The terminal alkyne C≡C stretch has a large Raman scattering cross section in the “silent” region for biomolecules. This has led to many Raman tag and probe studies using this moiety to study biomolecular systems. A computational investigation of these systems is vital to aid in the interpretation of these results. In this work, we develop a method for computing terminal alkyne vibrational frequencies and isotropic transition polarizabilities that can easily and accurately be applied to any terminal alkyne molecule. We apply the discrete variable representation method to a localized version of the C≡C stretch normal mode. The errors of (1) vibrational localization to the terminal alkyne moiety, (2) anharmonic normal mode isolation, and (3) discretization of the Born–Oppenheimer potential energy surface are quantified and found to be generally small and cancel each other. This results in a method with low error compared to other anharmonic vibrational methods like second-order vibrational perturbation theory and to experiments. Several density functionals are tested using the method, and TPSS-D3, an inexpensive nonempirical density functional with dispersion corrections, is found to perform surprisingly well. Diffuse basis functions are found to be important for the accuracy of computed frequencies. Finally, the computation of vibrational properties like isotropic transition polarizabilities and the universality of the localized normal mode for terminal alkynes are demonstrated.

## I. INTRODUCTION

Infrared and Raman spectroscopy are invaluable tools for investigating the structure and dynamics of a diverse array of chemically complex systems such as biomolecules, liquids, solutions, molecular clusters, and the atmosphere.^{1–10} When possible, effective vibrational probes can be used to investigate specific aspects of these systems.^{1} For instance, OH, OD, and nitrile moieties are effective probes of the availability of hydrogen bonding partners, and carbonyls and nitriles (in the absence of hydrogen bonding partners) have been used as probes of the electric fields of an environment.^{11–28} An optimal vibrational probe is non-perturbative to the system, and an investigation of the probe directly investigates a system property of interest. Recent work on the solvation of CO_{2} in ionic liquids exemplifies this; here, CO_{2} was the probe of its own solvation dynamics.^{29–31}

The triple bond C≡C stretch of alkynes has long been used as a Raman tag in biomolecular and surface enhanced Raman spectroscopy studies.^{32–38} This vibration appears in the biomolecular “Raman window,” a portion of the spectrum where biomolecular systems rarely produce Raman scattering.^{9,39–41} In the prior tagging studies, the alkyne vibration was used mainly to highlight the presence of a material. However, recent work has turned to exploring the terminal alkyne C≡C stretch as a possible solvatochromic vibrational probe.^{42–48} In a solvatochromic probe study, details of the lineshape, frequency, and intensity of the absorption are used to infer the alkyne probe’s molecular environment.

We follow recent work at Haverford College and elsewhere showing that the terminal alkyne C≡C stretch vibrational mode is (1) an effective probe of biomolecular structure and dynamics and (2) independently sensitive to both its solvent and substituent.^{42,44,45} A third question remains unanswered: *what, specifically, is the alkyne probe reporting on?* There are several possibilities, given the available experimental data. The work of Epstein *et al.* and Dong *et al.* implies that the terminal alkyne detects the hydrophilicity of or hydrogen bonding partners available in an environment.^{45,48} However, Romei *et al.* complicate this picture; they find that the terminal alkyne vibration is sensitive to a number of solvents with different properties.^{44} They correlate this sensitivity with empirical scales of the propensity of the solvent to donate electrons or polarize. Even so, it is possible that the factors that cause a probe’s frequency to shift *between solvents* are distinct from those that cause the frequency to shift *between configurations* in a single solvent, complicating the question of reporting. There are also possible effects arising from covalently bonded substituents; some experiments have shown that electronic conjugation with the alkyne triple bond or the addition of sulfur or silicon atoms can shift the scattering frequency, change the scattering intensity, or modify the population lifetime in both IR and Raman experiments.^{37,42,44,48–53}

In this paper, we take our first steps toward a full quantum chemistry and molecular dynamics investigation to understand the terminal alkyne vibration as a probe of its molecular environment. Here, we develop a new method for calculating the frequency of a terminal alkyne that is compatible with snapshots extracted from molecular dynamics simulations. Our goals are to

demonstrate that an isolated and localized discrete variable representation (DVR) method is sufficient to compute the terminal alkyne C≡C stretch normal mode vibrational frequency with good accuracy,

select one or more density functional theory (DFT) methods that show good accuracy and speed for use in the presence of solvents in future work, and

propose a universal set of terminal alkyne C≡C stretch localized normal mode atomic displacements and show that the computed vibrational properties have the expected features.

Our final method *localizes, isolates,* and *discretizes* the terminal alkyne C≡C-stretch vibrational normal mode. We quantify the errors associated with each of these assumptions; discretization is fairly benign, but localization and isolation errors are larger. Fortunately, these errors are themselves moderate, and each shifts the alkyne frequency in the opposite direction, nearly canceling out. We also attempt to lay out a clear theoretical description of our employed discrete variable method in Sec. II.

Here, alkynes are dealt with in the gas phase, providing a solid foundation for future work in the condensed phase, where there are substantial non-covalent interactions with a solvent. This also allows us to examine the effect of substituents on the alkyne frequency and other vibrational properties. We examine eight terminal alkyne molecules with diverse structures. Six are particularly small, having four or fewer atoms heavier than hydrogen. These molecules can be examined using highly accurate electronic and vibrational structure methods. These molecules will be used in the first section of our results to quantify errors associated with localization, isolation, and discretization. The remaining molecules are used to quantify substituent effects and explore the accuracy of a set of models of the electronic structure based on DFT.

## II. THEORY

*Q*, and $p\u0302Q$ is the momentum operator conjugate to that coordinate.

^{54}The normal mode coordinate can be written as a sum over atomic displacements away from the optimized geometry

*N*is the number of atoms (so 3

*N*is the number of Cartesian coordinates),

*c*

_{j}is the

*j*th atomic Cartesian coordinate, and

*c*

_{j,0}is the

*j*th optimized atomic Cartesian coordinate.

^{54}If all

*c*

_{j}=

*c*

_{j,0}, the molecular geometry is optimized, the energy is at a minimum, and

*Q*= 0. A vibration along a normal mode coordinate changes the molecular geometry according to the derivatives $\u2202Q\u2202cj$. The inverses of these derivatives, $\u2202cj\u2202Q$, are printed out by most quantum chemistry programs following a harmonic frequency analysis and are often called “normal mode atomic displacements.”

