrium geometry. In contrast, D 0 is the bond dissociation energy from the zero-point level, which is experimentally accessible (the subscript 0 denotes the ground vibrational–rotational state). D 0 is also the bond dissociation enthalpy at 0 K. Since D e cannot be directly obtained from experiment, we first evaluate D 0 using standard enthalpies of formation ( ∆ f H ° 0 ) of the parent molecule and its dissociation products:

(1) D 0 = ∑ ∆ f H ° 0 , fragments − ∆ f H ° 0 , molecule

The dissociation energy D e can then be obtained from D 0 by accounting for zero-point energies (ZPEs):

(2) D e = D 0 + ZPE molecule − ∑ ZPE fragments

Results and discussion

Table 1 presents the vertical excitation energies of the methyl radical (CH3) computed using XMS-CASPT2, CMS-PDFT with tPBE functional, and L-PDFT with the MC23, tPBE0, and tPBE functionals. The CASSCF calculations used to obtain orbitals and CASSCF eigenvectors for the CASPT2, CMS-PDFT and L-PDFT calculations have seven active electrons in 10 active orbitals.

Table 1:

Vertical excitation energies of CH3 in eVa.

Transition TBE (ref. 76) EOM-CCSD (ref. 77) XMS-CASPT2 CMS-PDFT: tPBE L-PDFT: MC23 L-PDFT: tPBE0 L-PDFT: tPBE
1 2 A 1 ′ –X 2 A 2 ″ 5.85 5.82 5.93 6.00 5.93 5.92 5.98
1 2E′–X 2 A 2 ″ 6.96 7.02 7.03 6.72 6.66 6.60 6.32
1 2E′–X 2 A 2 ″ 6.96 7.02 7.03 6.81 6.69 6.63 6.37
2 2E′–X 2 A 2 ″ 7.18 7.14 7.39 7.06 7.23 7.12 7.20
2 2E′–X 2 A 2 ″ 7.18 7.14 7.46 7.44 7.37 7.31 7.31
2 2 A 2 ″ –X 2 A 2 ″ 7.65 7.31 7.82 7.81 7.70 7.62 7.68
MUEb – 0.12 0.14 0.17 0.13 0.14 0.21

  1. aThe 2E′ states are doubly degenerate due to the high symmetry (threefold axis) of the methyl radical. However, in the calculations reported in this table, to be consistent with our photodissociation scans, we do not impose molecular symmetry; that is our standard procedure in photodynamics applications because we want the solutions to pass continuously through high-symmetry points (if one uses a symmetry-constrained calculation at high-symmetry points and no symmetry at unsymmetrical points (such as those encountered along the dissociation path), such continuity is not guaranteed). This lack of symmetry constraint leads to a splitting of degenerate 2E′ states into non-degenerate components, and this splitting is seen in the XMS-CASPT2, CMS-PDFT, and L-PDFT results. The observed splitting is therefore a consequence of symmetry breaking in the underlying wave functions and, thus, not a failure of the multistate methods. bMUEs are averaged over levels rather than states. In other words, the MUEs are defined as the average absolute deviations of the computed excitation energies of the various levels from the corresponding theoretical best estimates (TBEs), with weights assigned to account for degeneracy. Since the methyl radical has two doubly degenerate 2E′ excited states, including each component equally would overrepresent the excitation energies of these levels in the error metric. To avoid this, each component of the 2E′ states is assigned half-weight, such that the total contribution from each degenerate 2E′ level matches that of the non-degenerate 1 2 A 1 ′ and 2 2 A 2 ″ states. This gives an evaluation of average accuracy of the four excitation energies.

The table also contains the theoretical best estimate (TBE) reported by Loos et al., 76 as calculated using the aug-cc-pVTZ basis set, which serve as high-level multireference benchmarks for the vertical excitation energies. The EOM-CCSD values, obtained with the 6–311++G(d,p) basis set, are taken from the work of Mebel et al., 77 one of the earliest and most widely referenced benchmarking studies on the vertical excitation energies of the methyl radical.

XMS-CASPT2 shows good agreement with TBE values, with an MUE of only 0.14 eV, capturing both the qualitative ordering and quantitative energetics of the low-lying excited states of CH3 (The largest contributions to the MUE arise from the higher-energy transitions in Table 1).