^{54}The reduced mass for a normal mode is given by

*m*

_{j}is the mass of the atom with the Cartesian coordinate

*c*

_{j}, and the other symbols retain their meaning.

^{55}Several well-established anharmonic methods for solving the vibrational problem require the molecular geometry of interest to be optimized with the same method and basis set as is used in the frequency calculation.

^{56–59}However, we wish to eventually compute the vibrational frequencies of one specific normal mode for snapshots from molecular dynamics simulations at temperatures above absolute zero. The molecules sampled in this way will not necessarily even have geometries that are optimized according to the dynamical force field! Therefore, we require a method for calculating the vibrational Hamiltonian that does not assume or require that the first derivative of the potential energy is zero.

The discrete variable representation (DVR) method can be used to compute vibrational states without optimization.^{60} In this method, we *discretize* the normal mode—we select *P* normal mode coordinate points *Q*_{1}, *Q*_{2}, *Q*_{3}, …, *Q*_{P} equally spaced by Δ*Q* to investigate. These normal mode coordinate values each represent a particular molecular geometry (Fig. 1), and from them, we form a one-dimensional grid across the normal mode coordinate.

^{60}This basis has the important property that the potential energy operator is diagonal

*Q*

_{i}is the value of the normal mode coordinate at grid point

*i*and

*δ*

_{ij}is the Kronecker delta function. This is also true of all other operators that are functions of position. There are several ways to define such a basis, but this property is central to the DVR method.

^{60–63}Given such a basis, it can be shown that (in atomic units)

^{60,61,63}

*P*→ ∞) with infinite extent (i.e.,

*Q*

_{1}→ −∞ and

*Q*

_{P}→ ∞), resulting in a finite spacing, Δ

*Q*, between grid points.

^{60,62,63}These kinetic energy matrix elements are equivalent (within multiplicative constants) to an infinite-point finite difference approximation to the second derivative, so a smaller value of Δ

*Q*increases the accuracy, as is true for all finite difference approximations.

^{60,61}With this and the diagonal potential energy matrix from Eq. (4), we can construct the vibrational Hamiltonian matrix as

*E*

_{a}, and eigenstates, $\Psi a$. The vibrational eigenstates can be represented as weighted sums of the original grid point basis functions

*e*

_{ia}is the weight of the contribution of $Qi$ to eigenstate $\Psi a$. The absolute square of each

*e*

_{ia}is the probability of the molecule having normal mode coordinate value

*Q*

_{i}if it is in eigenstate $\Psi a$, i.e., $paQi=eia2=eia*eia$.

*P*grid points.

^{31,64,65}The isotropic transition polarizability

*α*

_{ab}is related to the probability of isotropic Raman scattering and can be computed as

*P*grid points.

^{64,66}Diagonalization of the vibrational Hamiltonian also produces a series of energy levels. The frequency of light resonant in a transition from vibrational state

*a*to state

*b*is

*E*

_{a}is the energy of state $\Psi a$ and ℏ is the reduced Plank’s constant.

## III. COMPUTATIONAL METHODS

### A. Electronic structure methods and basis sets

The quantum chemistry methods employed in this study include Hartree–Fock (HF),^{67} the post-Hartree–Fock wavefunction-based methods of second order Møller–Plesset perturbation theory (MP2)^{68} and coupled cluster with single excitations, double excitations, and perturbative triple excitations [CCSD(T)],^{69} and density functional theory (DFT) methods spanning several rungs of “Jacob’s ladder”: B3LYP (a hybrid density functional formed by combining Becke’s three-parameter exchange functional and Lee, Yang, and Parr’s correlation functional),^{70–72} ωB97M-V (an empirically optimized range separated hybrid functional from Head-Gordon and co-workers, including the dispersion correction of Vydrov and Van Voorhis),^{73,74} TPSS (a nonempirical density functional developed by Tao *et al.*),^{75} and PBEh-3c (a composite density functional based on the established Perdew–Burke–Ernzerhoff functional combined with the threefold corrected Hartree–Fock method).^{76} We modified B3LYP and TPSS with the third edition of the dispersion corrections of Gimme *et al.*, resulting in B3LYP-D3 and TPSS-D3.^{77} The comparison of HF to other methods allows us to understand the effect of exact exchange and the other method’s approximate electron correlation on the frequency calculation. We use CCSD(T) as a “gold standard” electronic structure method and MP2 as a sort of “silver standard”—not as accurate as CCSD(T) yet still reasonably applicable to some of the larger molecules examined in this study, and more accurate than DFT. We select TPSS-D3, B3LYP-D3, and ωB97M-V as representative, widely used, and economical meta generalized gradient approximation (GGA), global hybrid GGA, and range-separated hybrid meta-GGA functionals, respectively, with recommended empirical dispersion corrections. The composite method PBEh-3c (with the recommended def2-mSVP basis set) is included because of its impressive cost-to-accuracy ratio. These choices were influenced by recommendations from the quantum chemistry community, as were our basis-set choices.^{78–80} This search will allow us to find an economical method to use for our future condensed phase frequency calculations while uncovering some of the quantum effects on the frequency.

We also tested the effect of the basis set on the frequency by examining three families of basis sets: Pople, Dunning, and Ahlrichs.^{81–83} For each family, basis sets of the same zeta quality and with the same presence of diffuse functions were tested. The four combinations were: (1) double zeta with no diffuse functions [6-31G**, cc-pVDZ, and def2-SVP], (2) double zeta with diffuse functions on all atoms [6-31++G**, aug-cc-pVDZ, and def2-SVPD], (3) triple zeta with no diffuse functions [6-311G**, cc-pVTZ, and def2-TZVP], and (4) triple zeta with diffuse functions on all atoms [6-311++G**, aug-cc-pVTZ, and def2-TZVPD]. Because Raman intensities are strongly dependent on polarizability, we always included polarization functions on all atoms in our basis sets. For CCSD(T) and MP2, each of the largest triple zeta basis sets [6-311++G**, aug-cc-pVTZ, and def2-TZVPD] was used in order to obtain values with high accuracy for comparison to other methods.