The CMS-PDFT method with the tPBE functional also performs reasonably well, with an MUE of 0.17 eV, although it tends to slightly underestimate the energies of the 1 2E′ transitions compared to XMS-CASPT2 and TBE.

L-PDFT with L-MC23 and L-PDFT with L-tPBE0 deliver the closest agreement of any of the present calculations with TBE, yielding MUEs of 0.13 and 0.14 eV, respectively. In contrast, L-PDFT with L-tPBE slightly overestimates the 1 2 A 1 ′ transition and significantly underestimates the 1 2E′ transitions, resulting in a higher MUE of 0.21 eV.

In previous work on computing excitation energies by density functional theory, 78 , 79 , 80 , 81 we have labeled calculations with an MUE below 0.30 eV as “successful”. By that measure, all the methods considered here are successful for methyl radical vertical excitation energies with the chosen active space. It is especially encouraging that L-PDFT with MC23 and tPBE0 perform as well as or better than XMS-CASPT2.

If one considers the photodissociation of CH3, the potential surfaces will only be correct if the excitation energies of CH2 are also accurate. Table 2 compares the adiabatic excitation energies of CH2 computed using XMS-CASPT2, CMS-PDFT, and L-PDFT against completely renormalized equation-of-motion coupled cluster theory with single and double excitations (CR-EOM-CCSD/aug-cc-pVQZ) reference values from the work of Chien et al. 82 The table also includes the full configuration interaction (FCI/TZ2P) results of Sherrill et al., 83 which is the most widely referenced benchmarking study on the adiabatic excitation energies of CH2. XMS-CASPT2 reproduces all three low-lying singlet excitations to within 0.05 eV of CR-EOM-CCSD (with an MUE of 0.04 eV), which is excellent agreement.

Table 2:

Adiabatic excitation energies of CH2 in eV.

Transition CR-EOM-CCSD (ref. 82]) FCI (ref. 83]) XMS-CASPT2 CMS-PDFT: tPBE L-PDFT: MC23 L-PDFT: tPBE0 L-PDFT: tPBE
11A1–13B1 0.41 0.48 0.44 0.56 0.46 0.48 0.53
11B1–13B1 1.46 1.54 1.51 1.52 1.43 1.11 0.90
21A1–13B1 2.55 2.67 2.51 2.23 2.74 2.58 2.58
MUE – 0.09 0.04 0.18 0.09 0.15 0.24

CMS-PDFT:tPBE follows the CR-EOM-CCSD trends closely for the first two states, but it underestimates the highest state by more than 0.3 eV, yielding an overall MUE of 0.18 eV. Among the L-PDFT results, L-PDFT with MC23 matches the reference most closely, having an MUE of 0.09 eV; L-PDFT:tPBE0 is second best (MUE = 0.15 eV), and L-PDFT:tPBE shows the largest deviation (MUE = 0.24 eV), primarily due to a significant underestimation of the 1 1B1 excitation.

Table 3 gives the enthalpies of formation at 0 K( ∆ f H ° 0 ) and ZPEs for all the species in the reaction CH3 → CH2 + H. Since both singlet and triplet CH2 can be the products of the dissociation reaction, we include both species in the analysis. The Δ f H° for all the species at 0 K are obtained from the active thermochemical tables. 84 , 85 , 86 The ZPEs for all the species except the hydrogen atom was calculated using the Gaussian 16 program; 87 in particular, we calculate harmonic frequencies with M06-L/MG3S and corrected them with a scaling factor 88 of 0.978 to get the ZPEs in Table 3.

Table 3:

Δ f H° and ZPE values of all the species in the reaction CH3 → CH2 + H.

Species ∆ f H ° 0 (in eV)a ZPE (in eV)b
CH3 1.55 0.77
CH2 (triplet) 4.05 0.46
CH2 (singlet) 4.44 0.44
H 2.24 –

  1. a ∆ f H ° 0 (obtained from the active thermochemical tables). bZPEs calculated with M06-L/MG3S and corrected with a scaling factor of 0.978.

Using the values in Table 3 and eq. (1), D 0 for the reaction where CH3 dissociates to triplet CH2 and a hydrogen atom is 4.74 eV and is 5.13 eV for the dissociation to singlet CH2 and a hydrogen atom. Substituting these D 0 values and ZPE values from Table 3 in eq. (2) then gives D e values as 5.05 eV for the dissociation to triplet CH2 and a hydrogen atom and 5.46 eV for the dissociation to the singlet CH2 and a hydrogen atom. Table 4 shows the comparison of the calculated dissociation energies to these accurate values. The trend is different than the trend for excitation energies; here CMS-PDFT:tPBE performs the best out of all methods. The table shows that L-PDFT:tPBE is a close second. Both XMS-CASPT2 and L-PDFT with MC23 underestimate dissociation energies by more than 0.3 eV.

Table 4:

Dissociation energies, D e (in eV) for the reaction CH3 → CH2 + H.

Products accuratea XMS-CASPT2 CMS-PDFT: tPBE L-PDFT: MC23 L-PDFT: tPBE0 L-PDFT: tPBE
CH2(triplet) + H 5.05 4.74 5.12 4.55 4.76 5.18
CH2(singlet) + H 5.46 5.17 5.67 5.01 5.24 5.71
MUE – 0.30 0.14 0.48 0.26 0.19

  1. aAccurate D e calculated with eqs (1) and (2) with ZPEs from Table 3.

Figure 1 presents the potential energy curves for seven electronic states during the C–H bond dissociation of the methyl radical. All methods yield smooth and qualitatively reasonable dissociation profiles, indicating that the chosen active space is sufficient to describe the electronic structure along the reaction coordinate, even though a bond is broken. One notable feature in the curves is that the seventh state shows significant variation across different methods. This behavior is not particularly surprising, as high-lying excited states often exhibit irregularities due to strong mixing and interactions with other high-lying electronic states that are not calculated, such as the eighth state. Such features highlight the increased multiconfigurational character and complexity of accurately describing high-energy regions of the potential energy landscape.

Fig. 1: Seven-state potential energy surfaces for the C–H bond dissociation in methyl radical calculated with (a) XMS-CASPT2, (b) CMS-PDFT with tPBE, (c) L-PDFT with MC23, (d) L-PDFT with tPBE0, and (e) L-PDFT with tPBE.
Fig. 1:

Seven-state potential energy surfaces for the C–H bond dissociation in methyl radical calculated with (a) XMS-CASPT2, (b) CMS-PDFT with tPBE, (c) L-PDFT with MC23, (d) L-PDFT with tPBE0, and (e) L-PDFT with tPBE.

Concluding remarks

The CMS-PDFT and L-PDFT methods both offer an alternative to multi-state multireference perturbation methods such as XMS-CASPT2 and QD-NEVPT2. While the present study concerns a small system (CH3), where timing comparisons would not meaningfully reflect the scaling behavior of these methods on more complex systems, prior work has demonstrated the computational efficiency of MC-PDFT over CASPT2 for larger molecules. 89 , 90 , 91 , 92 The MC-PDFT method was originally proposed as a single-state method for ground and excited states (just as CASPT2 and NEVPT2 are single-state methods for ground and excited states), but one should use a multistate method for consistent treatment of a closely coupled set of states. The XMS-CASPT2 and QD-NEVPT2 methods are multistate versions of CASPT2 and NEVPT2, and CMS-PDFT and L-PDFT are multistate versions of MC-PDFT. Analytic gradients are available for both CMS-PDFT and L-PDFT, and this allows for geometry optimization of excited states with either method. As compared to CMS-PDFT, the L-PDFT method has the advantages of (i) using a Hamiltonian-like operator that is particularly convenient for taking advantage of symmetries, (ii) having a cost that scales as a constant with respect to the number of states included in the model space, 52 , 53 , 54 (unlike CMS-PDFT, whose cost scales linearly with the number of states in the model space), and (iii) not requiring the iterative determination of an intermediate basis that can lead to discontinuous gradients. The substantial cost advantage of L-PDFT over XMS-CASPT2 and QD-NEVPT2 makes it a good choice for performing nonadiabatic direct dynamics simulations of large, complex systems.