### B. Molecular analysis

All geometry optimizations, harmonic frequency calculations, single point energy calculations, and polarizability calculations were performed using the Q-Chem 5.4 software package.^{78} Single point energy calculations were typically performed using a self-consistent field (SCF) convergence criterion of 10^{−9} a.u. and a threshold for the neglect of two electron integrals of 10^{−14} a.u. The relaxed constraint algorithm (RCA) was used during early SCF iterations, and the Pulay Direct Inversion of the Iterative Subspace (DIIS) algorithm was used during later iterations.^{84,85} For DFT calculations, a quadrature grid with 99 radial points and 590 angular points was used for late SCF cycles, and a lower resolution Standard Grid 0 (SG-0) grid was used for early SCF cycles.^{86} We attempted to optimize all molecular geometries to a maximum gradient of 3 × 10^{−6} a.u.; if this was not possible, the restriction was loosened to 30 × 10^{−6} a.u. Optimization was confirmed by the absence of imaginary harmonic frequencies. Dipole moments were computed analytically based on the ground state electron density, and static polarizability tensors were computed using the finite field approach. Both computations were completed using the appropriate Q-Chem implementation.^{78}

We investigated four classes of alkyne containing molecules (Fig. 2). These choices were inspired by the work of Romei *et al.* and allow us to investigate a diverse array of alkyne containing molecular structures.^{44} We included simple alkynes with carbon-based R groups, simple alkynes with non-carbon substituents, complex alkynes, and synthetic amino acids. The synthetic amino acids were examined in their fully neutral forms since those forms are the most stable in the gas phase.^{87,88} We performed harmonic analyses on the zwitterionic forms and found no evidence that our results would be different if they had been examined instead.

### C. Localized vibrational frequency calculations

While the molecules we are investigating vary by R group, the atoms of the terminal alkyne moiety (–C≡C–H, hereafter CCH) are common between them. These atoms have much larger displacements in the C≡C normal mode vibration than the other atoms in each molecule (see Fig. 1 and Sec. IV A 2 c on localization error). Physically, this means the non-CCH atoms barely move during the vibration. One common approximation that can simplify a normal mode substantially is *localization*.^{89–94} In our localization approximation, we remove the motion of all non-CCH atoms in the molecule during the vibration so that only the CCH atoms move.

We do this in two separate but similar ways. In one method, we use the partial Hessian approximation.^{95,96} Here, the masses of non-CCH atoms are assumed to be so large that their mass-weighted Hessian entries can be neglected. This results in different values of the normal mode displacements than when the full mass-weighted Hessian is diagonalized. These displacements can be used to compute the reduced masses, scan the potential energy surface, and perform any other tasks that their full Hessian counterparts are used for in our DVR method. The second way we localize the normal mode is by simply setting the non-CCH atom displacements to zero and using the full Hessian CCH displacements, as in the partial Hessian case. Both of these methods result in similar DVR localized normal mode frequencies, reduced masses, and CCH displacements (Table S1). We preferred the partial Hessian-based method of localization if it was available in Q-Chem. Of the methods we explored, it is only unavailable for CCSD(T).

## IV. RESULTS AND DISCUSSION

### A. Performance of the isolated normal mode DVR method

#### 1. Overall performance

We performed anharmonic frequency calculations using the established second-order vibrational perturbation theory (VPT2) method implemented in the Q-Chem package with the “gold standard” CCSD(T) electronic structure method and triple zeta basis sets with diffuse functions to obtain benchmark anharmonic C≡C stretch frequencies.^{58} We were not able to complete a VPT2 calculation for BTY/CCSD(T)/aug-cc-pVTZ because of the large memory and time requirements of the calculation. We also computed harmonic frequencies with CCSD(T) and the same basis sets. Experimental values for all simple alkynes are taken from NIST sources.^{97–103} Nyquist reports 2185 cm^{−1} for the C≡C stretch frequency for POL.^{104} However, this disagrees with several other sources for the vapor and condensed phase frequencies of this molecule, including the pictured spectrum in the Nyquist reference itself.^{44,97,104–107} Because of the disagreement with all other sources and our otherwise highly accurate CCSD(T)/triple zeta/VPT2 calculations, we believe this is a misreport and that the sources stating the frequency is 2124 cm^{−1} are correct. We show in Table I that the calculated frequencies using VPT2 generally err by 10 cm^{−1} compared to the experimental frequencies for our simple alkynes. We observe a small basis set dependence on the calculated frequencies and find that the Dunning and Alrichs family basis sets are most consistent with the reported experimental gas-phase frequency.

. | . | aug-cc-pVTZ . | def2-TZVPD . | 6-311++G(d,p) . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Expt. . | HARM . | VPT2 . | DVR . | HARM . | VPT2 . | DVR . | HARM . | VPT2 . | DVR . | |

PPY | 2124 | +46 | +2 | +4 | +52 | +13 | +6 | +37 | +2 | −6 |

PAL | 2125 | +16 | −19 | −17 | +21 | −8 | −16 | +6 | −31 | −28 |

BTY | 2116 | +46 | ⋯ | +8 | +52 | +16 | +8 | +37 | −6 | −2 |

POL | 2124 | +37 | −3 | +1 | +43 | −1 | +1 | +29 | −11 | −8 |

EAM | 2155 | +42 | −2 | −29 | +48 | −4 | −17 | +35 | −1 | −37 |

EOL | 2198 | +33 | −11 | −39 | +37 | −19 | −28 | +30 | −20 | −43 |

RMSD_{R=C} | ⋯ | 38 | 11 | 10 | 44 | 11 | 9 | 30 | 17 | 15 |

RMSD_{all} | ⋯ | 38 | 10 | 21 | 44 | 12 | 15 | 31 | 16 | 26 |

. | . | aug-cc-pVTZ . | def2-TZVPD . | 6-311++G(d,p) . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Expt. . | HARM . | VPT2 . | DVR . | HARM . | VPT2 . | DVR . | HARM . | VPT2 . | DVR . | |

PPY | 2124 | +46 | +2 | +4 | +52 | +13 | +6 | +37 | +2 | −6 |

PAL | 2125 | +16 | −19 | −17 | +21 | −8 | −16 | +6 | −31 | −28 |

BTY | 2116 | +46 | ⋯ | +8 | +52 | +16 | +8 | +37 | −6 | −2 |

POL | 2124 | +37 | −3 | +1 | +43 | −1 | +1 | +29 | −11 | −8 |

EAM | 2155 | +42 | −2 | −29 | +48 | −4 | −17 | +35 | −1 | −37 |

EOL | 2198 | +33 | −11 | −39 | +37 | −19 | −28 | +30 | −20 | −43 |

RMSD_{R=C} | ⋯ | 38 | 11 | 10 | 44 | 11 | 9 | 30 | 17 | 15 |

RMSD_{all} | ⋯ | 38 | 10 | 21 | 44 | 12 | 15 | 31 | 16 | 26 |

VPT2 is known to fail catastrophically in cases where two or more vibrational transitions have nearly the same energy, so it is important to check if such transitions exist before using the method.^{56,58} We have checked this using the transition-optimized shifted Hermite (TOSH) method. TOSH is an approximation to VPT2 and does not share the concern of catastrophic failure.^{56} Because VPT2 and TOSH calculations are automatically performed simultaneously in Q-Chem, we have values from both methods for every calculation we performed. The frequencies we obtain from the TOSH method are always similar to those we obtain from VPT2, with a root-mean-square-deviation (RMSD) of 3.6 cm^{−1} (see Fig. S1). This is strong evidence that the VPT2 method is not suffering from catastrophic failure for the C≡C stretch. In the absence of these failures, the VPT2 calculations have the best cost-to-accuracy ratio of the anharmonic vibrational frequency methods packaged with Q-Chem.^{78} As such, we use VPT2 anharmonic frequency calculations as a benchmark against which to test our localized DVR anharmonic frequency calculations.

We tested the performance of our localized normal mode DVR method against VPT2. CCSD(T) was used for both calculations; using such an accurate electronic structure method allows us to isolate inaccuracies due to the vibrational method alone. The DVR method nearly matches the performance of the VPT2 method for simple alkynes, where the atom directly bonded to the alkyne group is a carbon. However, the DVR frequencies for EOL and EAM differ more substantially from the experiment and VPT2. The calculated DVR frequencies are compared to the experiment, VPT2, and harmonic frequencies in Table I.

#### 2. Sources of error

Our localized normal mode DVR method makes three major assumptions about the terminal alkyne C≡C vibration: (1) that the error of discretizing and limiting the potential energy surface is negligible, (2) that the C≡C stretch normal mode is not anharmonically coupled to other normal modes in the molecule, and (3) that the atoms in the R group do not move during the vibration. All of these are formally incorrect. In the following, we investigate each of these assumptions to determine the amount of error they introduce to our method.

##### a. Is the error from discretizing and limiting the potential energy surface negligible?

The true potential energy surface for the normal mode is continuous (*P* → ∞ and Δ*Q* → 0) and has an infinite domain (*Q*_{1} → −∞ and *Q*_{P} → ∞). However, our DVR method discretizes and limits this surface, and these approximations introduce some error. To determine this error, we produced a DVR potential energy surface with a very fine grid (*P* = 100 and Δ*Q* = 0.02 Å) and a very large domain (*Q*_{1} = −1.0 Å and *Q*_{100} = 1.0 Å) at the MP2/aug-cc-pVTZ level of theory for propyne (Fig. 3). The *ω*_{01} DVR frequency obtained using this PES was 2114.53 cm^{−1}. To ensure that this value was converged with respect to the number of grid points and the domain, PES grids with reduced resolution and domain were produced and used to compute the *ω*_{01} DVR frequency. No change greater than 0.2 cm^{−1} was observed until the grid spacing, Δ*Q*, was increased to more than 0.04 Å. In addition, the 100 grid point PES DVR results reproduce the behavior expected of a slightly anharmonic oscillator (Fig. 3); the energy level spacing decreases slowly as the energy eigenvalue increases, and the high energy wavefunctions are highly oscillatory. Therefore, we believe it is reasonable to treat this 100 grid point PES as nearly continuous and nearly unlimited.

We compared this nearly continuous and nearly unlimited surface to our preferred surface, containing *P* = 20 grid points between *Q*_{1} = −0.3 Å and *Q*_{20} = 0.5 Å (Δ*Q* = 0.04 Å). The *ω*_{01} frequency obtained using the near-continuous PES (2114.53 cm^{−1}) is consistent with the calculated *ω*_{01} frequency using the 20 grid point PES (2114.72 cm^{−1}). This happens because the low energy eigenvalues for both PESs are very similar (Fig. 3). The high energy eigenstates obtained from the 20 grid point PES are not similar to those from the near continuous surface, indicating an increasing error with increasing energy. The 20 grid point PES would be inadequate to compute, say, the *ω*_{09} frequency. However, we are most interested in the *ω*_{01} frequency, and the error from discretization for this value is 0.19 cm^{−1}. We find this error acceptable in exchange for computing one-fifth as many PES grid points.

We also interpolated the 20-point wavefunctions using the sinc basis functions of Colbert and Miller and compared them to the 100-point wavefunctions directly.^{60} The RMSD between the squared interpolated 20-point ground state wavefunction and the squared discrete 20-point ground state wavefunction was 0.1. The RMSD between the squared interpolated 20-point ground state wavefunction and the squared discrete 100-point ground state wavefunction was 0.3. Similar values were obtained for the first and second excited states, bolstering our confidence in the 20-point surface.

Other groups have shown that a reduced number of electronic structure calculations can be performed and fit to obtain high quality frequencies with much less computational cost.^{13,20,108} To explore this, we tried fitting the 20-point surface to an eighth order polynomial. The fit was extremely good, with a correlation coefficient of 1 and a root-mean-square residual of 1 × 10^{−6} a.u. We used this polynomial to quickly sample 1000 points between *Q*_{1} = −0.3 Å and *Q*_{20} = 0.5 Å. The DVR frequency obtained from this surface was 2114.53 cm^{−1}. In the future, we will explore the effect that a solvent environment has on this type of fitting procedure, including optimization of the polynomial order, reducing the number of electronic structure points, and verifying the best placement of those points in normal mode coordinate space.

##### b. Is the C≡C stretch normal mode weakly anharmonically coupled to other normal modes in the molecule?

Molecular normal modes are the completely orthogonal motions of a molecule under the harmonic oscillator approximation (this is, in fact, the definition of a normal mode). However, normal modes generally mix with each other once anharmonicity is considered. Our DVR method implicitly assumes that the C≡C stretch normal mode does not mix with others and is, therefore, *anharmonically isolated*. To determine the error due to this assumption, which we term *e*_{iso}, we performed anharmonic VPT2 frequency calculations for the isolated C≡C stretch normal mode. In this calculation, the third and fourth derivatives of the energy are only computed with respect to the C≡C stretch normal mode; no other anharmonic normal mode information is included in the calculation. In effect, this anharmonically isolates the C≡C stretch vibration in a similar way to our DVR method. *e*_{iso} is approximated as the difference between the isolated VPT2 value and the VPT2 frequency with normal mode coupling using CCSD(T) and triple zeta basis sets (i.e., *e*_{iso} = *ω*_{iso} − *ω*_{coup}). The error due to normal mode isolation is reported in Table II, and the average *e*_{iso} is +40 cm^{−1}.