All active space methods (e. g., L-PDFT, XMS-CASPT2, and QD-NEVPT2) suffer from the possibility of discontinuities that may occur if one cannot retain a continuous active space over the wide-amplitude region spanned by the dynamics. This, however, does not prevent their practical use when used with care, and L-PDFT has already been successfully applied in nonadiabatic dynamics simulations. In particular, Hennefarth et al. 54 reported that L-PDFT added only a modest cost over SA-CASSCF in nonadiabatic molecular dynamics simulations of azomethane.

The present study presents a detailed multistate electronic structure analysis of the methyl radical (CH3) and methylene (CH2) species using the XMS-CASPT2, CMS-PDFT and L-PDFT methods. Vertical excitation energies, adiabatic excitation energies, and dissociation energies, along with cuts through the potential surface along the dissociation coordinate, were computed with the (7, 10) active space. XMS-CASPT2 and L-PDFT with the MC23 functional usually demonstrate consistent and reliable performance, closely reproducing benchmark values and producing smooth, physically reasonable potential energy surfaces essential for nonadiabatic dynamics simulations; however, they both underestimate the dissociation energies. The L-PDFT calculations with the tPBE functional are more accurate for dissociation energies, but less accurate for excitation energies. These findings reinforce the importance of method choice and active space selection in multireference excited-state studies and highlight difficulty of finding globally accurate methods suitable for simulations.

Corresponding author: Donald G. Truhlar, Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455-0431, USA, e-mail:
Article note: A collection of invited papers to celebrate the UN’s proclamation of 2025 as the International Year of Quantum Science and Technology.

Funding source: Air Force Office of Scientific Research

Award Identifier / Grant number: FA9550-20-1-0360

Acknowledgments

The authors are grateful to Yinan Shu, Matthew Hennefarth, Matthew Hermes, and Laura Gagliardi for valuable discussions.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors declare no conflicts.

  6. Research funding: This work was supported in part by the Air Force Office of Scientific Research by grant FA9550-20-1-0360.

  7. Data availability: All data generated or analyzed during this study are included in this published article and its Supplementary information file.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/pac-2025-0546).

Received: 2025-06-15
Accepted: 2025-08-14
Published Online: 2025-08-27
Published in Print: 2025-10-27

© 2025 IUPAC & De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Review Articles
  3. Minimum energy path methods and reactivity for enzyme reaction mechanisms: a perspective
  4. The quantum revolution in enzymatic chemistry: combining quantum and classical mechanics to understand biochemical processes
  5. A quantum chemical perspective of photoactivated biological functions
  6. Does chemistry need more physics?
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  13. Electronic structure of methyl radical photodissociation
  14. Bridging experiment and theory: a computational exploration of UMG-SP3 dynamics
  15. Research Articles
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https://doi.org/10.1515/pac-2025-0546
Keywords for this article
chemistry; computer simulation; quantum chemistry; quantum mechanics; quantum science and technology; radical reactions; spectroscopy

Articles in the same Issue

  1. Frontmatter
  2. Review Articles
  3. Minimum energy path methods and reactivity for enzyme reaction mechanisms: a perspective
  4. The quantum revolution in enzymatic chemistry: combining quantum and classical mechanics to understand biochemical processes
  5. A quantum chemical perspective of photoactivated biological functions
  6. Does chemistry need more physics?
  7. Rotational dynamics of ATP synthase: mechanical constraints and energy dissipative channels
  8. Transforming dreams into reality: a fairy-tale wedding of chemistry with quantum mechanics
  9. The quantum chemistry revolution and the instrumental revolution as evidenced by the Nobel Prizes in chemistry
  10. Influence of symmetry on the second-order NLO properties: insights from the few state approximations
  11. The dichotomy between chemical concepts and numbers after almost 100 years of quantum chemistry: conceptual density functional theory as a case study
  12. How ‘de facto variational’ are fully iterative, approximate iterative, and quasiperturbative coupled cluster methods near equilibrium geometries?
  13. Electronic structure of methyl radical photodissociation
  14. Bridging experiment and theory: a computational exploration of UMG-SP3 dynamics
  15. Research Articles
  16. O–Li⋯O and C–Li⋯C lithium bonds in small closed shell and open shell systems as analogues of hydrogen bonds
  17. Metal–ligand bonding and noncovalent interactions of mutated myoglobin proteins: a quantum mechanical study
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