. | aug-cc-pVTZ . | def2-TZVPD . | 6-311++G(d,p) . |
---|---|---|---|

PPY | +37 | +40 | +30 |

PAL | +27 | +29 | +29 |

BTY | ⋯ | +38 | +37 |

POL | +33 | +45 | +34 |

EAM | +42 | +43 | +35 |

EOL | +44 | +51 | +51 |

. | aug-cc-pVTZ . | def2-TZVPD . | 6-311++G(d,p) . |
---|---|---|---|

PPY | +37 | +40 | +30 |

PAL | +27 | +29 | +29 |

BTY | ⋯ | +38 | +37 |

POL | +33 | +45 | +34 |

EAM | +42 | +43 | +35 |

EOL | +44 | +51 | +51 |

The total anharmonicity of the C≡C stretch normal mode can be approximated as the difference between harmonic and VPT2 frequencies (i.e., *ω*_{harm} − *ω*_{coup}) using the same CCSD(T) electronic structure theory. A strong correlation between *e*_{iso} and the total anharmonicity indicates that a significant portion of the variance in the total anharmonicity is due to normal mode coupling rather than *isolated* normal mode anharmonicity (harmonic frequency calculations do not capture either effect). We find a reasonably strong correlation (R = 0.80, p < 2 × 10^{−3}) between *e*_{iso} and the overall anharmonicity (Fig. 4). A linear regression treating the approximate total anharmonicity as the independent variable has a slope of 0.8 ± 0.3, quite close to 1. This, and the small y-intercept value of 3 ± 7 cm^{−1}, implies that a large portion of the overall magnitude of the anharmonicity (not just its variation) is due to normal mode coupling—if the slope were exactly 1, we could essentially say *e*_{iso} = *ω*_{harm} − *ω*_{coup}. This implies that the isolated C≡C stretch normal mode vibration itself is fairly harmonic. The isolated anharmonicity can be estimated by calculating *ω*_{harm} − *ω*_{iso} and is on average only 3 cm^{−1} (see also the highly harmonic PES in Fig. 3).

##### c. Do the R group atoms move only a small amount during the vibration?

Our localization method makes the approximation that only the CCH atoms move during the vibration. This cannot be exactly true—a pure vibration would not translate or rotate the molecule, but a CCH-only vibration would. This could be partially resolved by satisfying the appropriate Eckart conditions at each grid point.^{109} Unfortunately, our method cannot rely on satisfying these conditions in order to be truly universal—as the systems we investigate become larger, the conditions would become more difficult to satisfy. Consider, say, a terminal alkyne moiety covalently bonded to a protein.^{45} What effect does removing the R group displacement have on the vibrational motion and frequency, and is mixing translation and rotation into the vibration acceptable?

To quantify the effect on the motion, we analyzed the atomic normal mode displacements reported by harmonic frequency calculations for all small probes using CCSD(T) with the same series of basis sets. We found that on average, the magnitude of the R group atom displacement is 0.071 Å, while the average magnitude of ≡**C**–R, ≡**C**–H, and ≡C–**H** atom displacements was 0.559, 0.385, and 0.719 Å, respectively. The R group motion for EAM and EOL is larger, averaging 0.120 Å. The much larger CCH displacements show that these atoms are the most important in the vibrational mode. We used harmonic MP2/aug-cc-pVTZ calculations to quantify the localization error for our larger molecules (PAC, EBA, PEP, and HPG), some of which exhibit strong pi-conjugation between the terminal alkyne moiety and an aromatic ring. Localization is strong for these molecules as well. The average magnitude of R group atom displacement is 0.019 Å, while the average magnitudes of ≡**C**–R, ≡**C**–H, and ≡C–**H** atom displacements were 0.570, 0.391, and 0.709 Å, respectively. At MP2/aug-cc-pVTZ, the same values for the CCH atoms in the six simple alkyne probes are very similar, at 0.571, 0.388, and 0.707 Å, respectively. Even when pi-conjugation is present, the C≡C stretch vibration is local to the CCH atoms.

We also quantified the effect on the calculated frequencies. We performed both harmonic and DVR frequency analyses that capture the error due to localizing the vibration to the CCH atoms. Specifically, we compared full harmonic frequencies to localized harmonic frequencies, and separately, we compared full normal mode DVR frequencies to localized normal mode DVR frequencies. The localized harmonic frequencies were computed using the numerical second derivative of the energy as a function of the localized normal mode coordinate near *Q* = 0. All of these frequency calculations were performed using the CCSD(T) method with the same series of basis sets as above.

The error due to localization, *e*_{loc}, was estimated as the difference between the isolated normal mode frequency and the full normal mode frequency (i.e., *ω*_{loc} − *ω*_{full}) for each vibrational structure method. The individual *e*_{loc} values are reported in Table III. The average harmonic *e*_{loc} is −41 cm^{−1}, and the average anharmonic *e*_{loc} is −45 cm^{−1}. The overall average is −43 cm^{−1}. We find that the magnitude of *e*_{loc} is strongly correlated to the sum of the mass-weighted displacements of the R group atoms in the full normal mode (Fig. 5). In particular, the molecules with non-carbon-based R groups, EAM and EOL, have particularly large R group displacements and, therefore, large localization errors. When they are removed from the estimate of *e*_{loc}, it shrinks to −30 cm^{−1} for terminal alkynes with carbon-based R groups. The fact that this correlation exists, is strong, and is negative shows that molecules that more fully satisfy the partial Hessian assumption of zero R group motion will have less localization error, as might be expected. The average value of the localization error for the large probes (PAC, EBA, PEP, and HPG) based on harmonic MP2/aug-cc-pVTZ calculations is −18 cm^{−1} (standard deviation of 2 cm^{−1}), compared to −21 cm^{−1} (standard deviation of 7 cm^{−1}) for the six simple alkyne probes at MP2/aug-cc-pVTZ. From these data, it does not seem that pi-conjugation affects the localization error.

. | aug-cc-pVTZ . | def2-TZVPD . | 6-311++G(d,p) . | |||
---|---|---|---|---|---|---|

harm . | anharm . | harm . | anharm . | harm . | anharm . | |

PPY | −30 | −37 | −31 | −36 | −36 | −39 |

PAL | −26 | −26 | −22 | −26 | −21 | −28 |

BTY | −32 | −33 | −30 | −33 | −33 | −35 |

POL | −28 | −31 | −16 | −31 | −25 | −33 |

EAM | −70 | −70 | −66 | −69 | −68 | −72 |

EOL | −67 | −73 | −68 | −72 | −68 | −74 |

. | aug-cc-pVTZ . | def2-TZVPD . | 6-311++G(d,p) . | |||
---|---|---|---|---|---|---|

harm . | anharm . | harm . | anharm . | harm . | anharm . | |

PPY | −30 | −37 | −31 | −36 | −36 | −39 |

PAL | −26 | −26 | −22 | −26 | −21 | −28 |

BTY | −32 | −33 | −30 | −33 | −33 | −35 |

POL | −28 | −31 | −16 | −31 | −25 | −33 |

EAM | −70 | −70 | −66 | −69 | −68 | −72 |

EOL | −67 | −73 | −68 | −72 | −68 | −74 |

##### d. Fortuitous error cancellation.

In our localized normal mode DVR method, the error due to discretization is negligible. The errors due to isolation, e_{iso} ≈ +40 cm^{−1}, and localization, e_{loc} ≈ −43 cm^{−1}, are more substantial (though not individually enormous). Fortunately, they are nearly equal in magnitude and opposite in sign. This cancellation occurs not only on average across molecules and basis sets but also for most individual combinations of molecules and basis sets, as depicted in Fig. 6. This error cancellation is nearly complete across alkynes with carbon-based R groups and is even beneficial for alkynes with non-carbon-based R groups. As shown in Fig. 6, the co-cancellation of these errors explains the high accuracy of our isolated normal mode DVR method. Statistical t-tests indicate that our DVR method calculates frequencies that are indistinguishable from those calculated by the more expensive VPT2 vibrational method for molecules where the sum of the localization error and the isolation error is indistinguishable from zero. The DVR frequencies are distinct from VPT2 frequencies for molecules like EAM and EOL due to significant R group motion, resulting in high localization errors.

Because of the consistency of this error cancellation, the modest magnitudes of the errors, and the resulting accuracy, we believe it is appropriate to use this method for future alkyne frequency analyses in more complex systems. However, those using this method or a similar one should check that the error cancellation remains in their context, perhaps using the comparisons we have employed earlier. This method would need to be more carefully adjusted for use with alkynes having non-carbon-based R groups since, in their case, the error does not cancel out. The effects of pi-conjugation and sigma induction on localization and anharmonic isolation for the C≡C stretch will be more carefully assessed in future work. However, it is already apparent that sigma induction in EAM and EOL (and perhaps PAL) may change the balance between the two sources of error. As described earlier, we found that localization was an appropriate approximation for our larger pi-conjugated probes.

The electronic structure that a terminal alkyne might have in a biochemical context depends strongly on how the terminal alkyne is attached to the biomolecule. Several attachment mechanisms have been explored. Epstein *et al.* attached a terminal alkyne to an eight-carbon chain, which is then attached to an acyl carrier protein.^{45} Of our molecules, BTY probably best models this electronic structure. Romei *et al.* consider using unnatural amino acids, both of which are included in this work (HPG and PEP).^{44} Other strategies are broadly similar.^{34,110,111} In all these strategies, the terminal alkyne is bonded directly to a carbon atom. Therefore, the current results provide confidence that an accurate frequency would be obtained in most investigations of biomolecules vibrationally probed with terminal alkynes.

### B. Density functional theory calculations

CCSD(T) is a highly accurate electronic structure method. However, it is not accessible for use with our larger probes. In the future, we would prefer to use DFT methods because of their high speed and reasonable accuracy. We use the MP2 electronic structure method as a “silver standard” to evaluate the performance of various cheaper DFT methods. Lowering the level of theory to MP2 changes the frequency by about an average of 4 cm^{−1} and an RMSD of 22 cm^{−1} for our small probes (values in the supplementary material).

We also calculated localized normal mode DVR frequencies using four density functionals, PBEc-3h, TPSS-D3, B3LYP-D3, and ωB97M-V. These calculations were performed using Pople, Dunning, and Alrichs style basis sets fitting the descriptions “double zeta,” “double zeta with diffuse functions,” “triple zeta,” and “triple zeta with diffuse functions.” An exception was PBEc-3h, which was only used with the def2-mSVP basis set.

*et al.*

^{112}Scaling factors are commonly used to correct for inaccuracies in harmonic frequency calculations, which can be broadly sorted into two types. The first type of error is at the level of the vibrational calculation, especially the neglect of anharmonic effects. This is fundamental to the harmonic method. Our DVR calculations do not have exactly the same type of error but, as discussed earlier, our approximations introduce their own vibrational calculation errors. The second type is at the electronic structure level, where inadequacies in the description of electrons can introduce errors. Because of this, scaling factors are sensitive to the electronic structure method and basis set. A scaling factor,

*c*, can be computed using the formula

*x*

_{i}are their computed frequencies, and

*z*

_{i}are their experimental frequencies. The uncertainty in the scaling factor is given by

*c*is the previously computed scaling factor.

^{112}MP2 scaling factors and their uncertainties were computed for our set of simple alkynes for which experimental values are available. The average MP2 scaling factor across basis sets was 1.004 ± 0.003. These scaling factors were used to predict gas phase experimental values for the larger probes for which we did not have experimental data. The predicted experimental values (i.e., scaled MP2 values) are given in Table IV.

. | aug-cc-pVTZ . | def2-TZVPD . | 6-311++G(d,p) . | Average . |
---|---|---|---|---|

PAC | 2131 | 2132 | 2138 | 2134 |

EBA | 2099 | 2094 | ⋯ | 2097 |

HPG | 2115 | 2117 | 2114 | 2115 |

PEP | 2100 | 2100 | 2098 | 2099 |

. | aug-cc-pVTZ . | def2-TZVPD . | 6-311++G(d,p) . | Average . |
---|---|---|---|---|

PAC | 2131 | 2132 | 2138 | 2134 |

EBA | 2099 | 2094 | ⋯ | 2097 |

HPG | 2115 | 2117 | 2114 | 2115 |

PEP | 2100 | 2100 | 2098 | 2099 |

The averages of these predicted experimental values were used along with authentic experimental values for the simple alkyne probes to compute scaling factors and uncertainties for the density functional methods (Fig. 7). The results are nearly identical to those obtained if only simple alkynes with true experimental frequencies are used. If EAM and EOL are removed from the training set, the uncertainties become smaller, but the magnitudes of the scaling factors remain nearly the same.

A vibrational frequency method with no vibrational or electronic sources of error would have a scaling factor of 1.0. By this metric, MP2 performs very well, and so it serves as an effective benchmark. HF performs relatively poorly, with an average scaling factor of 0.900 ± 0.003, showing that electron correlation is an important factor in these calculations. Contrary to our prior expectations, TPSS-D3 is the cheapest *and* most accurate DFT method for calculating the terminal alkyne C≡C stretch frequencies. The best scaling factor observed for a DFT method was 0.989 ± 0.006 for TPSS-D3/6-311++G(d,p). ωB97M-V has surprisingly poor performance, though even its lowest scaling factor of 0.943 ± 0.007 for ωB97M-V/6-31G(d,p) is fairly reasonable. PBEc-3h/def2-mSVP has the worst performance of the density functionals at a scaling factor of 0.931 ± 0.007. All methods and all basis set families show a marked improvement in the frequency predictions when going from a double zeta basis set to a double zeta basis set with added diffuse functions. For the Pople and Dunning basis set families, performance is worsened whenever diffuse functions are removed, while the double/triple zeta distinction is less important. The Alrichs basis sets are more “stable” in a sense—changing the features of the basis set has the least effect on performance for this family. The Pople basis sets are the most erratic, so a small change in basis set character can have a large effect on performance. Based on these results, an ideal model of the electronic structure would include a functional like TPSS-D3 and a basis set of at least double zeta quality with diffuse functions included.

One reason that TPSS-D3 may perform better than the other functionals is because it was derived from fundamental density functional theory principles with very few fitted parameters.^{75} B3LYP-D3 and ωB97M-V include several adjustable parameters that were fit based on experimental data on several (or many) smaller molecules.^{70,73} The training sets for these empirical functionals did not include many alkynes or other triple bonded molecules relative to the overall size of the training set. This may harm their ability to characterize triple bonds compared to a less empirical method like TPSS. Presumably, an empirical DFT method based on many triple bonded molecules would perform even better than TPSS-D3. The importance of diffuse functions in the basis set is likely also related to the presence of the triple bond. In a triple bonded molecule, there is substantial electron density placed unusually far from the nuclei. A basis set without diffuse functions might have some difficulty fully describing this. Describing the triple bond well is particularly important when computing the C≡C normal mode frequencies, which directly depend on estimating the bond strength. A deeper understanding of these trends would likely require very careful electronic structure analysis, which is outside the scope of this work.

### C. Properties of the vibration

#### 1. Isotropic transition polarizabilities and transition dipole moments

Using this vibrational method, we can compute a number of useful vibrational properties. Here, we focus on the calculation of the transition dipole moment and the isotropic transition polarizability. To compute these, we first obtain the dipole moment and isotropic polarizability surfaces, shown in the left panels of Fig. 8. These surfaces were calculated for all our molecules at the grid point structures used in our localized normal mode DVR calculations at TPSS-D3/6-311++G(d,p). The dipole moment is reported as the overlap (dot product) between the full dipole moment vector and the unit vector pointing from the carbon bonded to the R group to the carbon bonded to the terminal hydrogen. The isotropic polarizability is reported as the trace of the full polarizability tensor. In general, the projection of the dipole moment along the bond slowly becomes more negative as the normal mode coordinate increases, which means the dipole points more toward the R group. For molecules where the R group does not begin with carbon, the decrease is more rapid; for PAL, the dipole moment projection actually increases with the normal mode coordinate. The polarizability increases along the normal mode for all molecules; this is expected since molecular bond lengths are generally increasing as the normal mode coordinate increases. The magnitude of this increase is generally fairly large, especially for the molecules containing aromatic rings conjugated to the triple bond.

^{64}These are shown in the right part of Fig. 8. We have also collected these values for a number of reasonably comparable moieties that have been used as vibrational probes, especially in biochemical contexts: azides,

^{113–115}nitriles,

^{116,117}and carbon dioxide.

^{118–120}These are found in Table V. Where a dipole or polarizability derivative was reported, it was converted to the appropriate transition property using the harmonic oscillator approximation,

Moiety . | $\mu 012$ (D^{2})
. | $\alpha 012$ (10^{−3} A^{6})
. | Notes . | Source . |
---|---|---|---|---|

Alkyne | 0–0.0280 | 12.6–104 | Gas phase, calculated | Present work |

Nitrile | 0.0016–0.01 | In a variety of solvents | Weaver et al., 2022 | |

Nitrile | 0.000 57–0.018 | 0–110 | In 2-methyl-THF glass | Andrews and Boxer, 2000 |

Azide | 0.264 | Gas phase, calculated | Botschwina, 1986 | |

Azide | 8.23 | β-NaN_{3} crystals | Fredrickson and Decius, 1975 | |

Azide | 181 | β-NaN_{3} crystals, aqueous solution | Bertsch et al., 1984 | |

Carbon dioxide | 0.106 | Asymmetric stretch, gas phase | Downing et al. 1975 | |

Carbon dioxide | 2.52, 3.73 | Symmetric stretch, gas phase | Tejeda et al. 1995 |

Moiety . | $\mu 012$ (D^{2})
. | $\alpha 012$ (10^{−3} A^{6})
. | Notes . | Source . |
---|---|---|---|---|

Alkyne | 0–0.0280 | 12.6–104 | Gas phase, calculated | Present work |

Nitrile | 0.0016–0.01 | In a variety of solvents | Weaver et al., 2022 | |

Nitrile | 0.000 57–0.018 | 0–110 | In 2-methyl-THF glass | Andrews and Boxer, 2000 |

Azide | 0.264 | Gas phase, calculated | Botschwina, 1986 | |

Azide | 8.23 | β-NaN_{3} crystals | Fredrickson and Decius, 1975 | |

Azide | 181 | β-NaN_{3} crystals, aqueous solution | Bertsch et al., 1984 | |

Carbon dioxide | 0.106 | Asymmetric stretch, gas phase | Downing et al. 1975 | |

Carbon dioxide | 2.52, 3.73 | Symmetric stretch, gas phase | Tejeda et al. 1995 |

As expected, the transition dipole moments of alkynes are relatively small. In magnitude, they span approximately the same range as nitriles. The largest transition dipoles occur for probes where the triple bond is attached to an atom that is not carbon, where significant sigma induction is possible. Most of the values are much smaller. The transition polarizabilities, however, are fairly large. These values are roughly on the same order of magnitude as those of azides and nitriles, which has already been observed in experimental relative Raman intensity comparisons.^{35,121,122} The transition polarizabilities are especially large for the alkyne molecules pi-conjugated to aromatic rings (EBA and PEP), for whom stronger Raman scattering has already been observed in experiments.^{35,44} In our future work, we plan to investigate the intentional design of terminal alkyne probes with high IR and Raman intensities.

#### 2. Universal displacements

The localized CCH atom displacements for the terminal alkyne C≡C normal mode vibration are remarkably similar across all of the molecules observed in this study. To take advantage of this similarity, we performed localized normal mode DVR on all molecules, with all basis sets and DFT functionals described in this work. These calculations were performed twice and then compared. In the first iteration, displacements were obtained individually, so each molecule, DFT method, and basis set combination had its own specific atomic normal mode displacements. In the second iteration, the localized displacements for the CCH atoms of propyne at MP2/aug-cc-pVTZ were used for all molecules, DFT methods, and basis sets. In Fig. 9, the strong correlation between these two methods of displacing the CCH atoms is shown. It is clear that the atomic displacements of the CCH atoms in the C≡C stretch are fairly *universal*—one could switch the specific values between any pair of combinations of molecule, DFT method, and basis set. In future work, we will use such universal C≡C displacements to simplify the calculation of terminal alkyne vibrational frequencies from molecular dynamics snapshots.

### D. The terminal alkyne C≡C stretch vibration

A number of fortunate occurrences allow us to substantially simplify the calculation of terminal alkyne C≡C stretch vibrations. First, the vibration is *localized* to only three atoms for a surprisingly diverse range of molecules, including those that exhibit pi-conjugation. Second, the vibration is also anharmonically *isolated*, regardless of the other properties of the molecule. Third, the modest error that exists between localization and isolation cancels, resulting in a localized DVR method with accuracy comparable to much more expensive methods like VPT2. Fourth, the accuracy carries over to DFT, where especially cheap functionals perform surprisingly well.

These fortunate occurrences allow us to use a relatively cheap DVR method where the same localized normal mode displacements can be used for any molecule. This DVR method can be used to calculate vibrational properties like the transition dipole moment and isotropic transition polarizability, and our results broadly match what has been seen in experiments and other theoretical work. Specifically, molecules exhibiting pi-conjugation have especially strong isotropic transition polarizabilities. Meanwhile, molecules with non-carbon atoms directly attached to the alkyne, likely exhibiting strong sigma induction, have strong transition dipole moments.

## V. CONCLUSION

Understanding the terminal alkyne C≡C stretch vibration is important for a number of biological and material questions. In this work, we develop and test an extension to the DVR method specific to this important molecular probe. Our resulting localized normal mode DVR method benefits from error cancellation between modest localization and isolation errors. It can be easily implemented using DFT methods. Finally, we are able to compute important vibrational properties such as isotropic transition polarizability, which are vital for eventually computing Raman spectra in realistic simulations.

The next step, of course, is to place alkynes into realistic solvent systems and use this method to investigate the vibrational properties. Prior work has found that the presence of a solvent can shift the alkyne vibrational frequency substantially compared to the gas phase.^{44} This may be due, in part, to the charge transfer. Such an effect could alter both localization and isolation errors. However, since charge transfer to the alkyne would likely make the vibration more anharmonic (making isolation error more positive) to a similar degree to how it weakens the partial Hessian approximation (making localization error more negative), it is unlikely that the cumulative effect on accuracy would be large. Most other effects that a solvent could have on an alkyne would also likely cancel out in a similar way.

This method can be used as the basis of a vibrational map. It can be used directly on snapshots from MD simulations to obtain the vibrational frequencies, transition polarizabilities, and transition dipole moments required to build effective spectroscopic maps. With such maps, it will become possible to calculate the IR and Raman spectra of alkynes in a variety of solvents, depending on what types of MD snapshots are used in the map training set.

For carbon-based substituents to the alkyne, the magnitude of the localization error (∼30 cm^{−1}) is comparable to that of the harmonic approximation (∼40 cm^{−1}) or anharmonic isolation (∼40 cm^{−1}). However, this is not a universally applicable approximation; for EAM and EOL, this source of error was dominant. This will likely lead to a requirement for different spectroscopic maps for molecules with carbon-based R groups than for those with non-carbon-based R groups like EAM and EOL.

This method provides a launchpad for the development of a vibrational spectroscopic map for these terminal alkyne vibrational probes and for further investigation of the effects of pi-conjugation and sigma induction on the vibration.

## SUPPLEMENTARY MATERIAL

Included in the main PDF of our supplementary material is a more complete explanation of the theory behind our and other closely related methods, a graph of the comparison between our VPT2 and TOSH frequencies, and a table comparing partial and full Hessian localization methods. XYZ files of the investigated molecules optimized with CCSD(T) or MP2 and triple zeta basis sets are also included because of the large computational cost of obtaining these structures. JSON data files and instructions for interacting with them in Python are included. These JSON files contain all frequencies and localized normal mode CCH displacements computed in this work. Finally, text files with the PESs, wavefunctions, and energy levels obtained at the CCSD(T) and MP2 levels are included, along with code amenable to any electronic structure program for computing these frequencies.

## ACKNOWLEDGMENTS

Computational resources were provided in part by the MERCURY consortium (https://mercuryconsortium.org) under NSF Grant Nos. CHE-1229354, CHE-1662030, and CHE-2018427. We acknowledge Haverford College for startup funding and the NSF for support under Grant No. CHE-2213339. We also acknowledge Professor Casey Londergan, Professor Steve Corcelli, Professor Lou Charkoudian, and Professor Jeffrey Woodford for helpful comments on and discussion about this manuscript.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Kristina Streu**: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Sara Hunsberger**: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). **Jeanette Patel**: Investigation (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). **Xiang Wan**: Formal analysis (equal); Methodology (equal). **Clyde A. Daly, Jr.**: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The vibrational frequencies and localized normal mode CCH displacements calculated in this work are available as supplementary material, as are the CCSD(T) and MP2 structures and localized normal mode PESs of the investigated molecules. The general methods and code used to complete the calculations are available on GitHub at https://github.com/Daly-Lab-at-Haverford/code_examples and in the supplementary material. Other data supporting the findings of this study are available upon reasonable request.

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